This document provides a lesson on indirect proofs and inequalities in one triangle from Holt Geometry. It includes examples of writing indirect proofs, ordering triangle side lengths and angle measures, applying the triangle inequality theorem, finding possible side lengths of a triangle, and using triangle inequalities in a travel application problem. Practice problems are provided to check understanding. The key concepts covered are indirect proof, triangle inequalities, and relating triangle side lengths to angle measures.
Word problem is often used to refer to any mathematical exercise where significant background information on the problem is presented as text rather than in mathematical notation. As word problems often involve a narrative of some sort, they are occasionally also referred to as story problems and may vary in the amount of language used.
-http://en.wikipedia.org/wiki/Word_problem_(mathematics_education)
This document defines arithmetic sequences and provides examples. It explains that an arithmetic sequence is a series of numbers where the difference between consecutive terms is constant. It gives the formula for finding the nth term (an) of a sequence given the first term (a1) and common difference (d). Several examples are worked out applying this formula to find specific terms of arithmetic sequences. The objectives are to define arithmetic sequences, find the common difference and nth term given the first few terms.
The document discusses multiplying polynomials, including multiplying monomials, combining like terms, and special cases such as the sum and difference of binomials, squares of binomials, and cubes of binomials. Examples are provided for multiplying polynomials with 2, 3, or 4 terms. Formulas and step-by-step workings are shown for finding products of binomial expressions.
Algebraic Mathematics of Linear Inequality & System of Linear InequalityJacqueline Chau
A brief, yet thorough look into the Linear Inequality & System of Linear Inequality and how these Math Concepts would be useful in solving our everyday life problems.
The document contains multiple word problems involving linear equations with one unknown. The problems cover a range of topics including train speeds, plane speeds accounting for wind, mixing solutions of different concentrations, coin values, investment returns, filling pools, properties of triangles, rectangle areas, digit sums, and numbers with reversed digits.
2/27/12 Special Factoring - Sum & Difference of Two Cubesjennoga08
The document is about factoring polynomials, specifically factoring the sum and difference of cubes. It provides the formulas for factoring the sum and difference of cubes, along with examples of factoring expressions using those formulas. It also discusses factoring out the greatest common factor from polynomials.
This document discusses simplifying complex rational expressions, which have numerators or denominators containing fractions. It provides two methods for simplification:
1) Multiplying the numerator and denominator by the lowest common denominator to clear fractions. Examples and steps are shown.
2) Dividing the numerator by the denominator after collecting like terms. An example problem is worked through to demonstrate the process. Objectives and learning outcomes are stated to guide readers.
This document introduces a graphic organizer called tic-tac-toe factoring to help students factor second degree trinomials step-by-step. It uses a tic-tac-toe grid to systematically arrange the terms of the trinomial and then find two binomial factors by filling in the grid with pairs of numbers that multiply to give the terms. An example is worked through to demonstrate the process of using the tic-tac-toe grid to factor the trinomial 8x^2 - 14x + 3 into the two binomial factors (4x - 1)(2x - 3).
Word problem is often used to refer to any mathematical exercise where significant background information on the problem is presented as text rather than in mathematical notation. As word problems often involve a narrative of some sort, they are occasionally also referred to as story problems and may vary in the amount of language used.
-http://en.wikipedia.org/wiki/Word_problem_(mathematics_education)
This document defines arithmetic sequences and provides examples. It explains that an arithmetic sequence is a series of numbers where the difference between consecutive terms is constant. It gives the formula for finding the nth term (an) of a sequence given the first term (a1) and common difference (d). Several examples are worked out applying this formula to find specific terms of arithmetic sequences. The objectives are to define arithmetic sequences, find the common difference and nth term given the first few terms.
The document discusses multiplying polynomials, including multiplying monomials, combining like terms, and special cases such as the sum and difference of binomials, squares of binomials, and cubes of binomials. Examples are provided for multiplying polynomials with 2, 3, or 4 terms. Formulas and step-by-step workings are shown for finding products of binomial expressions.
Algebraic Mathematics of Linear Inequality & System of Linear InequalityJacqueline Chau
A brief, yet thorough look into the Linear Inequality & System of Linear Inequality and how these Math Concepts would be useful in solving our everyday life problems.
The document contains multiple word problems involving linear equations with one unknown. The problems cover a range of topics including train speeds, plane speeds accounting for wind, mixing solutions of different concentrations, coin values, investment returns, filling pools, properties of triangles, rectangle areas, digit sums, and numbers with reversed digits.
2/27/12 Special Factoring - Sum & Difference of Two Cubesjennoga08
The document is about factoring polynomials, specifically factoring the sum and difference of cubes. It provides the formulas for factoring the sum and difference of cubes, along with examples of factoring expressions using those formulas. It also discusses factoring out the greatest common factor from polynomials.
This document discusses simplifying complex rational expressions, which have numerators or denominators containing fractions. It provides two methods for simplification:
1) Multiplying the numerator and denominator by the lowest common denominator to clear fractions. Examples and steps are shown.
2) Dividing the numerator by the denominator after collecting like terms. An example problem is worked through to demonstrate the process. Objectives and learning outcomes are stated to guide readers.
This document introduces a graphic organizer called tic-tac-toe factoring to help students factor second degree trinomials step-by-step. It uses a tic-tac-toe grid to systematically arrange the terms of the trinomial and then find two binomial factors by filling in the grid with pairs of numbers that multiply to give the terms. An example is worked through to demonstrate the process of using the tic-tac-toe grid to factor the trinomial 8x^2 - 14x + 3 into the two binomial factors (4x - 1)(2x - 3).
This document discusses relationships between sides and angles of triangles. It provides examples of ordering triangle sides by length and angles by measure. The triangle inequality theorem states that the sum of any two side lengths must be greater than the third side length for a triangle to exist. Examples show applying this theorem to determine if a triangle can exist with given side lengths, and to find the possible range of a third side given two other side lengths. The document also includes practice problems ordering triangle parts and applying the triangle inequality theorem.
This document provides examples and explanations of indirect proofs. It begins with examples of writing indirect proofs to show that a triangle cannot have two right angles or that a number is greater than 0. It then discusses using inequalities in indirect proofs involving triangles. Examples demonstrate ordering triangle sides and angles, applying the triangle inequality theorem to determine if a triangle can exist with given side lengths, and using indirect proofs to find possible side lengths. The document concludes with practice problems applying these concepts.
