LESSON

Triangles are congruent if they have the same size and shape. Their corresponding parts, the angles and sides that are in the same positions, are congruent.

Corresponding Parts

Congruent Angles Congruent Sides

�A � �J�B � �K�C � �L

_

AB � _

JK _

BC � _

KL _

CA � _

LJ

congruence statement such as �ABC � �JKL.

1. �Z � 2. _

YZ �

_

3. �P � 4. �X �

5. _

NQ � 6. _

PN �

4-3

Congruent TrianglesName: _______________________Geometry

Page 1 of 6

Equation to find x: ________________________ x = _______

Equation to find y: ________________________ y = _______

Equation to find z: ________________________ z = _______

K

To identify corresponding parts of congruent triangles, look at the order of the vertices in the

Given: �XYZ � �NPQ. Identify the congruent corresponding parts.

Given: �EFG � �RST. Find each value below.

LESSON

You can prove triangles congruent by using the definition of congruence.

Given: �D and �B are right angles. �DCE � �BCA C is the midpoint of

_ DB .

_ ED �

_ AB , _

EC � _

AC

Prove: �EDC � �ABCProof:

Statements Reasons

1. �D and �B are rt. �. 1. Given

2. �D � �B 2. Rt. � � Thm.

3. �DCE � �BCA 3. Given

4. �E � �A

5. C is the midpoint of _

DB . 5. Given

6. _

DC � _

BC 6. Def. of mdpt.

7. _

ED � _

AB , _

EC � _

AC 7. Given

8. �EDC � �ABC 8. Def. of � �s

Given: �Q � �R

P is the midpoint of _

QR .

_

NQ � _

SR , _

NP � _

SP

Prove: �NPQ � �SPR

Proof:

Statements Reasons

1. �Q � �R 1. Given

2. �NPQ � �SPR 2. a. Vert. � Thm.

3. �N � �S 3. b. Third � Thm._ Given

_ _

6. _

NQ � _

SR , _

NP � _

SP 6. e. Given

7. �NPQ � �SPR 7. f. Def. of � �s

4-3

Congruent Triangles Page 2 of 6

11. Complete the proof.

4. P is the midpoint of QR . 4. c.

5. d5. QP RP___ ___

4. Third � Thm.

Third Angles TheoremIf two angles of one triangle arecongruent to two angles in a second triangle, then the thirdangles are also congruent.

LESSON

4-3 Practice A

1. angles and sides are in

the same position in polygons with an equal number of sides.

2. Two polygons are polygons if and only if their

corresponding angles and sides are congruent.

3. Name the three pairs of corresponding sides.

_ _ _ _ _ _

4. Name the three pairs of corresponding angles.

51°°

39°

8.1

114.3

7. Etienne flies a kite. When the kite is flying well, the tail sticks out straight so the indicated angles at V are congruent. Use the phrases from the word bank to complete this two-column proof.

Given: �S and �U are right angles. �SVW � �UVW,

_ SV �

_ UV , _

ST � _

UT

Prove: �STV � �UTV

�S � �U Third � Thm.Given�SVT � �UVT

Proof:

Statements Reasons

1. _

SV � _

UV , _

ST � _

UT 1. Given

2. _

TV � _

TV 2. Reflex. Prop. of �

3. �S and �U are right angles. 3. a. Given

4. b. �S � �U 4. Rt. � � Thm.

5. �SVW � �UVW 5. Given

6. �SVW and �SVT are supplementary, �UVW are �UVT are supplementary.

6. Lin. Pair Thm.

7. c. �SVT � �UVT 7. � Suppls. Thm.

8. �STV � �UTV 8. d. Third � Thm.9. �STV � �UTV 9. Def. of � �s

Congruent Triangles Page 3 of 6

5. x = _______ 6. x = _______

Given: RCQ BXO

7. CQ = _______ 8. m R = m _______ 9. XB = _______ 10. m C = m _______

11. QRC __________ 12. BOX ___________

4. �E � 5. PQ � 6. TS �

LESSON

Practice B4-3In baseball, home plate is a pentagon. Pentagon ABCDE is a diagram of a regulation home plate. The baseball rules are very specific about the exact dimensions of this pentagon so that every home plate is congruent to every other home plate. If pentagon PQRST is another home plate, identify each congruent corresponding part.

