TechMathI - 4.4 - Isosceles and Right Triangle Theorems
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The document is a math lesson on isosceles triangles that includes definitions, theorems, examples and practice problems. It defines isosceles triangles as triangles with two congruent sides and base angles as the two angles adjacent to the base. It presents the Isosceles Triangle Theorem stating that if two sides are congruent, the angles opposite are also congruent. Examples show applying the theorem to find missing angle measures and side lengths. The Hypotenuse-Leg Congruence Theorem is introduced for right triangles. Practice problems have students identify if triangles can be proven congruent.
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- Adjacent angles, angle bisectors, and the angle addition postulate
- Supplementary, complementary, linear pair, and vertical angles
- How to name angles and identify different angle relationships
- Steps to solve problems involving unknown angle measures using properties of angles
Geometry unit 4.5
This document covers properties of isosceles and equilateral triangles. It defines key terms like legs, vertex angle, and base of an isosceles triangle. It presents theorems like if a triangle is isosceles, the vertex angle is equal to the base angles, and the bisector of the vertex angle is the perpendicular bisector of the base. Examples demonstrate using these properties to find missing angle measures. The connection between equilateral and equiangular triangles is also explained. Coordinate proofs may be used to show triangles are isosceles. Practice problems are included to assess understanding.
Gch04 l8
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.
Lecture 4.1 4.2
The document discusses classifying and finding angles of triangles. It defines different types of triangles based on sides or angles, such as equilateral, isosceles, scalene, acute, obtuse, right. The triangle sum theorem states the sum of interior angles is 180 degrees. The exterior angle theorem relates exterior to interior angles. Examples show using these theorems to find missing angle measures in various triangles.
Triangles 101
1. Triangles can be classified based on their sides as isosceles, scalene, or equilateral. The sum of the interior angles of any triangle is always 180 degrees.
2. The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles.
3. True or false questions about triangle properties include: if two angles are congruent then the third must be; each angle of an equiangular triangle is 60 degrees; a triangle can have at most one right or obtuse angle.
Geometry Test Review
The document provides a review of trigonometric ratios such as sine, cosine, and tangent that are used to solve problems involving right triangles, includes examples of applying these ratios to solve practice problems involving missing side lengths and angle measures, and reviews special right triangles and the SOH CAH TOA method for remembering the trigonometric functions.
Geometry Test Review
The document provides a review of trigonometric ratios such as sine, cosine, and tangent that are used to solve problems involving right triangles, includes examples of applying these ratios to solve practice problems involving missing side lengths and angle measures, and discusses special right triangles and the mnemonic device SOH CAH TOA to remember the trigonometric functions.
law_of_sines.ppt
This document discusses the Law of Sines and Law of Cosines for solving oblique triangles. It provides examples of using these laws to:
- Solve triangles given two angles and one side (AAS) or two sides and an angle opposite (SSA) using the Law of Sines.
- Determine if a triangle is valid or ambiguous given two sides and an angle (SSA) using the Law of Sines.
- Find the area of oblique triangles using the relationship between a side and its opposite angle from the Law of Sines or by Heron's formula when all three sides are given.
law_of_sines.ppt
This document discusses using the Law of Sines and Law of Cosines to solve oblique triangles. It covers the four cases for solving triangles: two angles and a side (AAS/ASA), two sides and an angle opposite (SSA), three sides (SSS), and two sides and their included angle (SAS). The Law of Sines can be used for AAS/ASA and SSA cases, while the Law of Cosines is needed for SSS and SAS cases. It also discusses finding the area of triangles using the Law of Sines and Heron's formula for SSS cases.
Unit_4_-_S4 (1).ppt
This document provides information about circle theorems developed by Euclid of Alexandria. It begins with introductory terminology for parts of a circle such as diameter, radius, circumference. It then presents 8 theorems with examples and explanations: 1) The angle in a semi-circle is a right angle, 2) The angle subtended at the centre is twice the angle at the circumference, 3) Angles subtended by an arc or chord in the same segment are equal, 4) The angle between a tangent and radius is 90 degrees, 5) The alternate segment theorem relating angles of tangents and chords, 6) The cyclic quadrilateral theorem stating opposite angles sum to 180 degrees, 7) The two tangent
1.5 describe angle pair relationships
1. The document provides examples and explanations of different types of angle relationships including: complementary angles, supplementary angles, adjacent angles, linear pairs, and vertical angles. Examples are worked through step-by-step to find the measures of unknown angles using properties of these relationships.
2. Guided practice problems similar to the examples are provided for students to work through. The correct answers are given.
3. Key concepts covered include writing equations to represent angle relationships and solving them algebraically to find unknown angle measures. Diagrams are also used to illustrate and identify different angle pairs.
Chapter activity plus-in-mathematics-10
This document introduces the Basic Proportionality Theorem and discusses activities to verify it using parallel line boards and triangle cut-outs. It also discusses the Centroid and how to illustrate that the medians of a triangle intersect at a point inside the triangle, called the centroid. Finally, it discusses an activity to verify the Pythagorean Theorem using paper folding, cutting, and pasting to construct squares based on the sides of a right triangle.
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This geometry review document contains questions about various geometry topics including:
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TechMathI - Point slope form
The document provides examples and explanations of finding the rate of change and linear equations from tables of data points and given slopes and points. It includes examples of using the point-slope formula to write linear equations from a given point and slope, as well as examples of writing linear equations from two given points. Students are prompted to write linear equations for several examples.
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The document discusses finding the x-intercept and y-intercept of linear equations by setting either x or y equal to 0 and solving for the other variable. It provides examples of finding the intercepts of equations and graphing the resulting point coordinates. Students are asked to practice finding and graphing intercepts for several equations.