This document discusses using inequalities to compare angles and side lengths in two triangles. It begins with examples that use the hinge theorem and its converse to determine relationships between angles and sides. Application examples are provided, including comparing distances traveled from school. Proofs are presented to demonstrate triangle relationships using statements and reasons. The document concludes with a lesson quiz to assess understanding.
1) The document discusses inequalities in two triangles using theorems like the hinge theorem and its converse. It includes examples comparing angles and side lengths in different triangles.
2) One example asks students to determine if John or Luke is farther from school based on the distances and angles of their routes. Another example proves relationships between angles and sides of triangles.
3) The document concludes with a lesson quiz testing students' ability to compare angles and sides in triangles and write two-column proofs of triangle relationships.
This document contains a lesson on using proportional relationships from Holt Geometry. It includes examples of using ratios and proportions to make indirect measurements, solve problems involving scale drawings, and find perimeters and areas of similar figures. The lesson defines key terms like indirect measurement, scale, and scale drawing. It presents examples and practice problems demonstrating techniques for setting up and solving proportions to determine unknown lengths and dimensions.
This document provides a lesson on ratio and proportion from Holt Geometry. It includes examples of writing ratios, using ratios to solve problems, writing and solving proportions using the cross products property, and applying proportions to solve real-world problems. Key concepts covered are ratios, proportions, extremes, means, and cross products. Worked examples demonstrate how to set up and solve proportions to find missing values.
This document is from a Holt Geometry textbook and covers identifying and finding measures of pairs of angles such as adjacent angles, vertical angles, complementary angles, and supplementary angles. It includes examples of identifying angle pairs, finding measures of complements and supplements, using relationships between angles to solve problems, and a lesson quiz assessing these concepts. The key ideas are identifying different types of angle pairs and using their defining relationships to determine unknown angle measures.
The document discusses several key theorems regarding triangles:
1) The triangle inequality theorem states that the length of any side of a triangle must be less than the sum of the lengths of the other two sides and greater than their difference.
2) The exterior angle theorem states that an exterior angle of a triangle is equal to the sum of the remote interior angles.
3) The hinge theorem states that if two triangles have two congruent sides but not congruent included angles, then the third sides will be unequal, with the longer side opposite the larger angle.
The document discusses several theorems related to triangles:
1) The triangle inequality theorem states that the length of any side of a triangle must be less than the sum of the lengths of the other two sides and greater than their difference.
2) The exterior angle theorem states that an exterior angle of a triangle is equal to the sum of the remote interior angles.
3) The hinge theorem states that if two triangles have two congruent sides but not congruent included angles, then the third sides will be unequal, with the longer side opposite the larger angle.
The document discusses various theorems and properties related to triangles, including:
1) The exterior angle theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles.
2) The triangle inequality theorem states that the sum of the measures of any two sides of a triangle is greater than the measure of the third side.
3) Properties relating the lengths of sides and measures of angles in a triangle, such as if sides are unequal then angles will be unequal as well.
This document is a lesson on the Pythagorean theorem from a Holt Geometry textbook. It begins with examples of using the Pythagorean theorem to solve for missing sides of right triangles. It then discusses how to identify if a set of three sides forms a Pythagorean triple. The lesson continues with examples of using the converse of the Pythagorean theorem and Pythagorean inequalities to classify triangles as right, acute, or obtuse based on their side lengths. It concludes with a two-part lesson quiz reviewing the material.
This document provides an overview of indirect proofs in mathematics. It defines indirect reasoning, indirect proof, and proof by contradiction as forms of proof that assume the opposite of what is being proved in order to derive a contradiction. The document gives examples of stating assumptions for indirect proofs, writing indirect proofs algebraically and geometrically, and identifying the key steps of writing an indirect proof by first assuming the opposite, then deriving a contradiction. It provides fully worked examples of indirect proofs for algebraic, number theory, and geometric statements. Finally, it assigns practice problems for readers to work.
1. The document discusses geometric proofs in geometry. It covers topics like writing justifications for each step, using definitions and theorems as reasons, and completing two-column proofs.
2. Examples are provided to demonstrate writing justifications, filling in blanks in two-column proofs, and writing full proofs from given plans.
3. The key aspects of writing geometric proofs are to justify each logical step and use appropriate reasons like definitions, postulates, properties, and theorems. Students must also draw diagrams and state the given information and what is to be proved.
This document discusses solving right triangles using trigonometric ratios. It includes examples of identifying angles from trigonometric ratios, calculating angle measures from ratios using inverse functions, solving right triangles when given side lengths or an angle measure, and solving triangles in the coordinate plane. One example problem solves for the angle of a road given its grade in percentage form.
This document provides examples and exercises on using inequalities to solve problems involving triangles. It defines the triangle inequality theorem and shows how to set up and solve inequalities to find possible lengths of triangle sides. Examples demonstrate drawing diagrams to visualize solutions and labeling angles and sides in order of measure. Practice problems have students list triangle sides or angles from least to greatest, determine if given side lengths can form a triangle using inequalities, and find the possible length of a third side of a triangle given the other two sides.
This document provides instruction on solving right triangles using trigonometric ratios. It includes examples of identifying angles from trigonometric ratios, calculating angle measures from ratios using inverse functions, solving right triangles when given side lengths or an angle measure, solving triangles in the coordinate plane, and applying trigonometry to problems involving road grades. Students are guided through applying concepts to multiple examples and practice problems.
This document discusses classifying angles formed when parallel lines are cut by a transversal. It defines key terms like alternate interior angles, alternate exterior angles, and corresponding angles. It then provides examples of classifying angle pairs based on their relationship. Sample problems are worked through, applying the concepts to find missing angle measures by justifying answers using angle properties of parallel lines cut by a transversal.
This document discusses classifying angles formed when parallel lines are cut by a transversal. It defines key terms like interior angles, exterior angles, alternate interior angles, alternate exterior angles, and corresponding angles. Examples are provided to illustrate each type of angle relationship. The document also provides step-by-step examples of classifying angle pairs and solving for missing angle measures using properties of parallel lines cut by a transversal.
