Triangle Congruence and Proofs

Textbook Pages for Unit 4:

Regular- Chapter 4-

Sec. 4.1 Pgs.230-236 and Sec 4.2 pgs.238-245

Sec. 4.3 pgs. 246-255 and Sec 4.4pgs. 260-264

Sec 4.6 pgs. 274-280

Honors- Chapter 4-

Sec. 4.1 Pgs.218-224 and Sec 4.2 pgs. 226-233

Sec. 4.3 pgs. 234-242 and Sec 4.4 pgs.244-248

Sec 4.6 pgs. 258-264

Unit 4- Triangle CongruenceSaturday, October 31, 2015 11:25 AM

Unit 4 Page 1

Congruent Figureshave the same shape and size

Congruent Polygons-Have congruent corresponding parts They have matching sides and anglesWhen naming congruent polygons you must list corresponding vertices in the same order.

LMWZ ZYMN NOYX LOWX LW ZM XO NY

Third Angle Theorem-

If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent.

ABDE BCFE CAFD

AD CF

Example #1: Find the values of the variables

Step #1: Define the triangles; The angles in a triangle =180. Thus, since the triangles are congruent then:

5x+(3x+2)+74=180

8x+76=180

8x=104

x=13

Lesson 1- Congruent Figures Wednesday, November 4, 2015 12:45 PM

Unit 4 Page 2

Example #2: Find the values of the variables

What are the values of the x and y if ∆MAS∆NER?

Step #1: Define the triangles; Thus, since the triangles are congruent then:

16x=112

x=7

3x+5y=41

3(7)+5y=41

21+5y=41

5y=20

y=4

Unit 4 Page 3

Side-Side-Side (SSS)PostulatesIf…

ABDE BCEF ACDF Then… ∆ABC∆DEF

If the three sides of one triangle are congruent to the three sides of another triangle, then the two triangles are congruent.

•

Side-Angle-Side (SAS) PostulateIf…

MPSR PS and PNSQ Then…∆MPN∆QSR

If two sides and one angle of one triangle are congruent to the two sides and one angle of another triangle, then the two triangles are congruent.

•

Example #1: Recognizing Congruent triangles

Would you use SSS or SAS to prove the triangles congruent? If there is not enough information to prove the triangles congruent by SSS or SAS, write not enough information. Explain your answer.

Lesson 2-Triangle Congruence by SSS and SASWednesday, November 4, 2015 12:46 PM

Unit 4 Page 4

Angle-Side-Angle (ASA) PostulateIf…BE CF and BCFE Then… ∆ABC∆DEF

If two angles and the included side of one triangle are congruent to the two angles and included side of another triangle, then the two triangles are congruent.

Angle-Angle-Side (AAS) TheoremIf…BE CF and ABDE Then… ∆BCA∆DFE

If two angles and a non-included side of one triangle are congruent to the two angles and corresponding non-included side of another triangle, then the two triangles are congruent.

Example #1 Finding congruent triangles

Name two triangles that are congruent by ASA

Lesson 3-Triangle Congruence by ASA and AASWednesday, November 4, 2015 12:46 PM

Unit 4 Page 5

CPCTC is a short hand acronym for the phrase 'corresponding parts of congruent triangles are congruent.

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It means that once we know that two triangles are congruent, we know that all corresponding sides and angles are congruent!

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Which congruent triangle method do you think is used in this example?

Which congruent triangle method do you think

Lesson 4-Corresponding Parts of Congruent Triangles Wednesday, November 4, 2015 12:47 PM

Unit 4 Page 6

Which congruent triangle method do you think is used in this example?

Which congruent triangle method do you think is used in this example?

Which congruent triangle method do you think is used in this example?

Unit 4 Page 7

Hypotenuse Leg Theorem

If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of a second right triangle, then the triangles are congruent.

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AB = XZ, CB = XY, ACB = ZYX = 90° •

Example#1: HL theoremCalculate the Y value for the following triangle shapes.Step#1: Write down all given information to set up problem

Side AB/DE= AC/DF

ΔDEF, DE side length is 4.4 cm

ΔABC, AC side length is 7.6 cm

Δ DEF, DF side length is 3.8 cm

Y = 8.8

The Y value of similar triangle is 8.8

Lesson 5-HL Theorem Wednesday, November 4, 2015 12:50 PM

Unit 4 Page 8

Triangle Congruence Theorems

Helpful Video about theorems

How to Write Proofs

1. Mark given information•

2. Look at "parts" of the triangles that can be proven congruent. •

3.Look to see what else you might know about the diagram. •

4. Know your definitions!•

5. Stay open-minded.•

6. Look to see if your triangles "share" parts.•

Remember that proving triangles congruent is like solving a puzzle. Look carefully at the "puzzle" and use all of your geometrical strategies to arrive at an answerwwq

Developing Proof:

Use the information given in the diagram. Give a reason that each statement is true.

a. b. c.

d.

e.

Lesson 6-Proofs Saturday, November 14, 2015 12:39 PM

Unit 4 Page 9