Proving Triangles Congruent

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!

Built – In Information in Triangles  

Identify the ‘built-in’ part

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

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

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. ASA HL

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

Let’s Practice B  D AC  FE A  F Indicate the additional information needed to enable us to apply the specified congruence postulate. For ASA: B  D For SAS: AC  FE A  F For 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 4 G I H J K 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. B A C E D Ex 5 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 K L M 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 L K V U Not possible

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

Problem #5 HL Statements Reasons Given Given Reflexive Property Given ABC, ADC right s, Prove: Statements Reasons Given 1. ABC, ADC right s Given Reflexive Property HL Postulate

Congruence Proofs 1. Mark the Given. 2. Mark … Reflexive Sides or Angles / Vertical Angles Also: mark info implied by given info. 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?

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

Example Problem

Step 1: Mark the Given … and what it implies

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

Step 3: Choose a Method SSS SAS ASA AAS HL

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

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

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

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?

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.

Corresponding Parts of Congruent Triangles For example, can you prove that sides AD and BC are congruent in the figure at right? The sides will be congruent if triangle ADM is congruent to triangle BCM. Angles A and B are congruent because they are marked. Sides MA and MB are congruent because they are marked. Angles 1 and 2 are congruent because they are vertical angles. So triangle ADM is congruent to triangle BCM by ASA. This means sides AD and BC are congruent by CPCTC.

Corresponding Parts of Congruent Triangles A two column proof that sides AD and BC are congruent in the figure at right is shown below: Statement Reason MA @ MB Given ÐA @ ÐB Ð1 @ Ð2 Vertical angles DADM @ DBCM ASA AD @ BC CPCTC

Corresponding Parts of Congruent Triangles A two column proof that sides AD and BC are congruent in the figure at right is shown below: Statement Reason MA @ MB Given ÐA @ ÐB Ð1 @ Ð2 Vertical angles DADM @ DBCM ASA AD @ BC CPCTC

Corresponding Parts of Congruent Triangles Sometimes it is necessary to add an auxiliary line in order to complete a proof For example, to prove ÐR @ ÐO in this picture Statement Reason FR @ FO Given RU @ OU UF @ UF reflexive prop. DFRU @ DFOU SSS ÐR @ ÐO CPCTC

Corresponding Parts of Congruent Triangles Sometimes it is necessary to add an auxiliary line in order to complete a proof For example, to prove ÐR @ ÐO in this picture Statement Reason FR @ FO Given RU @ OU UF @ UF Same segment DFRU @ DFOU SSS ÐR @ ÐO CPCTC