The document discusses paragraph proofs and two-column proofs in geometry. A paragraph proof states the given information, what is to be proven, then uses a logical chain of statements and reasons to justify the conclusion. A two-column proof separates the statements from the reasons into two columns, with the given information and conclusion stated. Both proof styles use definitions, properties, and theorems to logically justify each step. The document provides examples of each type of proof to illustrate the process.
3008 perpendicular lines an theoremsno quizjbianco9910
This document contains a geometry drill with examples and explanations of perpendicular lines. It begins with warm up problems solving inequalities and equations. It then covers the key concepts that the perpendicular bisector of a segment is perpendicular to the segment at its midpoint, and the shortest distance from a point to a line is the perpendicular segment. Examples demonstrate writing proofs about perpendicular and parallel lines and applying the concept to carpentry. The homework assigned is to complete problems from the textbook.
This document discusses spheres and their volumes and surface areas. It includes examples of calculating volumes and surface areas of spheres and composite figures containing spheres. Formulas used include the volume of a sphere equaling (4/3)πr^3 and the surface area equaling 4πr^2. Examples show multiplying or dividing the radius changes the volume and surface area. The document contains warm up questions, objectives, vocabulary, lesson material and examples, and a quiz on spheres.
This document discusses calculating the volumes of pyramids, cones, and composite three-dimensional figures. It provides examples of finding volumes of different shapes, such as rectangular and square pyramids, cylinders, and cones. It also explores how changing the dimensions of these figures affects their volumes. Formulas are given for calculating the volumes of pyramids and cones based on their base areas and heights.
This document discusses relationships between sides and angles of triangles. It provides examples of ordering triangle sides by length and angles by measure. The triangle inequality theorem states that the sum of any two side lengths must be greater than the third side length for a triangle to exist. Examples show applying this theorem to determine if a triangle can exist with given side lengths, and to find the possible range of a third side given two other side lengths. The document also includes practice problems ordering triangle parts and applying the triangle inequality theorem.
This document provides examples and explanations of indirect proofs. It begins with examples of writing indirect proofs to show that a triangle cannot have two right angles or that a number is greater than 0. It then discusses using inequalities in indirect proofs involving triangles. Examples demonstrate ordering triangle sides and angles, applying the triangle inequality theorem to determine if a triangle can exist with given side lengths, and using indirect proofs to find possible side lengths. The document concludes with practice problems applying these concepts.
This document discusses using inequalities to compare angles and side lengths in two triangles. It begins with examples that use the hinge theorem and its converse to determine relationships between angles and sides. Application examples are provided, including comparing distances traveled from school. Proofs are presented to demonstrate triangle relationships using statements and reasons. The document concludes with a lesson quiz to assess understanding.
1) The document discusses inequalities in two triangles using theorems like the hinge theorem and its converse. It includes examples comparing angles and side lengths in different triangles.
2) One example asks students to determine if John or Luke is farther from school based on the distances and angles of their routes. Another example proves relationships between angles and sides of triangles.
3) The document concludes with a lesson quiz testing students' ability to compare angles and sides in triangles and write two-column proofs of triangle relationships.
This document contains a lesson on using proportional relationships from Holt Geometry. It includes examples of using ratios and proportions to make indirect measurements, solve problems involving scale drawings, and find perimeters and areas of similar figures. The lesson defines key terms like indirect measurement, scale, and scale drawing. It presents examples and practice problems demonstrating techniques for setting up and solving proportions to determine unknown lengths and dimensions.
This document provides a lesson on ratio and proportion from Holt Geometry. It includes examples of writing ratios, using ratios to solve problems, writing and solving proportions using the cross products property, and applying proportions to solve real-world problems. Key concepts covered are ratios, proportions, extremes, means, and cross products. Worked examples demonstrate how to set up and solve proportions to find missing values.
This document is from a Holt Geometry textbook and covers identifying and finding measures of pairs of angles such as adjacent angles, vertical angles, complementary angles, and supplementary angles. It includes examples of identifying angle pairs, finding measures of complements and supplements, using relationships between angles to solve problems, and a lesson quiz assessing these concepts. The key ideas are identifying different types of angle pairs and using their defining relationships to determine unknown angle measures.
The document discusses several key theorems regarding triangles:
1) The triangle inequality theorem states that the length of any side of a triangle must be less than the sum of the lengths of the other two sides and greater than their difference.
2) The exterior angle theorem states that an exterior angle of a triangle is equal to the sum of the remote interior angles.
3) The hinge theorem states that if two triangles have two congruent sides but not congruent included angles, then the third sides will be unequal, with the longer side opposite the larger angle.
The document discusses several theorems related to triangles:
1) The triangle inequality theorem states that the length of any side of a triangle must be less than the sum of the lengths of the other two sides and greater than their difference.
2) The exterior angle theorem states that an exterior angle of a triangle is equal to the sum of the remote interior angles.
3) The hinge theorem states that if two triangles have two congruent sides but not congruent included angles, then the third sides will be unequal, with the longer side opposite the larger angle.
The document discusses various theorems and properties related to triangles, including:
1) The exterior angle theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles.
2) The triangle inequality theorem states that the sum of the measures of any two sides of a triangle is greater than the measure of the third side.
3) Properties relating the lengths of sides and measures of angles in a triangle, such as if sides are unequal then angles will be unequal as well.
This document is a lesson on the Pythagorean theorem from a Holt Geometry textbook. It begins with examples of using the Pythagorean theorem to solve for missing sides of right triangles. It then discusses how to identify if a set of three sides forms a Pythagorean triple. The lesson continues with examples of using the converse of the Pythagorean theorem and Pythagorean inequalities to classify triangles as right, acute, or obtuse based on their side lengths. It concludes with a two-part lesson quiz reviewing the material.
This document provides an overview of indirect proofs in mathematics. It defines indirect reasoning, indirect proof, and proof by contradiction as forms of proof that assume the opposite of what is being proved in order to derive a contradiction. The document gives examples of stating assumptions for indirect proofs, writing indirect proofs algebraically and geometrically, and identifying the key steps of writing an indirect proof by first assuming the opposite, then deriving a contradiction. It provides fully worked examples of indirect proofs for algebraic, number theory, and geometric statements. Finally, it assigns practice problems for readers to work.
1. The document discusses geometric proofs in geometry. It covers topics like writing justifications for each step, using definitions and theorems as reasons, and completing two-column proofs.
