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1

Chapter 4 – Congruent Triangles

4.2 and 4.9– Classifying Triangles and Isosceles, and Equilateral Triangles.

Match the letter of the figure to the correct vocabulary word in Exercises 1–4.

1. right triangle __________

2. obtuse triangle __________

3. acute triangle __________

4. equiangular triangle __________

Classify each triangle by its angle measures as acute, equiangular, right, or obtuse. (Note: Give two classifications for

Exercise 7.)

5. 6. 7.

For Exercises 8–10, fill in the blanks to complete each definition.

8. An isosceles triangle has ____________________ congruent sides.

9. An ____________________ triangle has three congruent sides.

10. A ____________________ triangle has no congruent sides.

Classify each triangle by its side lengths as equilateral, isosceles, or scalene. (Note: Give two classifications in

Exercise 13.)

11. 12. 13.

2

Isosceles Triangles

Remember…

Isosceles triangles are triangles with at least two congruent sides.

The two congruent sides are called legs.

The third side is the base.

The two angles at the base are called base angles.

Isosceles Triangle Theorem: If two sides of a triangle are congruent, then the angles opposite them are congruent.

Converse is true!

3

4

Find the value of x.

1. 2. 3.

4. 5.

x°

50° 5x

3x + 20

21

3x

60°

100°

x°

72°

x°

5

Corollary 4.3 Angle Relationships in Triangles

The interior is the set of all points inside the figure.

The exterior is the set of all points outside the figure.

An interior angle is formed by two sides of a triangle.

An exterior angle is formed by one side of the triangle

and extension of an adjacent side. It forms alinear pair

with an angle of the triangle.

Each exterior angle has two remote interior angles. A

remote interior angle is an interior angle that is not

adjacent to the exterior angle.

Exterior Angles: Find each angle measure.

37. mB ___________________ 38. mPRS

39.In LMN, the measure of an exterior angle at N measures 99.

1m

3L x

and 2

m3

M x . Find mL, mM, and mLNM. ____________________

40.mE and mG ___________________ 41. mT and mV ___________________

6

42. In ABC and DEF, mA mD and mB mE. Find mF if an exterior

angle at A measures 107, mB (5x 2) , and mC (5x 5) . _______________

43. The angle measures of a triangle are in the ratio 3 : 4 : 3. Find the angle measures of the triangle.

____________________

44. One of the acute angles in a right triangle measures 2x°. What is the measure of the other acute angle?

___________________

45. The measure of one of the acute angles in a right triangle is 63.7°. What is the measure of the other acute angle?

46. The measure of one of the acute angles in a right triangle is x°. What is the measure of the other acute angle?

47. Find mB 48. Find m<ACD

49. Find mK and mJ 50. Find m<P and m<T

7

Use the figure at the right for problems 1-3.

1. Find m3 if m5 = 130 and m4 = 70.

2. Find m1 if m5 = 142 and m4 = 65.

3. Find m2 if m3 = 125 and m4 = 23.

Use the figure at the right for problems 4-7.

4. m6 + m7 + m8 = _______.

5. If m6 = x, m7 = x – 20, and m11 = 80,

then x = _____.

6. If m8 = 4x, m7 = 30, and m9 = 6x -20,

then x = _____.

7. m9 + m10 + m11 = _______.

For 8 – 12, solve for x.

8.

9.

1 3

5 2 4

6

8

7 10

11

9

120

x°

(5x)°

x° 140°

35°

8

4.4 Congruent Triangles

Polygons are congruent if all

of their corresponding sides

and all of their corresponding

angles are congruent.

Consecutive vertices of a polygon–

the endpoints of a side

Ex. P and Q are consecutive

vertices

Opposite vertices of a polygon-

vertices that are not consecutive

Congruent riangles: Two ’s are if they can be matched up so that corresponding angles and sides of the ’s are .

Congruence Statement: A congruence statement matches up the parts in the same order.

