Proving Triangles Congruent Triangle Congruency ShortCuts If you

Proving Triangles Congruent

Triangle Congruency Short-Cuts If you can prove one of the following short cuts, you have two congruent triangles 1. 2. 3. 4. 5. SSS (side-side) SAS (side-angle-side) ASA (angle-side-angle) AAS (angle-side) HL (hypotenuse-leg) right triangles only!

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? 3

Built – In Information in Triangles •

Identify the ‘built-in’ part

Shared side SSS Parallel lines -> AIA Vertical angles SAS Shared side SAS

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

Given implies Congruent Parts midpoint parallel segment bisector segments angles segments angle bisector angles perpendicular angles 8

This is called a common side. It is a side for both triangles. We’ll use the reflexive property.

HL ( hypotenuse leg ) is used only with right triangles, BUT, not all right triangles. HL ASA

Given ABC, ADC right s, Prove: Statements 1. ABC, ADC right s Reasons Given Reflexive Property HL Postulate 12

Name That Postulate (when possible) Reflexive Property SAS Vertical Angles SAS Reflexive Property SSA

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 K I H J ΔGIH ΔJIK by AAS

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 5 B A C D E ΔABC ΔEDC by ASA

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

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 M K L ΔJMK ΔLKM by SAS or ASA

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 K L V Not possible U

Problem #4 Statements AAS Reasons Given Vertical Angles Thm Given AAS Postulate 41

Example Problem 42

Step 1: Mark the Given … and what it implies 43

Step 2: Mark. • Reflexive Sides. . • Vertical Angles … if they exist. 44

Step 3: Choose a Method SSS SAS ASA AAS HL 45

Step 4: List the Parts STATEMENTS REASONS S A S … in the order of the Method 46

Step 5: Fill in the Reasons STATEMENTS REASONS S A S (Why did you mark those parts? ) 47

Step 6: Is there more? STATEMENTS S 1. 2. A 3. S 4. 5. REASONS 1. 2. 3. 4. 5. 48

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. 54