2. Examples are provided to demonstrate writing justifications, filling in blanks in two-column proofs, and writing full proofs from given plans.
3. The key aspects of writing geometric proofs are to justify each logical step and use appropriate reasons like definitions, postulates, properties, and theorems. Students must also draw diagrams and state the given information and what is to be proved.
This document discusses solving right triangles using trigonometric ratios. It includes examples of identifying angles from trigonometric ratios, calculating angle measures from ratios using inverse functions, solving right triangles when given side lengths or an angle measure, and solving triangles in the coordinate plane. One example problem solves for the angle of a road given its grade in percentage form.
This document provides examples and exercises on using inequalities to solve problems involving triangles. It defines the triangle inequality theorem and shows how to set up and solve inequalities to find possible lengths of triangle sides. Examples demonstrate drawing diagrams to visualize solutions and labeling angles and sides in order of measure. Practice problems have students list triangle sides or angles from least to greatest, determine if given side lengths can form a triangle using inequalities, and find the possible length of a third side of a triangle given the other two sides.
This document provides instruction on solving right triangles using trigonometric ratios. It includes examples of identifying angles from trigonometric ratios, calculating angle measures from ratios using inverse functions, solving right triangles when given side lengths or an angle measure, solving triangles in the coordinate plane, and applying trigonometry to problems involving road grades. Students are guided through applying concepts to multiple examples and practice problems.
This document discusses classifying angles formed when parallel lines are cut by a transversal. It defines key terms like alternate interior angles, alternate exterior angles, and corresponding angles. It then provides examples of classifying angle pairs based on their relationship. Sample problems are worked through, applying the concepts to find missing angle measures by justifying answers using angle properties of parallel lines cut by a transversal.
This document discusses classifying angles formed when parallel lines are cut by a transversal. It defines key terms like interior angles, exterior angles, alternate interior angles, alternate exterior angles, and corresponding angles. Examples are provided to illustrate each type of angle relationship. The document also provides step-by-step examples of classifying angle pairs and solving for missing angle measures using properties of parallel lines cut by a transversal.
The document discusses paragraph proofs and two-column proofs in geometry. A paragraph proof states the given information, what is to be proven, then uses a logical chain of statements and reasons to justify the conclusion. A two-column proof separates the statements from the reasons into two columns, with the given information and conclusion stated. Both proof styles use definitions, properties, and theorems to logically justify each step. The document provides examples of each type of proof to illustrate the process.
3008 perpendicular lines an theoremsno quizjbianco9910
This document contains a geometry drill with examples and explanations of perpendicular lines. It begins with warm up problems solving inequalities and equations. It then covers the key concepts that the perpendicular bisector of a segment is perpendicular to the segment at its midpoint, and the shortest distance from a point to a line is the perpendicular segment. Examples demonstrate writing proofs about perpendicular and parallel lines and applying the concept to carpentry. The homework assigned is to complete problems from the textbook.
This document discusses spheres and their volumes and surface areas. It includes examples of calculating volumes and surface areas of spheres and composite figures containing spheres. Formulas used include the volume of a sphere equaling (4/3)πr^3 and the surface area equaling 4πr^2. Examples show multiplying or dividing the radius changes the volume and surface area. The document contains warm up questions, objectives, vocabulary, lesson material and examples, and a quiz on spheres.
This document discusses calculating the volumes of pyramids, cones, and composite three-dimensional figures. It provides examples of finding volumes of different shapes, such as rectangular and square pyramids, cylinders, and cones. It also explores how changing the dimensions of these figures affects their volumes. Formulas are given for calculating the volumes of pyramids and cones based on their base areas and heights.
This document discusses calculating the volumes of prisms and cylinders. It provides examples of using the formulas for volume of a prism (V=lwh) and cylinder (V=πr^2h) to find volumes. It also explores how changing the dimensions of prisms and cylinders affects their volumes. For prisms, doubling the length, width, and height increases the volume by 8 times. For cylinders, doubling the radius and height quadruples the volume.
This document provides instruction on calculating the surface areas of pyramids and cones. It begins with key vocabulary terms like vertex, slant height, and altitude. Examples are then given of using the surface area formulas to find the lateral and total surface areas of regular pyramids and right cones when given dimensions like base length and slant height. The effects of changing dimensions on surface area are explored, and a manufacturing application involving cones is presented.
This document provides instruction on calculating the surface areas of prisms and cylinders. It includes definitions of key terms like lateral face and altitude. Formulas are provided for calculating the surface area of a right rectangular prism and a right cylinder. Examples show how to use the formulas to find the surface area of various prisms and cylinders. Effects of changing dimensions on surface area are also explored through examples.
This document is from a Holt Geometry textbook and covers solid geometry. It defines three-dimensional figures like prisms, pyramids, cylinders and cones. It discusses how to classify solids based on their vertices, edges and bases. It also covers nets, which are diagrams that can be folded to form three-dimensional figures, and cross sections, which are shapes revealed when a solid is cut by a plane. Examples are provided to identify solids from nets and describe cross sections. Food applications involving cutting cheese and watermelon are also discussed.
This document is from a Holt Geometry textbook and covers angles of elevation and depression. It includes examples of classifying angles as elevation or depression, using tangent ratios to find distances given an elevation or depression angle, and multi-step word problems combining these concepts. Warm-up questions, vocabulary, lesson objectives, examples with step-by-step solutions, and a lesson quiz are provided.
This document discusses trigonometric ratios and their use in solving problems involving right triangles. It begins with examples of writing trigonometric ratios (sine, cosine, tangent) as fractions and decimals for given angles. It then demonstrates using trigonometric ratios to find side lengths in right triangles when one or two sides and the angle between them is known. Several examples are provided to illustrate finding unknown side lengths by setting up and solving equations based on the appropriate trigonometric ratio. The document concludes with an example of applying trigonometric ratios to solve a real-world problem involving finding the length of an inclined railway track given the rise and angle of inclination.
This document is from a Holt Geometry textbook and covers applying special right triangles. It includes examples of using 45-45-90 and 30-60-90 triangles to find unknown side lengths. Key relationships covered are that for a 45-45-90 triangle the hypotenuse is √2 times the leg and for a 30-60-90 triangle the hypotenuse is 2 times the shorter leg. Examples include finding side lengths, checking solutions, and solving applied problems involving diagonal lengths of squares and equilateral triangles. The lesson ends with a quiz to test understanding of using these special right triangle relationships.
