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Similar triangles can be identified using angle-angle similarity, where if two angles of one triangle are congruent to two angles of another triangle, the two triangles are considered similar.
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4.6 terms
This document defines key terms and theorems related to triangles, including isosceles, equilateral, and right triangles. It defines base angles, vertex angles, the base angle theorem, converse of the base angle theorem, and corollaries relating equilateral and equiangular triangles. It also defines the hypotenuse-leg congruence theorem for right triangles.
1.2.2 Pairs of Angles
Identify linear pairs, vertical angles, complementary angles, and supplementary angles, and use their relationships to set up and solve equations.
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This document defines and describes several geometric shapes including linear pairs of angles, octagons, parallel lines, parallelograms, arcs, circles, trapezoids, and isosceles triangles. It provides brief definitions for each shape stating their key properties such as number of sides, angles, and relationships between sides or angles.
Vocab lessons 7 1 through 7-5
- A point is named by a capital letter. A line is named by two points on the line. A plane is named by three non-collinear points.
- A line segment is named by its two endpoints. A ray has one endpoint and extends without end in one direction. An angle is named using the vertex in the middle and the two rays.
- Angles are measured and can be acute, right, or obtuse. A protractor is used to measure angles. Vertical angles, adjacent angles, complementary angles, and supplementary angles are special angle relationships.
All about triangles (2)
This document discusses triangle similarity and how to determine if triangles are similar. It defines triangle similarity as when triangles only differ in size, with all angles being equal and corresponding sides having the same ratio. It then outlines several theorems - SAS, SSS, AAS, AA - that can be used to prove triangles are similar by comparing their sides and angles. Examples are provided to illustrate each theorem, and an interactive model is referenced for students to explore triangle similarity on their own.
Clil.geometry presentation-
This document defines and describes various geometric shapes and concepts. It defines points, lines, line segments, rays, angles, parallel lines, triangles, quadrilaterals such as parallelograms, rectangles, squares, trapezoids, and rhombi. It also covers circles, areas, volumes, and properties of angles, triangles, parallelograms and polygons. Key concepts include classifying angles as acute, obtuse, right or straight, and properties of parallel lines, perpendicular lines, complementary and supplementary angles.
Geovocab
This document defines and provides examples of geometric shapes and their properties. It discusses polygons like triangles and quadrilaterals, describing their classifications based on sides or angles. It also defines three-dimensional shapes like prisms, pyramids, cylinders, and cones, noting properties like parallel bases and lateral faces. Key properties discussed include symmetry, parallelism of sides, congruence of sides and angles, and right angles.
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4.6 terms
This document defines key terms and theorems related to triangles, including isosceles, equilateral, and right triangles. It defines base angles, vertex angles, the base angle theorem, converse of the base angle theorem, and corollaries relating equilateral and equiangular triangles. It also defines the hypotenuse-leg congruence theorem for right triangles.
1.2.2 Pairs of Angles
Identify linear pairs, vertical angles, complementary angles, and supplementary angles, and use their relationships to set up and solve equations.
Savanah presentatiom
This document defines and describes several geometric shapes including linear pairs of angles, octagons, parallel lines, parallelograms, arcs, circles, trapezoids, and isosceles triangles. It provides brief definitions for each shape stating their key properties such as number of sides, angles, and relationships between sides or angles.
Vocab lessons 7 1 through 7-5
- A point is named by a capital letter. A line is named by two points on the line. A plane is named by three non-collinear points.
- A line segment is named by its two endpoints. A ray has one endpoint and extends without end in one direction. An angle is named using the vertex in the middle and the two rays.
- Angles are measured and can be acute, right, or obtuse. A protractor is used to measure angles. Vertical angles, adjacent angles, complementary angles, and supplementary angles are special angle relationships.
All about triangles (2)
This document discusses triangle similarity and how to determine if triangles are similar. It defines triangle similarity as when triangles only differ in size, with all angles being equal and corresponding sides having the same ratio. It then outlines several theorems - SAS, SSS, AAS, AA - that can be used to prove triangles are similar by comparing their sides and angles. Examples are provided to illustrate each theorem, and an interactive model is referenced for students to explore triangle similarity on their own.
