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Proving Triangles Congruent

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1 Proving Triangles Congruent

2 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!

3 Built – In Information in Triangles

4 Identify the ‘built-in’ part

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

6 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

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

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

9 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

10 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:

11 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

12 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

13 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

14 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

15 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

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

17 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

18 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?

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

20 Example Problem

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

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

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

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

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

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

27 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?

28 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.

29 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.

30 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 MB Given ÐB Ð2 Vertical angles DBCM ASA BC CPCTC

31 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 MB Given ÐB Ð2 Vertical angles DBCM ASA BC CPCTC

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

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


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