This document discusses properties of isosceles and equilateral triangles. It begins with examples calculating angle measures using properties of isosceles triangles, such as the fact that the two base angles of an isosceles triangle are congruent. It then proves theorems about isosceles and equilateral triangles, including that an equilateral triangle is also equiangular. Coordinate proofs are presented to show properties such as the midpoint of the two sides of an isosceles triangle bisecting the base. The document concludes with a lesson quiz reviewing the material.
This document discusses different methods for proving triangles congruent:
- ASA (Angle-Side-Angle) congruence can be used if two angles and the included side are congruent.
- AAS (Angle-Angle-Side) congruence can be used if two angles of one triangle are congruent to two angles of another triangle and one side included between the angles is congruent.
- HL (Hypotenuse-Leg) congruence can be used if two right triangles have one leg and the hypotenuse congruent.
Examples are provided to demonstrate applications of each method of triangle congruence.
This document discusses triangle congruence using the Side-Side-Side (SSS) and Side-Angle-Side (SAS) postulates. It provides examples of using SSS and SAS to prove that two triangles are congruent. Key concepts covered include triangle rigidity, included angles, and applying SSS and SAS to solve problems and construct congruent triangles. Worked examples demonstrate using SSS and SAS, along with step-by-step proofs of triangle congruence.
The document is from a geometry textbook and covers congruent triangles. It defines congruent triangles as those that are the same size and shape, with corresponding angles and sides being equal. Examples are provided of naming corresponding parts of congruent triangles and using properties of congruent triangles to prove other triangles are congruent or find missing angle measures. An engineering application example proves two triangles congruent based on bisecting lines and equal corresponding parts.
This document is from a geometry textbook and covers angle relationships in triangles. It begins with examples of finding missing angle measures using the Triangle Sum Theorem, Linear Pair Theorem, and exterior angle theorem. It then discusses properties of interior and exterior angles and applying the Third Angle Theorem. The document includes examples, practice problems, and key vocabulary terms like interior, exterior, and remote interior angles. It provides instruction and practice for finding missing angle measures in different types of triangles.
This document provides a lesson on classifying triangles based on their angle measures and side lengths. Triangles can be classified as acute, obtuse, right, equiangular, equilateral, isosceles or scalene. Examples show how to determine the type of triangle based on the given information. The lesson concludes with examples of applying triangle classification to find missing side lengths or determine how many triangles can be made from a given amount of material.
This document discusses properties of parallelograms from a geometry textbook. It includes examples of using properties such as opposite sides being equal, opposite angles being equal, and diagonals bisecting each other to solve problems about parallelograms. It also covers using parallelogram properties to write two-column proofs and finding coordinates of the fourth vertex of a parallelogram given three vertices.
The document discusses conditions for identifying whether a quadrilateral is a parallelogram. It provides examples of showing quadrilateral figures are parallelograms using different criteria such as having two pairs of parallel opposite sides, two pairs of congruent opposite sides, or diagonals bisecting each other. Theorems are presented for determining parallelograms based on angles or side lengths. Students are given practice identifying parallelograms and justifying their reasoning.
This document discusses formulas for finding measurements of triangles, quadrilaterals, and other geometric shapes. It includes examples of using formulas to find heights, bases, areas, and perimeters given certain measurements. Formulas are provided for triangles, parallelograms, rectangles, squares, trapezoids, rhombuses, kites, and applying these formulas to problems involving geometric grids. Lesson objectives and examples with step-by-step solutions demonstrate how to use the formulas to calculate missing values.
The document discusses formulas for circles and regular polygons. It defines key terms like radius, diameter, circumference, area, apothem, and central angle. Examples are provided for calculating circle measurements like circumference and area using formulas like C=2πr and A=πr^2. A formula is developed for the area of a regular polygon by dividing it into congruent triangles: A=(1/2)aps, where a is the apothem, p is the perimeter, and s is the side length. Worked examples demonstrate calculating measurements for circles and finding areas of regular polygons.
This document discusses finding the areas of composite figures, which are made up of simple shapes. It provides examples of adding or subtracting the areas of individual shapes to find the total area of composite figures. Estimating the area of irregular shapes using composite figures is also demonstrated. The key steps are to divide composite figures into simple shapes, calculate the area of each, and then add or subtract areas as needed using the Area Addition Postulate.
Leveraging Generative AI to Drive Nonprofit InnovationTechSoup
In this webinar, participants learned how to utilize Generative AI to streamline operations and elevate member engagement. Amazon Web Service experts provided a customer specific use cases and dived into low/no-code tools that are quick and easy to deploy through Amazon Web Service (AWS.)
This document provides an overview of wound healing, its functions, stages, mechanisms, factors affecting it, and complications.
A wound is a break in the integrity of the skin or tissues, which may be associated with disruption of the structure and function.
Healing is the body’s response to injury in an attempt to restore normal structure and functions.
Healing can occur in two ways: Regeneration and Repair
There are 4 phases of wound healing: hemostasis, inflammation, proliferation, and remodeling. This document also describes the mechanism of wound healing. Factors that affect healing include infection, uncontrolled diabetes, poor nutrition, age, anemia, the presence of foreign bodies, etc.
Complications of wound healing like infection, hyperpigmentation of scar, contractures, and keloid formation.
Chapter wise All Notes of First year Basic Civil Engineering.pptxDenish Jangid
Chapter wise All Notes of First year Basic Civil Engineering
Syllabus
Chapter-1
Introduction to objective, scope and outcome the subject
Chapter 2
Introduction: Scope and Specialization of Civil Engineering, Role of civil Engineer in Society, Impact of infrastructural development on economy of country.
Chapter 3
Surveying: Object Principles & Types of Surveying; Site Plans, Plans & Maps; Scales & Unit of different Measurements.
Linear Measurements: Instruments used. Linear Measurement by Tape, Ranging out Survey Lines and overcoming Obstructions; Measurements on sloping ground; Tape corrections, conventional symbols. Angular Measurements: Instruments used; Introduction to Compass Surveying, Bearings and Longitude & Latitude of a Line, Introduction to total station.