Clil.geometry presentation-
This document defines and describes various geometric shapes and concepts. It defines points, lines, line segments, rays, angles, parallel lines, triangles, quadrilaterals such as parallelograms, rectangles, squares, trapezoids, and rhombi. It also covers circles, areas, volumes, and properties of angles, triangles, parallelograms and polygons. Key concepts include classifying angles as acute, obtuse, right or straight, and properties of parallel lines, perpendicular lines, complementary and supplementary angles.
Geovocab
This document defines and provides examples of geometric shapes and their properties. It discusses polygons like triangles and quadrilaterals, describing their classifications based on sides or angles. It also defines three-dimensional shapes like prisms, pyramids, cylinders, and cones, noting properties like parallel bases and lateral faces. Key properties discussed include symmetry, parallelism of sides, congruence of sides and angles, and right angles.
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The document discusses different types of triangles based on the lengths of their sides: equilateral triangles have all three sides equal; isosceles triangles have two equal sides; scalene triangles have no equal sides. It also describes criteria for determining if two triangles are congruent, including side-angle-side (SAS), angle-side-angle (ASA), and side-side-side (SSS). Properties of triangles are outlined, such as the angle sum property that the interior angles sum to 180 degrees and the exterior angle property relating an exterior angle to the two interior angles. The Pythagorean theorem relating the sides of a right triangle is presented along with Heron's formula for calculating the area of any triangle.
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The document discusses several theorems and properties related to circles:
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2) Parallel chords in a circle cut congruent arcs, and congruent chords are equidistant from the circle's center.
3) When two chords intersect, the product of one chord's segments equals the product of the other's segments.
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There are four types of triangles: equilateral triangles have three equal sides and three equal angles; isosceles triangles have two equal sides and two equal angles; scalene triangles have no equal sides and all different angles; right-angled triangles contain one 90 degree angle and can be either isosceles or scalene.
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Lines and angles
LINES AND ANGLES
content:
Introduction
Basic terms and definitions
Intersecting lines and Non-intersecting lines
Perpendicular lines
Angles
Types of angles
Parallel lines
Transversal
Triangles
1) A triangle is a three-sided polygon with three vertices and three edges.
2) Triangles can be classified based on side lengths (equilateral, isosceles, scalene) or interior angles (right, acute, obtuse).
3) The interior angles of any triangle always sum to 180 degrees. Congruent triangles have the same shape and size, while similar triangles have the same angle measures but sides proportional in length.
Classifying Triangles
The document defines and provides examples of different types of triangles based on their interior angles and side lengths. It explains that triangles can be classified as right, obtuse, or acute based on their interior angles, and as equilateral, isosceles, or scalene based on their side lengths. Examples are given of right scalene triangles, obtuse isosceles triangles, and acute scalene triangles to demonstrate how triangles can be classified based on both their angles and side lengths.
Quadrilaterals
This document defines and classifies different types of quadrilaterals. It introduces quadrilaterals and defines special types including parallelograms, rectangles, rhombuses, squares, and trapezoids. Quadrilaterals can have multiple names because some properties overlap between types. The best name is the most specific one. The document shows how the types are related through Venn diagrams and concept maps, and provides examples of classifying quadrilaterals and identifying the best name.
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This document discusses properties of isosceles triangles. It states that an isosceles triangle has at least two sides of equal length. The base angles of an isosceles triangle are always congruent. If a triangle has two congruent angles, then it is an isosceles triangle. An equilateral triangle is a special type of isosceles triangle where all three sides are congruent and all three angles are congruent.
triangle
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Lines and angles
Two angles are complementary if their measures sum to 90 degrees. Two angles are supplementary if their measures sum to 180 degrees. Adjacent angles have a common vertex and one common side but no interior points in common, such as the angles formed by scissors. Vertically opposite angles are formed without a common side when two lines intersect. A linear pair is formed when the non-common sides of adjacent angles lie in a straight line. A transversal is a line that intersects two or more other lines at distinct points, and intersecting lines cut each other at a common point.
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lines and angles
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- Angle theorems about parallel lines cut by a transversal and the angle sums of triangles
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A document defines and compares different types of quadrilaterals:
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- A rectangle has four right angles and opposite sides that are parallel and congruent. Like a square, its diagonals are congruent and bisect each other.
- A parallelogram has two pairs of parallel opposite sides. Its diagonals bisect each other and opposite angles are congruent.
