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Proving Triangles Congruent
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Triangle Congruency Short-Cuts
If you can prove one of the following short cuts, you have two congruent triangles SSS (side-side-side) SAS (side-angle-side) ASA (angle-side-angle) AAS (angle-angle-side) HL (hypotenuse-leg) right triangles only!
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Congruent Triangles Proofs
1. Mark the Given and what it implies. 2. Mark … Reflexive Sides / Vertical Angles 3. Choose a Method. (SSS , SAS, ASA) 4. List the Parts … in the order of the method. 5. Fill in the Reasons … why you marked the parts. 6. Is there more?
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Built – In Information in Triangles
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Identify the ‘built-in’ part
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SAS SAS SSS Shared side Vertical angles Parallel lines -> AIA
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SOME REASONS For Indirect Information
Def of midpoint Def of a bisector Vert angles are congruent Def of perpendicular bisector Reflexive property (shared side) Parallel lines ….. alt int angles Property of Perpendicular Lines
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Given implies Congruent Parts
segments midpoint angles parallel segments segment bisector angles angle bisector angles perpendicular
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This is called a common side. It is a side for both triangles.
We’ll use the reflexive property.
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HL ( hypotenuse leg ) is used
only with right triangles, BUT, not all right triangles. ASA HL
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Statements Reasons Given Given Reflexive Property HL Postulate
Given ABC, ADC right s, Prove: Statements Reasons Given 1. ABC, ADC right s Given Reflexive Property HL Postulate
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Name That Postulate SAS SAS SSA SAS Vertical Angles Reflexive Property
(when possible) Vertical Angles Reflexive Property SAS SAS Vertical Angles Reflexive Property SSA SAS
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Determine if whether each pair of triangles is congruent by SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write not possible. Ex 4 G I H J K ΔGIH ΔJIK by AAS
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Determine if whether each pair of triangles is congruent by SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write not possible. B A C E D Ex 5 ΔABC ΔEDC by ASA
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Determine if whether each pair of triangles is congruent by SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write not possible. Ex 6 E A C B D ΔACB ΔECD by SAS
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Determine if whether each pair of triangles is congruent by SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write not possible. Ex 7 J K L M ΔJMK ΔLKM by SAS or ASA
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Determine if whether each pair of triangles is congruent by SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write not possible. J T Ex 8 L K V U Not possible
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Problem #4 AAS Statements Reasons Given Given AAS Postulate
Vertical Angles Thm Given AAS Postulate
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Example Problem
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Step 1: Mark the Given … and what it implies
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Step 2: Mark . . . Reflexive Sides Vertical Angles … if they exist.
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Step 3: Choose a Method SSS SAS ASA AAS HL
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Step 4: List the Parts S A … in the order of the Method STATEMENTS
REASONS S A … in the order of the Method
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Step 5: Fill in the Reasons
STATEMENTS REASONS S A S (Why did you mark those parts?)
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Step 6: Is there more? STATEMENTS REASONS S 1. 2. 3. 4. 5. A S
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Using CPCTC in Proofs According to the definition of congruence, if two triangles are congruent, their corresponding parts (sides and angles) are also congruent. This means that two sides or angles that are not marked as congruent can be proven to be congruent if they are part of two congruent triangles. This reasoning, when used to prove congruence, is abbreviated CPCTC, which stands for Corresponding Parts of Congruent Triangles are Congruent.
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