Levelling: Instrument used Object of levelling, Methods of levelling in brief, and Contour maps.
Chapter 4
Buildings: Selection of site for Buildings, Layout of Building Plan, Types of buildings, Plinth area, carpet area, floor space index, Introduction to building byelaws, concept of sun light & ventilation. Components of Buildings & their functions, Basic concept of R.C.C., Introduction to types of foundation
Chapter 5
Transportation: Introduction to Transportation Engineering; Traffic and Road Safety: Types and Characteristics of Various Modes of Transportation; Various Road Traffic Signs, Causes of Accidents and Road Safety Measures.
Chapter 6
Environmental Engineering: Environmental Pollution, Environmental Acts and Regulations, Functional Concepts of Ecology, Basics of Species, Biodiversity, Ecosystem, Hydrological Cycle; Chemical Cycles: Carbon, Nitrogen & Phosphorus; Energy Flow in Ecosystems.
Water Pollution: Water Quality standards, Introduction to Treatment & Disposal of Waste Water. Reuse and Saving of Water, Rain Water Harvesting. Solid Waste Management: Classification of Solid Waste, Collection, Transportation and Disposal of Solid. Recycling of Solid Waste: Energy Recovery, Sanitary Landfill, On-Site Sanitation. Air & Noise Pollution: Primary and Secondary air pollutants, Harmful effects of Air Pollution, Control of Air Pollution. . Noise Pollution Harmful Effects of noise pollution, control of noise pollution, Global warming & Climate Change, Ozone depletion, Greenhouse effect
Text Books:
1. Palancharmy, Basic Civil Engineering, McGraw Hill publishers.
2. Satheesh Gopi, Basic Civil Engineering, Pearson Publishers.
3. Ketki Rangwala Dalal, Essentials of Civil Engineering, Charotar Publishing House.
4. BCP, Surveying volume 1
LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UPRAHUL
This Dissertation explores the particular circumstances of Mirzapur, a region located in the
core of India. Mirzapur, with its varied terrains and abundant biodiversity, offers an optimal
environment for investigating the changes in vegetation cover dynamics. Our study utilizes
advanced technologies such as GIS (Geographic Information Systems) and Remote sensing to
analyze the transformations that have taken place over the course of a decade.
The complex relationship between human activities and the environment has been the focus
of extensive research and worry. As the global community grapples with swift urbanization,
population expansion, and economic progress, the effects on natural ecosystems are becoming
more evident. A crucial element of this impact is the alteration of vegetation cover, which plays a
significant role in maintaining the ecological equilibrium of our planet.Land serves as the foundation for all human activities and provides the necessary materials for
these activities. As the most crucial natural resource, its utilization by humans results in different
'Land uses,' which are determined by both human activities and the physical characteristics of the
land.
The utilization of land is impacted by human needs and environmental factors. In countries
like India, rapid population growth and the emphasis on extensive resource exploitation can lead
to significant land degradation, adversely affecting the region's land cover.
Therefore, human intervention has significantly influenced land use patterns over many
centuries, evolving its structure over time and space. In the present era, these changes have
accelerated due to factors such as agriculture and urbanization. Information regarding land use and
cover is essential for various planning and management tasks related to the Earth's surface,
providing crucial environmental data for scientific, resource management, policy purposes, and
diverse human activities.
Accurate understanding of land use and cover is imperative for the development planning
of any area. Consequently, a wide range of professionals, including earth system scientists, land
and water managers, and urban planners, are interested in obtaining data on land use and cover
changes, conversion trends, and other related patterns. The spatial dimensions of land use and
cover support policymakers and scientists in making well-informed decisions, as alterations in
these patterns indicate shifts in economic and social conditions. Monitoring such changes with the
help of Advanced technologies like Remote Sensing and Geographic Information Systems is
crucial for coordinated efforts across different administrative levels. Advanced technologies like
Remote Sensing and Geographic Information Systems
9
Changes in vegetation cover refer to variations in the distribution, composition, and overall
structure of plant communities across different temporal and spatial scales. These changes can
occur natural.
Temple of Asclepius in Thrace. Excavation resultsKrassimira Luka
The temple and the sanctuary around were dedicated to Asklepios Zmidrenus. This name has been known since 1875 when an inscription dedicated to him was discovered in Rome. The inscription is dated in 227 AD and was left by soldiers originating from the city of Philippopolis (modern Plovdiv).
বাংলাদেশের অর্থনৈতিক সমীক্ষা ২০২৪ [Bangladesh Economic Review 2024 Bangla.pdf] কম্পিউটার , ট্যাব ও স্মার্ট ফোন ভার্সন সহ সম্পূর্ণ বাংলা ই-বুক বা pdf বই " সুচিপত্র ...বুকমার্ক মেনু 🔖 ও হাইপার লিংক মেনু 📝👆 যুক্ত ..
আমাদের সবার জন্য খুব খুব গুরুত্বপূর্ণ একটি বই ..বিসিএস, ব্যাংক, ইউনিভার্সিটি ভর্তি ও যে কোন প্রতিযোগিতা মূলক পরীক্ষার জন্য এর খুব ইম্পরট্যান্ট একটি বিষয় ...তাছাড়া বাংলাদেশের সাম্প্রতিক যে কোন ডাটা বা তথ্য এই বইতে পাবেন ...
তাই একজন নাগরিক হিসাবে এই তথ্য গুলো আপনার জানা প্রয়োজন ...।
বিসিএস ও ব্যাংক এর লিখিত পরীক্ষা ...+এছাড়া মাধ্যমিক ও উচ্চমাধ্যমিকের স্টুডেন্টদের জন্য অনেক কাজে আসবে ...
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...PECB
Denis is a dynamic and results-driven Chief Information Officer (CIO) with a distinguished career spanning information systems analysis and technical project management. With a proven track record of spearheading the design and delivery of cutting-edge Information Management solutions, he has consistently elevated business operations, streamlined reporting functions, and maximized process efficiency.
Certified as an ISO/IEC 27001: Information Security Management Systems (ISMS) Lead Implementer, Data Protection Officer, and Cyber Risks Analyst, Denis brings a heightened focus on data security, privacy, and cyber resilience to every endeavor.