TechMathI - Slope intercept form
This document discusses linear equations and slope-intercept form. It provides examples of:
1) Finding the slope and y-intercept of linear equations in slope-intercept form like y = 4x + 2.
2) Writing equations of lines given the slope and y-intercept, such as an equation with m = 3 and y-intercept of 7.
3) Finding the slope and y-intercept from two points on a line, then writing the equation in slope-intercept form.
TechMathI - Rate of change and slope
The document contains information about rates of change, slope, and lines. It includes examples of calculating rates of change from tables of data and slope from ordered pairs using the slope formula. It discusses horizontal and vertical lines having slopes of 0 and undefined, and positive and negative slopes indicating the direction a line slants.
TechMathI - Ch6 Test
This document contains a 36 question multiple choice test on various math concepts including:
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The questions require skills like setting up and solving proportions, unit conversion, percent calculation, and percent change.
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This document contains 30 math word problems testing foundations concepts like rates, proportions, percentages, and percent change. The problems can be solved by setting up and solving proportions, finding unit rates, calculating percentages, or determining percent change between two amounts. Key information includes rates, costs, percentages, initial and final amounts, and measurements involved in rate and percent problems.
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This document contains a 15 question quiz on math concepts like proportions, ratios, rates, similar triangles, and unit conversions. The questions require setting up and solving proportions, finding unit rates, calculating rates per hour, and using similar triangles to determine unknown lengths.
TechMathI - 6.2 - Similarity examples
Similar figures are two-dimensional shapes that have the same shape but can differ in size. To be similar, the corresponding sides of the figures must be proportional, meaning they differ by a constant scale factor. Similar figures have the same angle measures but different side lengths. Problems involving similar figures can be solved by identifying the scale factor between corresponding sides and using proportional reasoning.
TechMathI - 6.2 - Similarity
This document contains 4 math word problems involving similar triangles and scaling factors to determine unknown side lengths or heights. It also provides examples of finding the height of objects like trees using the similarity of triangles formed by shadows of objects of known height.
TechMathI - 6.1 - Ratio and Proportion
1. The document provides examples of ratios, proportions, rates, unit rates, and examples of setting up and solving proportions using cross multiplication.
2. It includes multi-step proportion examples involving fractions with variables in both the numerator and denominator.
3. Additional practice problems are provided for readers to try setting up and solving different types of proportions.
TechMathI - Ch5 Test
This document contains a 33 question test on polygons and their properties. The test covers identifying polygons by name and type, finding missing side lengths and angle measures of polygons, calculating sums of interior and exterior angles, determining which statements about polygon properties are true or false, and identifying polygons based on given properties. The test includes multiple choice, true/false, and short answer questions.
TechMathI - Ch5 Vocab Quiz
This document is a vocabulary quiz that tests knowledge of polygon terms. It contains 17 multiple choice questions that ask the learner to match polygon names like triangle, square, pentagon, and hexagon with their definitions relating to number of sides, side properties, angles, and geometric shapes. The word bank at the bottom provides additional polygon shape names as potential answers.
TechMathI - 5.6 - Trapezoids
This document discusses properties and definitions related to trapezoids. It defines a trapezoid as a quadrilateral with one pair of parallel sides, and defines an isosceles trapezoid as one where the non-parallel sides are congruent. It states that the diagonals of an isosceles trapezoid are congruent and that the median of any trapezoid is parallel to the bases and has a length that is half the sum of the base lengths. It includes examples of finding missing side lengths using the median property.
TechMathI - 5.5 - Special Parallelograms
1. The document discusses different types of quadrilaterals including trapezoids, parallelograms, rectangles, squares, and rhombuses.
2. It provides steps to identify these shapes based on their properties like the number of parallel sides and whether the angles are right angles or the sides are congruent.
3. Tests are described to determine if a quadrilateral is a parallelogram based on properties like opposite sides being parallel and congruent.
TechMathI - 5.4 - Parallelograms
This document provides examples and practice problems for identifying angle measures in parallelograms. It gives the side lengths of parallelogram ABCD and asks to find the three other angle measures. It also provides a practice problem to find missing angle measures for an additional parallelogram.
TechMathI - 5.3 - Properties of Quadrilaterals
This document provides information about different types of quadrilaterals (four-sided polygons) through a foldable activity. It defines and lists properties of parallelograms, rectangles, rhombuses, squares, and trapezoids. Key quadrilaterals are distinguished by their parallel sides, right angles, and congruent sides or angles. The foldable allows students to draw and label examples of each quadrilateral type.
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TechMathI - 4.4 - Isosceles and Right Triangle Theorems
- 4. Definitions - Review ABC is an isosceles triangle. Name each item(s): Vertex Angle Base Legs Base Angles AC AB, CB Side opposite C AB Angle opposite BC
- 5. A C B A B C A C B Yes Yes No
- 6. Isosceles Triangle Theorem If two sides of a triangle are congruent, then the angles opposite those sides are congruent.
- 7. Example 1 Find the measure of each angle. x + x + 30 o = 180 o 2x + 30 o = 180 o 2x = 150 o x = 75 o
- 8. Example 2 Find the length of each side. 3x – 6 2x 6 3x – 6 = 6 3x = 12 x = 4 EF = 6 EG = 8
- 9. Example 3 Find the measure of each angle. (2x – 4) + (x + 2) + (x + 2) = 180 o 4x = 180 o x = 45 o
- 11. Practice - What do we still need to know to prove the triangles are congruent? C A B M N F P
- 14. 2x 2x x 62