Introduction to Triangle Congruence
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The document introduces the concept of triangle congruence. Two triangles are congruent if their corresponding sides and angles are equal in measure. The document provides examples of naming corresponding parts of congruent triangles and questions to test understanding of triangle congruence.
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The document defines and discusses triangles. It states that a triangle has three sides, three angles, and three vertices. It then defines two triangles as congruent if all corresponding sides and angles are equal. The criteria for congruence are then listed as SAS, ASA, AAS, SSS, and RHS. Properties of triangles are also discussed, such as isosceles triangles having two equal sides and equal angles opposite the equal sides, and the theorem that the angle opposite the longer side of a triangle is larger.
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This document provides information about proving triangles congruent using various postulates and theorems:
1. It describes the SSS, SAS, ASA, and AAS postulates that can be used to prove any triangles congruent, as well as the HL postulate that applies only to right triangles.
2. Examples are given to demonstrate applying each of the postulates to determine if two triangles are congruent and to find missing corresponding parts.
3. Additional theorems like CPCTC (corresponding parts of congruent triangles are congruent) and properties of vertical angles, isosceles triangles, and midpoints are also explained.
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1) The document discusses the concept of congruence in geometry, specifically how congruence relates to triangles. It states that two triangles are congruent if specifying two sides and the angle between them (SAS), two angles and the side between them (ASA), or two angles and a corresponding adjacent side (AAS).
2) It also discusses properties of congruence, including that corresponding parts of congruent triangles (sides and angles) are congruent to each other.
3) The document provides examples justifying statements about angles and sides of triangles using properties of equality and congruence.
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- AAS (angle-angle-side) theorem
- Hypotenuse-Leg theorem
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The document discusses different methods for proving triangles are congruent: SSS, SAS, ASA, AAS, and HL. It states that if the specified corresponding parts of two triangles are congruent using one of these methods, then the triangles are congruent. The document provides examples of congruence proofs using these methods and their reasoning.
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This document provides information on proving triangle congruence using various postulates and properties. It discusses the six corresponding parts used to determine if two triangles are congruent, as well as five postulates for proving congruence: SSS, SAS, ASA, SAA/AAS, and the third angle theorem. Examples are given of applying each postulate, along with exercises to identify the postulate used and complete triangle congruence proofs. Key details include identifying the six corresponding parts of triangles as sides and angles, discussing the five postulates for proving congruence based on sides and angles, and providing examples of setting up triangle congruence proofs.
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- The ASA (Angle-Side-Angle) rule, which states two triangles are congruent if two angles and the included side of one triangle are equal to two angles and the included side of the other triangle. An example proof of this rule is provided.
- The SSS (Side-Side-Side) rule, which states two triangles are congruent if three sides of one triangle are equal to the corresponding three sides of the other triangle. An example proof is also provided.
- The Hypotenuse-Leg rule, which states two right triangles are congruent if the hypotenuse and one side of one
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This document defines and describes different types of triangles based on side lengths and angle measures. It also discusses properties of triangles related to similarity, including the AAA, SSS, SAS, and RHS similarity criteria. Properties of special right triangles and the Pythagorean theorem are also covered. Types of triangles described include scalene, isosceles, equilateral, acute, obtuse, and right triangles.
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This document discusses different types of triangle parts including:
- Angle bisectors, which divide any angle of a triangle into two equal angles
- Altitudes, which are perpendicular segments from a vertex to the opposite side
- Medians, which connect vertices to midpoints of the opposite sides
It also covers triangle congruence, noting triangles are congruent if corresponding sides and angles are equal. Five postulates for triangle congruence are presented relating equal sides, angles, and combinations of these. The midsegment theorem is introduced, stating a midsegment is parallel to the third side and half as long. Examples demonstrate using midsegments and finding unknown sides.
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- Properties of triangle congruence include reflexive, symmetric, and transitive properties. The lesson teaches students to identify corresponding angles and sides to determine if two triangles are congruent.
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This document provides definitions and examples of triangle congruence concepts including corresponding angles and sides, included angles, and included sides. It defines corresponding angles and sides as angles or sides in different triangles that are in the same relative position. Included angles are defined as the angle between two sides of a triangle, and included sides are defined as the side between two angles of a triangle. Examples are given to illustrate these concepts and how to identify corresponding angles and sides, as well as included angles and sides in different triangles.
Parallelogram
This document defines and provides properties of parallelograms, rectangles, squares, and rhombuses. It states that a parallelogram is a quadrilateral with both pairs of opposite sides parallel. A rectangle is a parallelogram with right angles, and a square is a rectangle with all sides congruent. A rhombus is a parallelogram with all sides equal. It then provides 5 theorems about parallelograms, including that diagonals bisect each other and opposite angles are congruent. An example problem applies these properties to find missing angle and side measures.
Corresponding parts of congruent figures.
This document section discusses identifying and naming congruent figures. It defines congruent figures as two figures that have the same size and shape. Congruent polygons have matching sides and angles. Corresponding parts of congruent figures are equal. Examples demonstrate writing congruence statements and identifying corresponding congruent parts of triangles.