His expertise extends across a diverse spectrum of reporting, database, and web development applications, underpinned by an exceptional grasp of data storage and virtualization technologies. His proficiency in application testing, database administration, and data cleansing ensures seamless execution of complex projects.
What sets Denis apart is his comprehensive understanding of Business and Systems Analysis technologies, honed through involvement in all phases of the Software Development Lifecycle (SDLC). From meticulous requirements gathering to precise analysis, innovative design, rigorous development, thorough testing, and successful implementation, he has consistently delivered exceptional results.
Throughout his career, he has taken on multifaceted roles, from leading technical project management teams to owning solutions that drive operational excellence. His conscientious and proactive approach is unwavering, whether he is working independently or collaboratively within a team. His ability to connect with colleagues on a personal level underscores his commitment to fostering a harmonious and productive workplace environment.
Date: May 29, 2024
Tags: Information Security, ISO/IEC 27001, ISO/IEC 42001, Artificial Intelligence, GDPR
-------------------------------------------------------------------------------
Find out more about ISO training and certification services
Training: ISO/IEC 27001 Information Security Management System - EN | PECB
ISO/IEC 42001 Artificial Intelligence Management System - EN | PECB
General Data Protection Regulation (GDPR) - Training Courses - EN | PECB
Webinars: https://pecb.com/webinars
Article: https://pecb.com/article
-------------------------------------------------------------------------------
For more information about PECB:
Website: https://pecb.com/
LinkedIn: https://www.linkedin.com/company/pecb/
Facebook: https://www.facebook.com/PECBInternational/
Slideshare: http://www.slideshare.net/PECBCERTIFICATION
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...
Gch5 l5
1. Holt Geometry
5-5
Indirect Proof and Inequalities
in One Triangle5-5
Indirect Proof and Inequalities
in One Triangle
Holt Geometry
Warm UpWarm Up
Lesson PresentationLesson Presentation
Lesson QuizLesson Quiz
2. Holt Geometry
5-5
Indirect Proof and Inequalities
in One Triangle
Warm Up
1. Write a conditional from the sentence “An
isosceles triangle has two congruent sides.”
2. Write the contrapositive of the conditional “If it
is Tuesday, then John has a piano lesson.”
3. Show that the conjecture “If x > 6, then 2x >
14” is false by finding a counterexample.
If a ∆ is isosc., then it has 2 ≅ sides.
If John does not have a piano lesson, then it is
not Tuesday.
x = 7
3. Holt Geometry
5-5
Indirect Proof and Inequalities
in One Triangle
Write indirect proofs.
Apply inequalities in one triangle.
Objectives
5. Holt Geometry
5-5
Indirect Proof and Inequalities
in One Triangle
So far you have written proofs using direct reasoning.
You began with a true hypothesis and built a logical
argument to show that a conclusion was true. In an
indirect proof, you begin by assuming that the
conclusion is false. Then you show that this
assumption leads to a contradiction. This type of
proof is also called a proof by contradiction.
7. Holt Geometry
5-5
Indirect Proof and Inequalities
in One Triangle
When writing an indirect proof, look for a
contradiction of one of the following: the given
information, a definition, a postulate, or a
theorem.
Helpful Hint
8. Holt Geometry
5-5
Indirect Proof and Inequalities
in One Triangle
Example 1: Writing an Indirect Proof
Step 1 Identify the conjecture to be proven.
Given: a > 0
Step 2 Assume the opposite of the conclusion.
Write an indirect proof that if a > 0, then
Prove:
Assume
9. Holt Geometry
5-5
Indirect Proof and Inequalities
in One Triangle
Example 1 Continued
Step 3 Use direct reasoning to lead to a contradiction.
However, 1 > 0.
1 ≤ 0
Given, opposite of conclusion
Zero Prop. of Mult. Prop. of Inequality
Simplify.
10. Holt Geometry
5-5
Indirect Proof and Inequalities
in One Triangle
Step 4 Conclude that the original conjecture is true.
Example 1 Continued
The assumption that is false.
Therefore
11. Holt Geometry
5-5
Indirect Proof and Inequalities
in One Triangle
Check It Out! Example 1
Write an indirect proof that a triangle cannot
have two right angles.
Step 1 Identify the conjecture to be proven.
Given: A triangle’s interior angles add up to 180°.
Prove: A triangle cannot have two right angles.
Step 2 Assume the opposite of the conclusion.
An angle has two right angles.
12. Holt Geometry
5-5
Indirect Proof and Inequalities
in One Triangle
Check It Out! Example 1 Continued
Step 3 Use direct reasoning to lead to a contradiction.
However, by the Protractor Postulate, a triangle
cannot have an angle with a measure of 0°.
m∠1 + m∠2 + m∠3 = 180°
90° + 90° + m∠3 = 180°
180° + m∠3 = 180°
m∠3 = 0°
13. Holt Geometry
5-5
Indirect Proof and Inequalities
in One Triangle
Step 4 Conclude that the original conjecture is true.
The assumption that a triangle can have
two right angles is false.
Therefore a triangle cannot have two right
angles.
Check It Out! Example 1 Continued
14. Holt Geometry
5-5
Indirect Proof and Inequalities
in One Triangle
The positions of the longest and shortest sides of
a triangle are related to the positions of the
largest and smallest angles.
15. Holt Geometry
5-5
Indirect Proof and Inequalities
in One Triangle
Example 2A: Ordering Triangle Side Lengths and
Angle Measures
Write the angles in order from
smallest to largest.
The angles from smallest to largest are ∠F, ∠H and ∠G.
The shortest side is , so the
smallest angle is ∠F.
The longest side is , so the largest angle is ∠G.
16. Holt Geometry
5-5
Indirect Proof and Inequalities
in One Triangle
Example 2B: Ordering Triangle Side Lengths and
Angle Measures
Write the sides in order from
shortest to longest.
m∠R = 180° – (60° + 72°) = 48°
The smallest angle is ∠R, so the
shortest side is .
The largest angle is ∠Q, so the longest side is .
The sides from shortest to longest are
17. Holt Geometry
5-5
Indirect Proof and Inequalities
in One Triangle
Check It Out! Example 2a
Write the angles in order from
smallest to largest.
The angles from smallest to largest are ∠B, ∠A, and ∠C.
The shortest side is , so the
smallest angle is ∠B.