10.1 terms
The document defines key terms related to circles and tangents, including: a circle as points equidistant from a center point; a radius and diameter; a chord as a segment with endpoints on the circle; a secant as a line intersecting a circle at two points; and a tangent as a line intersecting a circle at exactly one point. It states that a tangent line is perpendicular to the radius drawn to the point of tangency, and that for two segments from the same exterior point to both be tangent to a circle, they must be congruent.
6.4 notes
This document discusses proving triangle similarity using the AA similarity postulate. It states that if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. The document contains notes from a class on November 18th discussing this postulate and assigning homework problems from page 4 to page 20 that are even numbers and problems 26 through 28.
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The document discusses several theorems and properties related to circles:
1) Tangent segments drawn from the same point outside a circle to the circle are equal in length.
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There are four types of triangles: equilateral triangles have three equal sides and three equal angles; isosceles triangles have two equal sides and two equal angles; scalene triangles have no equal sides and all different angles; right-angled triangles contain one 90 degree angle and can be either isosceles or scalene.
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Lines and angles
LINES AND ANGLES
content:
Introduction
Basic terms and definitions
Intersecting lines and Non-intersecting lines
Perpendicular lines
Angles
Types of angles
Parallel lines
Transversal
Triangles
1) A triangle is a three-sided polygon with three vertices and three edges.
2) Triangles can be classified based on side lengths (equilateral, isosceles, scalene) or interior angles (right, acute, obtuse).
3) The interior angles of any triangle always sum to 180 degrees. Congruent triangles have the same shape and size, while similar triangles have the same angle measures but sides proportional in length.
Classifying Triangles
The document defines and provides examples of different types of triangles based on their interior angles and side lengths. It explains that triangles can be classified as right, obtuse, or acute based on their interior angles, and as equilateral, isosceles, or scalene based on their side lengths. Examples are given of right scalene triangles, obtuse isosceles triangles, and acute scalene triangles to demonstrate how triangles can be classified based on both their angles and side lengths.
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This document discusses properties of isosceles triangles. It states that an isosceles triangle has at least two sides of equal length. The base angles of an isosceles triangle are always congruent. If a triangle has two congruent angles, then it is an isosceles triangle. An equilateral triangle is a special type of isosceles triangle where all three sides are congruent and all three angles are congruent.
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This document discusses triangles and their properties. It defines a triangle as a three-sided polygon with three angles and vertices. It describes the three main types of triangles based on side lengths (equilateral, isosceles, scalene) and angles (acute, right, obtuse). Heron's formula for calculating the area of a triangle given the side lengths is presented. Key properties of triangles like the angle sum property, exterior angle property, and congruency criteria (SSS, SAS, ASA) are outlined. Important triangle centers such as the incenter, circumcenter, centroid, and orthocenter are defined.
Lines and angles
Two angles are complementary if their measures sum to 90 degrees. Two angles are supplementary if their measures sum to 180 degrees. Adjacent angles have a common vertex and one common side but no interior points in common, such as the angles formed by scissors. Vertically opposite angles are formed without a common side when two lines intersect. A linear pair is formed when the non-common sides of adjacent angles lie in a straight line. A transversal is a line that intersects two or more other lines at distinct points, and intersecting lines cut each other at a common point.
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This document discusses isosceles and equilateral triangles. It defines isosceles triangles as triangles with two congruent sides and equilateral triangles as triangles with three congruent sides. The Isosceles Triangle Theorem and its converse state that if two sides or angles of a triangle are congruent, then the opposite angles or sides are also congruent. Similarly, the Equilateral Triangle Corollary and its converse state that if a triangle is equilateral, it is also equiangular, and vice versa. Examples are given to demonstrate using these properties to solve for missing angle measures.
lines and angles
This document discusses various geometry concepts related to angles and lines including:
- Types of angles such as acute, obtuse, straight, and reflex angles
- Relationships between pairs of angles such as complementary, supplementary, vertical, corresponding, alternate interior, and alternate exterior angles
- Properties of lines such as intersecting lines, parallel lines, and transversals
- Angle theorems about parallel lines cut by a transversal and the angle sums of triangles
Classifying triangles mary marks.2pptx
The document discusses classifying triangles based on their sides and angles. It explains that triangles can be equilateral, isosceles, or scalene based on their side lengths. Triangles can also be acute, obtuse, or right based on their angle measures. The document provides examples of finding the measure of the third angle of a triangle when two angles are given.
