TEKS (6)(B) Prove two triangles are congruent by applying the Side-Angle-Side, Angle-Side-Angle, Side-Side-Side, Angle-Angle-Side, and Hypotenuse-Leg congruence conditions.

TEKS (1)(G) Display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.

TEKS FOCUSJustify – explain with logical reasoning. You can justify a mathematical argument.

Argument – a set of statements put forth to show the truth or falsehood of a mathematical claim

VOCABULARY

If you know two triangles are congruent, then you know that every pair of their corresponding sides and angles is also congruent.

ESSENTIAL UNDERSTANDING

Problem 1

Proving Parts of Triangles Congruent

Given: ∠KBC ≅ ∠ACB, ∠K ≅ ∠A

Prove: KB ≅ AC

K

B

C

A

Proof

! KBC ! ! ACB" K ! " A

Given

Given

Reflexive Property of !

AAS Theorem

BC ! BC" KBC ! " ACB

Corresp. parts of ! are !.

KB ! AC

4-4 Using Corresponding Parts of Congruent Triangles

In the diagram, which congruent pair is not marked?The third angles of both triangles are congruent. But there is no AAA congruence rule. So, find a congruent pair of sides.

164 Lesson 4-4 Using Corresponding Parts of Congruent Triangles

S R

T

L

Proving Triangle Parts Congruent to Measure Distance

Measurement Thales, a Greek philosopher, is said to have developed a method to measure the distance to a ship at sea. He made a compass by nailing two sticks together. Standing on top of a tower, he would hold one stick vertical and tilt the other until he could see the ship S along the line of the tilted stick. With this compass setting, he would find a landmark L on the shore along the line of the tilted stick. How far would the ship be from the base of the tower?

Given: ∠TRS and ∠TRL are right angles, ∠RTS ≅ ∠RTL

Prove: RS ≅ RL

Statements Reasons1) ∠RTS ≅ ∠RTL 1) Given

2) TR ≅ TR 2) Reflexive Property of Congruence

3) ∠TRS and ∠TRL are right angles. 3) Given

4) ∠TRS ≅ ∠TRL 4) All right angles are congruent.

5) △TRS ≅ △TRL 5) ASA Postulate

6) RS ≅ RL 6) Corresponding parts of ≅ △s are ≅.

The distance between the ship and the base of the tower would be the same as the distance between the base of the tower and the landmark.

Proof

TEKS Process Standard (1)(G)

STEM

Problem 2

Which congruency rule can you use?You have information about two pairs of angles. Guess-and-check AAS and ASA.

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PRACTICE and APPLICATION EXERCISES

ONLINE

HO M E W O R

K

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1. Explain Mathematical Ideas (1)(G) Tell why the two triangles are congruent. Give the congruence statement. Then list all the other corresponding parts of the triangles that are congruent.

2. Given: ∠ABD ≅ ∠CBD, ∠BDA ≅ ∠BDC

Prove: AB ≅ CB

3. Given: OM ≅ ER, ME ≅ RO

Prove: ∠M ≅ ∠R

4. Justify Mathematical Arguments (1)(G) A balalaika is a stringed instrument. Prove that the bases of the balalaikas are congruent.

Given: RA ≅ NY , ∠KRA ≅ ∠JNY , ∠KAR ≅ ∠JYN

Prove: KA ≅ JY

Proof: It is given that two angles and the included side of one triangle are congruent to two angles and the included side of the other. So, a. ? ≅ △JNY by b. ? . KA ≅ JY because c. ? .

5. Given: ∠SPT ≅ ∠OPT , 6. Given: YT ≅ YP, ∠C ≅ ∠R, SP ≅ OP ∠T ≅ ∠P

Prove: ∠S ≅ ∠O Prove: CT ≅ RP

Analyze Mathematical Relationships (1)(F) Copy and mark the figure to show the given information. Explain how you would prove jP @ jQ.

7. Given: PK ≅ QK , KL bisects ∠PKQ

8. Given: KL is the perpendicular bisector of PQ.

9. Given: KL # PQ, KL bisects ∠PKQ

L

JK O

N

M

Proof

A

B

D

C

Proof

M

O

R

E

K

R

J YA

N

Proof Proof

Y

P

RC

T

S OT

P

P

K

L Q

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166 Lesson 4-4 Using Corresponding Parts of Congruent Triangles

10. Justify Mathematical Arguments (1)(G) The construction of a line perpendicular to line / through point P on line / is shown. Explain why you can conclude that

<CP

> is perpendicular to /.

11. The construction of ∠B congruent to given ∠A is shown. AD ≅ BF because they are congruent radii. DC ≅ FE because both arcs have the same compass settings. Explain why you can conclude that ∠A ≅ ∠B.

12. Given: BE # AC, DF # AC, BE ≅ DF , AF ≅ CE

Prove: AB ≅ CD

13. Given: JK } QP, JK ≅ PQ

Prove: KQ bisects JP

14. Apply Mathematics (1)(A) Rangoli is a colorful design pattern drawn outside houses in India, especially during festivals. Vina plans to use the pattern at the right as the base of her design. In this pattern, RU , SV , and QT bisect each other at O. RS = 6, RU = 12, RU ≅ SV , ST } RU , and RS } QT . What is the perimeter of the hexagon?

In the diagram at the right, BA @ KA and BE @ KE.

15. Prove: S is the midpoint of BK .

16. Prove: BK # AE

A B

C

P!

A BD

C

F

E

Proof

A

B

E F

D

C

Proof

K

JQ

P

M

AS

K

E

B

Proof

Proof

TEXAS Test Practice

For Exercises 17 and 18, use the diagram at the right. TM # BDand TM bisects jBTD and jATC.

17. Suppose BD = 17 and AM = 5. What is the length of CD?

18. Suppose m∠ATC = 64, and m∠BTA = 16. What is m∠B?

19. Two parallel lines q and s are cut by a transversal t. ∠1 and ∠2 are a pair of alternate interior angles and m∠2 = 38. ∠1 and ∠3 are vertical angles. What is m∠3?

20. △ABC has vertices A(1, 9), B(4, 3), and C(x, 6). For what value of x is △ABC a right triangle with right ∠B?

B A M

T

C D

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