Triangles
1. Two triangles are congruent if their corresponding sides and angles are equal. This can be determined through Side-Angle-Side (SAS), Angle-Side-Angle (ASA), Angle-Angle-Side (AAS), Side-Side-Side (SSS), or Right-Angle-Hypotenuse-Side (RHS) criteria.
2. Triangles have properties related to equal sides and angles, including that angles opposite equal sides are equal, and sides opposite equal angles are equal.
3. Inequalities in a triangle follow patterns such as the angle opposite the longer side being larger, and the side opposite the larger angle being longer. The sum of any two sides is also greater
ppt on Triangles Class 9
1. The document defines triangles and their properties including three sides, three angles, and three vertices.
2. It explains five criteria for determining if two triangles are congruent: side-angle-side (SAS), angle-side-angle (ASA), angle-angle-side (AAS), side-side-side (SSS), and right-angle-hypotenuse-side (RHS).
3. Some properties of triangles discussed are: angles opposite equal sides are equal, sides opposite equal angles are equal, and the sum of any two sides is greater than the third side.
LET’S DO MATH.pptx
This document discusses geometry concepts like congruence, triangles, and their properties. It contains examples of:
- Using tangram pieces to complete a puzzle based on congruent shapes
- How congruence is used in mass production by ensuring identical copies are made
- Properties of triangle congruence like the reflexive, symmetric, and transitive properties
- Identifying sides, angles, and vertices of triangles in examples
- Defining terms and listing properties of congruence
The document provides instruction and activities to help students learn about geometric concepts like congruence through examples, definitions, and identifying parts of shapes.
2.5.2 Isosceles and Equilateral Triangles
Identify isosceles and equilateral triangles by side length and angle measure
Use the Isosceles Triangle Theorem to solve problems
Use the Equilateral Triangle Corollary to solve problems
Triangles and Types of triangles&Congruent Triangles (Congruency Rule)
This document defines and describes different types of triangles:
- Equilateral triangles have three equal sides and three equal angles.
- Isosceles triangles have at least two equal sides.
- Scalene triangles have no equal sides.
- Right triangles have one 90 degree angle.
- Acute triangles have all angles less than 90 degrees.
- Obtuse triangles have one angle greater than 90 degrees.
It also describes three theorems used to prove triangle congruence: SSS (three equal sides), SAS (two equal sides and the included angle), and ASA (two equal angles and one included side).
7.Congruence of Triangles.pdf
This document defines and provides examples of congruent shapes in geometry. It states that two objects are congruent if they have the same size and shape. Specifically:
- Two line segments are congruent if they have the same length.
- Two angles are congruent if they have the same measure.
- Two triangles are congruent if they are identical copies that overlap completely when placed on top of each other.
- Having equal corresponding angles alone is not enough to determine if two triangles are congruent, as one could be an enlarged copy of the other.
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Introduction to Triangle Congruence
- 1. REVIEW: ● What is a triangle? ● What are the parts of a triangle?
- 3. What is the idea of Congruence?
- 4. Two geometric figures with exactly the same size and shape.
- 5. ΔABC is congruent to ΔXYZ The corresponding parts of these triangles are congruent.
- 6. ΔABC is congruent to ΔXYZ The corresponding parts of these triangles are congruent. ∠A ≅ ∠X
- 7. ΔABC is congruent to ΔXYZ The corresponding parts of these triangles are congruent. ∠B ≅ ∠Y
- 8. ΔABC is congruent to ΔXYZ The corresponding parts of these triangles are congruent. ∠C ≅ ∠Z
- 9. ΔABC is congruent to ΔXYZ The corresponding parts of these triangles are congruent. AB ≅ XY
- 10. ΔABC is congruent to ΔXYZ The corresponding parts of these triangles are congruent. BC ≅ YZ
- 11. ΔABC is congruent to ΔXYZ The corresponding parts of these triangles are congruent. AC ≅ XZ
- 13. ΔBAD is congruent to ΔTHE Name all the corresponding parts.
- 14. ΔQRS is congruent to ΔBRX Name all the corresponding parts.
- 15. ΔEFG is congruent to ΔHFG Name all the corresponding parts.
- 16. TRIANGLE CONGRUENCE Two triangles are congruent if their vertices can be paired so that corresponding sides are congruent and corresponding angles are congruent. Two triangles are congruent if they have the same size and shape. Two triangles are congruent when all corresponding sides and interior angles are congruent.
- 17. TRIANGLE CONGRUENCE ΔABC ≅ ΔDEF read as “triangle ABC is congruent to triangle DEF”. ≅ - symbol for congruency Δ - symbol for triangle
- 18. 1. Complete the statement: ΔARM ≅ Δ______. 2. What is the angle congruent to ∠E in ΔARM? 3. What is the angle congruent to ∠L in ΔARM? 4. What is the angle congruent to ∠R in ΔLEG? 5. What is the name of the side congruent to RM in ΔLEG?