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Triangles are three-sided polygons that come in three types based on their side lengths: equilateral triangles have all three sides the same length, isosceles triangles have two sides the same length, and scalene triangles have all three sides of different lengths. The area of any triangle can be calculated using Hero's Formula, which uses the triangle's side lengths and semi-perimeter. While Pascal's Triangle is named after Blaise Pascal, he did not invent the triangle - other mathematicians studied and used triangular patterns before him.
Symmetry Presentation By Sharanya
The document discusses different types of symmetry, including line symmetry and rotational symmetry. Line symmetry occurs when one half of an object is a mirror image of the other half. Rotational symmetry is when an object looks the same after being rotated around a center point by a certain number of degrees. Examples are given of different shapes and objects that demonstrate line symmetry, such as butterflies and smiley faces, as well as rotational symmetry, including triangles, pizzas, and flowers.
Kite a special quadrilateral
The document defines and compares different types of quadrilaterals. A quadrilateral is a closed shape with four sides whose interior angles sum to 360 degrees. The main quadrilaterals discussed are: parallelograms, which have opposite sides that are parallel and equal; rectangles, which have four right angles; rhombi, which have four equal sides; squares, which are rhombi with four right angles; kites, which have two pairs of equal sides that meet at a right angle; and trapezoids, which have one set of parallel sides. Key differences between kites, squares, rhombi and parallelograms are outlined.
Quadrilaterals
A document defines and compares different types of quadrilaterals:
- A square has four equal sides and four right angles. Its diagonals are congruent and bisect each other.
- A rectangle has four right angles and opposite sides that are parallel and congruent. Like a square, its diagonals are congruent and bisect each other.
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Solve Lines and Angles Question Paper for CBSE Class 9
Solve Lines and Angles Question Paper for CBSE Class 9
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6.4 notes
This document discusses proving triangle similarity using the AA similarity postulate. It states that if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. The document contains notes from a class on November 18th discussing this postulate and assigning homework problems from page 4 to page 20 that are even numbers and problems 26 through 28.
Geometry unit 7.3
This document discusses proving triangles are similar using the AA, SSS, and SAS similarity criteria. It provides examples of using these criteria to prove triangles are similar, find missing side lengths in similar triangles using proportionality, and write proofs involving similar triangles. Applications to engineering problems are also presented. The key methods covered are using corresponding angles or sides that are equal to prove triangles are similar, setting up proportions between corresponding sides in similar triangles to solve for unknowns, and structuring multi-statement proofs involving similar triangles.
6.4 prove triangles similar by aa
This document provides examples and explanations for proving triangles similar using the Angle-Angle (AA) criterion. It includes examples of showing two triangles are similar by showing they have two pairs of congruent angles. It also includes examples of writing similarity statements and using proportions to find missing side lengths when corresponding angles and one pair of corresponding sides are given. Guided practice problems allow students to practice determining if triangles are similar and writing similarity statements.
3.8.4 Triangle Similarity
* Prove certain triangles are similar by using AA~, SSS~, and SAS~
* Use triangle similarity to solve problems.
Similar Triangles Notes
1) Congruent and similar triangles can be used to simplify design and calculations. Congruent triangles have equal sides and angles, while similar triangles have the same shape but not necessarily the same size.
2) Corresponding sides and angles of similar triangles have the same ratios. Ratios can be used to determine unknown side lengths.
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This document provides information about a mathematics module on similarity for grade 9 learners. It was collaboratively developed by educators from various educational institutions in the Philippines. The module aims to teach learners about proportions, similarity of polygons, conditions for similarity of triangles using various theorems, applying similarity to solve real-world problems involving proportions and similarity. It includes a module map, pre-assessment questions to gauge learners' prior knowledge, and covers topics like proportions, similarity of polygons and triangles, and applying similarity concepts to solve problems.
Triangles
This document discusses various theorems and properties related to triangles. It explains the Basic Proportionality Theorem, also known as Thales' Theorem, which states that if a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally. It also covers similarity criteria like AAA, SSA, and SSS. The Area Theorem demonstrates that the ratio of areas of similar triangles equals the square of the ratio of corresponding sides. Additionally, it proves Pythagoras' Theorem, which relates the sides of a right triangle, and its converse. In summary, the document outlines key triangle theorems regarding proportional division, similarity, areas, and the Pythagorean relationship between sides.