1. �S � 2. �B � 3. _

EA �

_ _

Given: �DEF � �LMN. Find each value.

7. m�L �

8. EF �

9. Write a two-column proof.

Given: �U � �UWV � �ZXY � �Z, _

UV � _

WV � _

XY � _

ZY , _

UX � _

WZ

Prove: �UVW � �XYZ

Proof:

11. Given: �CDE � �HIJ, m�D � (5y � 1)�, and m�I � (6y � 25)�.

17 in.

8.5 in. 8.5 in.

12.02 in. 12.02 in.

53

25.41.5 1.3

2 3( 15)°

120°

5 °

001-082_Go07an_CRB_c04.indd 20001-082_Go07an_CRB_c04.indd 20 4/2/07 4:25:54 PM4/2/07 4:25:54 PM

Equation to find x: _______________________ x = _______ DE = _______ IJ = _______

Equation to find y: _______________________ y = _______ m D = _______ m I = _______

10. Given: �CDE � �HIJ, DE � 9x, and IJ � 7x � 3.

Congruent Triangles Page 4 of 6

complementary. What is the measure

7. The area of a square is 16 square units.

8. The midpoint of a segment is (�8, 5). If

between (7, �4) and (�3, �1)?

10. Which coordinate pair represents the image of (9, 10) reflected over the

11. What is the next figure in the pattern?

A C

B D

F If Mary can swim, then she can swim

G If it is raining outside, then the temperature is above freezing.

H If Greg has two children, then he has one son and one daughter.

J If Carolyn can stand up, then she can walk.

6. Two vertical angles are also

Page 5 of 64-3

freestyle.

x-axis?

9. In SRF, what is the distance

endpoint?one endpoint is (0, 1), what is the other

What is the perimeter?

of each of the two vertical angles?

12. For which statement is the converse false?

A If a triangle has no congruent angles, then it is not an isosceles triangle.

B If a triangle is an isosceles triangle, then it has at least two congruent angles.

C If a triangle does not have at least two congruent angles, then it is an isosceles triangle.

D If a triangle is not an isosceles triangle, then it does not have at least two congruent angles.

F penguin H duck

G robin J rabbit

F If a number is divisible by 6, then it is

G If an angle is formed by two opposite

A 10x � 90 C x � 13 � �1

B � x __ 3 � 4 D x 2 � 8

18. If _ _ _ _

_ _

F Reflexive Property of Congruence

G Transitive Property of Congruence

H Symmetric Property of Congruence

J Segment Addition Postulate

19. Which completes the proof?

Given: m�TKV � m�UKW

Prove: m�TKU � m�VKW

Proof:

Statements Reasons

1. m�TKV � m�UKW 1. Given

2. ? 2. Reflex. Prop. of �

3. m�TKV � m�TKU � m�UKV

3. � Add. Post.

4. m�UKW � m�VKW � m�VKU

4. � Add. Post.

5. m�TKU � m�UKV � m�VKW � m�VKU

5. Subst. Prop of �

6. m�TKU � m�VKW 6. Subtr. Prop. of �

A Substitution Property of �

B m�VKW � m�TKU

C m�TKW � m�WKT

D m�UKV � m�VKU

Page 6 of 64-3

If a triangle has at least two congruent angles, then it is an isosceles triangle.

divisible by 3.

rays, then the angle is a straight angle.

H If the month is August in Kansas, then the season is summer.

J If you are in California, then the nearest ocean is to the west.

_____________________________________________________________

_____________________________________________________________

If an animal has wings, then it can fly.

If a musician practices, then she gets better.

AB � EF ? AB � CD and CD � EF , why is

16. Which conditional statement can be used to write a true biconditional?

15. Write a logical conclusion by the Law of Syllogism.

14. Which is a counterexample of the statement?

17. Which equation can be solved by using the Subtraction Property of Equality?

13. What is the contrapositive of the statement?