The longest side is , so the largest angle is ∠C.
18. Holt Geometry
5-5
Indirect Proof and Inequalities
in One Triangle
Check It Out! Example 2b
Write the sides in order from
shortest to longest.
m∠E = 180° – (90° + 22°) = 68°
The smallest angle is ∠D, so the shortest side is .
The largest angle is ∠F, so the longest side is .
The sides from shortest to longest are
19. Holt Geometry
5-5
Indirect Proof and Inequalities
in One Triangle
A triangle is formed by three segments, but not
every set of three segments can form a triangle.
20. Holt Geometry
5-5
Indirect Proof and Inequalities
in One Triangle
A certain relationship must exist among the lengths
of three segments in order for them to form a
triangle.
21. Holt Geometry
5-5
Indirect Proof and Inequalities
in One Triangle
Example 3A: Applying the Triangle Inequality
Theorem
Tell whether a triangle can have sides with the
given lengths. Explain.
7, 10, 19
No—by the Triangle Inequality Theorem, a triangle
cannot have these side lengths.
22. Holt Geometry
5-5
Indirect Proof and Inequalities
in One Triangle
Example 3B: Applying the Triangle Inequality
Theorem
Tell whether a triangle can have sides with the
given lengths. Explain.
2.3, 3.1, 4.6
Yes—the sum of each pair of lengths is greater
than the third length.
23. Holt Geometry
5-5
Indirect Proof and Inequalities
in One Triangle
Example 3C: Applying the Triangle Inequality
Theorem
Tell whether a triangle can have sides with the
given lengths. Explain.
n + 6, n2
– 1, 3n, when n = 4.
Step 1 Evaluate each expression when n = 4.
n + 6
4 + 6
10
n2
– 1
(4)2
– 1
15
3n
3(4)
12
24. Holt Geometry
5-5
Indirect Proof and Inequalities
in One Triangle
Example 3C Continued
Step 2 Compare the lengths.
Yes—the sum of each pair of lengths is greater
than the third length.
25. Holt Geometry
5-5
Indirect Proof and Inequalities
in One Triangle
Check It Out! Example 3a
Tell whether a triangle can have sides with the
given lengths. Explain.
8, 13, 21
No—by the Triangle Inequality Theorem, a triangle
cannot have these side lengths.
26. Holt Geometry
5-5
Indirect Proof and Inequalities
in One Triangle
Check It Out! Example 3b
Tell whether a triangle can have sides with the
given lengths. Explain.
6.2, 7, 9
Yes—the sum of each pair of lengths is greater
than the third side.
27. Holt Geometry
5-5
Indirect Proof and Inequalities
in One Triangle
Check It Out! Example 3c
Tell whether a triangle can have sides with the
given lengths. Explain.
t – 2, 4t, t2
+ 1, when t = 4
Step 1 Evaluate each expression when t = 4.
t – 2
4 – 2
2
t2
+ 1
(4)2
+ 1
17
4t
4(4)
16
28. Holt Geometry
5-5
Indirect Proof and Inequalities
in One Triangle
Check It Out! Example 3c Continued
Step 2 Compare the lengths.
Yes—the sum of each pair of lengths is greater
than the third length.
29. Holt Geometry
5-5
Indirect Proof and Inequalities
in One Triangle
Example 4: Finding Side Lengths
The lengths of two sides of a triangle are 8
inches and 13 inches. Find the range of
possible lengths for the third side.
Let x represent the length of the third side. Then
apply the Triangle Inequality Theorem.
Combine the inequalities. So 5 < x < 21. The length
of the third side is greater than 5 inches and less
than 21 inches.
x + 8 > 13
x > 5
x + 13 > 8
x > –5
8 + 13 > x
21 > x
30. Holt Geometry
5-5
Indirect Proof and Inequalities
in One Triangle
Check It Out! Example 4
The lengths of two sides of a triangle are 22
inches and 17 inches. Find the range of possible
lengths for the third side.
Let x represent the length of the third side. Then
apply the Triangle Inequality Theorem.
Combine the inequalities. So 5 < x < 39. The length
of the third side is greater than 5 inches and less
than 39 inches.
x + 22 > 17
x > –5
x + 17 > 22
x > 5
22 + 17 > x
39 > x
31. Holt Geometry
5-5
Indirect Proof and Inequalities
in One Triangle
Example 5: Travel Application
The figure shows the
approximate distances
between cities in California.
What is the range of distances
from San Francisco to Oakland?
Let x be the distance from San Francisco to Oakland.
x + 46 > 51
x > 5
x + 51 > 46
x > –5
46 + 51 > x
97 > x
5 < x < 97 Combine the inequalities.
Δ Inequal. Thm.
Subtr. Prop. of
Inequal.
The distance from San Francisco to Oakland is
greater than 5 miles and less than 97 miles.
32. Holt Geometry
5-5
Indirect Proof and Inequalities
in One Triangle
Check It Out! Example 5
The distance from San Marcos to Johnson City is
50 miles, and the distance from Seguin to San
Marcos is 22 miles. What is the range of
distances from Seguin to Johnson City?
Let x be the distance from Seguin to Johnson City.
x + 22 > 50
x > 28
x + 50 > 22
x > –28
22 + 50 > x
72 > x
28 < x < 72 Combine the inequalities.
Δ Inequal. Thm.
Subtr. Prop. of
Inequal.
The distance from Seguin to Johnson City is greater
than 28 miles and less than 72 miles.
33. Holt Geometry
5-5
Indirect Proof and Inequalities
in One Triangle
Lesson Quiz: Part I
1. Write the angles in order from smallest to
largest.
2. Write the sides in order from shortest to
longest.
∠C, ∠B, ∠A
34. Holt Geometry
5-5
Indirect Proof and Inequalities
in One Triangle
Lesson Quiz: Part II
3. The lengths of two sides of a triangle are 17 cm
and 12 cm. Find the range of possible lengths for
the third side.
4. Tell whether a triangle can have sides with
lengths 2.7, 3.5, and 9.8. Explain.
No; 2.7 + 3.5 is not greater than 9.8.
5 cm < x < 29 cm
5. Ray wants to place a chair so it is
10 ft from his television set. Can
the other two distances
shown be 8 ft and 6 ft? Explain.
Yes; the sum of any two lengths is
greater than the third length.