Triangles
Triangles are three-sided polygons that have three angles and three sides. There are three main types of triangles based on side lengths: equilateral (all sides equal), isosceles (two sides equal), and scalene (no sides equal). The interior angles of any triangle always sum to 180 degrees. Important triangle properties include the exterior angle theorem, Pythagorean theorem, and congruency criteria like SSS, SAS, ASA. Common secondary parts are the median, altitude, angle bisector, and perpendicular bisector. The area of triangles can be found using Heron's formula or other formulas based on side lengths and types of triangles.
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Two methods for proving triangles are similar are presented: SSS similarity, where all corresponding sides between two triangles are proportional, proving the triangles are similar; and SAS similarity, where two corresponding sides of two triangles are proportional and the included angle is congruent, also proving the triangles are similar.
8.2 terms
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6.2 terms
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6.4 terms
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The document discusses key concepts relating to arcs and chords in circles. It defines a central angle as an angle whose vertex is the center of the circle, with the measure of its corresponding arc being equal to the angle's degrees. Arcs are classified as minor, major, or semicircles based on their angle measures. Adjacent arcs can be added together to find the measure of a larger arc. There are several theorems stated about relationships between arcs, chords, diameters, and angles, such as congruent arcs corresponding to congruent chords, diameters bisecting chords and angles if perpendicular to a chord, and chords being congruent if equidistant from the center.
9.6 terms
This document provides instructions for solving right triangles using trigonometric functions. It explains that solving a right triangle involves finding the measures of the three angles and lengths of the three sides. It notes that a calculator's inverse trigonometric functions allow working backwards from a trig ratio to the corresponding angle measure. Finally, it offers guidance for determining which inverse function to use based on the given information, such as using tangent when the hypotenuse is unknown or cosine when the opposite side is missing.
6.5 terms
This document defines and provides key details about trapezoids, isosceles trapezoids, the midsegment of a trapezoid, and kites. It states that a trapezoid has one pair of parallel sides called bases, and the other sides are called legs. An isosceles trapezoid has congruent legs and congruent base angles. The midsegment of a trapezoid connects the midpoints of the legs and is parallel to the bases. A kite has two pairs of consecutive congruent sides but opposite sides are not congruent, and its diagonals are perpendicular with one pair of opposite angles being congruent.
11.5 terms
The document discusses calculating the areas of circles and sectors. It states that the area of a circle is calculated as A = πr2, and the area of a sector is calculated as (Arc Measure/360)(area of the full circle), which finds the percentage of the total circular area that the sector represents by dividing the arc measure in degrees by 360.
11.1 terms
The document discusses angle measures in polygons. It states that the sum of interior angles in any polygon is (n-2)180, where n is the number of sides. It also explains that the sum of exterior angles is always 360, and each exterior angle is calculated as 360/n, where n is again the number of sides.
10.3 terms
This document discusses inscribed angles and their relationship to circles. It states that an inscribed angle is an angle whose vertex is on a circle and whose sides contain chords of the circle. It also notes that an inscribed angle measures half the degrees of its intercepted arc and that two inscribed angles intercepting the same arc together measure the full arc. Additionally, it mentions that if a right triangle is inscribed in a circle, its hypotenuse is the diameter of the circle and that a quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary.
11.4 terms
This document discusses formulas for circumference and arc length. It states that circumference C equals 2πr or πd, where r is the radius and d is the diameter. It also explains that arc length is calculated as a percentage of the circumference by taking the arc measure in degrees and dividing by 360, then multiplying that fraction by the circumference.
9.5 terms
This document provides information about trigonometric ratios and using a calculator to evaluate trigonometric functions. It defines the sine, cosine, and tangent ratios using the SOHCAHTOA mnemonic. It also explains that calculators may require typing the trig function before or after the number, and some may not require hitting the equals sign, so it is important to know how to use the specific calculator.
10.6 terms
The standard equation for a circle is (x-h)2 + (y-k)2 = r2, where (h,k) represents the center of the circle and r represents the radius. When solving for the center or radius, the terms in the parentheses change signs - the x and y terms change to the opposite of the center coordinates h and k, and the radius r is the square root of the number after the equals sign.
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