RED FOX

List the corresponding ’s: corresponding sides:

R ___ RE ____

E ___ ED ____

D ___ RD ____

Examples:

1. The two ’s shown are .

a) ABO _____ b) A ____

c) AO _____ d) BO = ____

2. The pentagons shown are .

a) B corresponds to ____ b) BLACK _______

c) ______ = mE d) KB = ____ cm

e) If CA LA , name two right ’s in the figures.

3. Given BIG CAT, BI = 14, IG = 18, BG = 21, CT = 2x + 7. Find x.

D C

O

B A

B

L A

C

K

H

O

R

S

4 cm

E

9

The following ’s are , complete the congruence statement:

Parts of a Triangle in terms of their relative positions.

1. Name the opposite side to C.

2. Name the included side between A and B.

3. Name the opposite angle to BC .

4. Name the included angle between AB and AC .

4.5-4.7 Proving Triangles Congruent

Ways to Prove ’s :

SSS Postulate: (side-side-side) Three sides of one are to three sides of a second ,

Given: AS bisects PW ; AWPA

SAS Postulate: (side-angle-side) Two sides and the included angle of one are to two sides

and the included angle of another .

Given: PX bisects AXE; XEAX

ASA Postulate: (angle-side-angle) Two angles and the included side of one are to two angles

and the included side of another .

Given: MHAT

THMA

//

//

A

B

C

A

P W S

A

X

P

E

A

M

T

H

10

AAS Theorem: (angle-angle-side) Two angles and a non-included side of one are to two

angles and a non-included side of another .

Given: CAtsbiUZ sec

ZAUZCUUZ ;

HL Theorem: (hypotenuse-leg) The hypotenuse and leg of one right are to the hypotenuse

and leg of another right .

Given: FCAT

Isosceles FAC with legs ACFA,

CPCTC – Corresponding Parts of Congruent Triangles are Congruent

State which congruence method(s) can be used to prove the ∆s . If no method applies, write none. All markings must

correspond to your answer.

1. 2.

3. 4.

5. 6.

7. 8.

C

R U Z

A

A

F T C

A

B

D

C

E G

H

F

I

T

Q

S

R

10

A B

D C

10

12 12

R

P Q

S

S T

V

R

U

11

State which congruence method(s) can be used to prove the ∆s . If no method applies, write none. All markings must

correspond to your answer.

1. 2. 3.

4. 5. 6.

7. 8. 9.

Fill in the congruence statement and then name the postulate that proves the ∆s are . If the ∆s are not , write “not possible” in

second blank. (Leave first blank empty)*Markings must go along with your answer**

Some may have more than one postulate

1. 2.

∆ABC _____ by ________ ∆ABC ________ by __________

3. 4.

∆ABC ________ by __________ ∆ABC ________ by _________

D

E

F

A

B C

A

B

C

D

F E

A

B C

D

E

F

A

B C

D

12

5. 6.

∆ABC ________ by _________ ∆ABC _______ by __________

7. 8.

∆ABC ________ by ___________ ∆ABC ________ by ____________

9. 10.

∆ABC ________ by _________ ∆ABC _______ by ___________

11. 12.

∆ABC _________ by ___________ ∆ABC _________ by __________

13. 14.

∆ABC _________ by ___________ ∆ABC __________ by ___________

A

B

C

D

E A

B

C

P Q

R

60

70

50

60

C D

A B

30

60 A

B

C

D

A B

C D

A

B C

D

C

A

B

D

B

A

C

U

N

A

B

C

D

A

D

B

C

13

Proofs!

#1 Given: USUTSRUTSR ;//; Prove: UVST //

1. USUTSRUTSR ;//; 1. _____________________________

2. 1 4 2. __________________________________________

3. ∆RST ∆TUV 3. __________________________________________

4. 3 2 4. __________________________________________

5. UVST // 5. __________________________________________

#2 Given: D is the midpoint of CBCAAB ; Prove: CD bisects ACB.

1. D is the midpoint of CBCAAB ; 1. _________________________________________

2. DBAD 2. __________________________________________

3. CDCD 3. __________________________________________

4. ∆ACD ∆BCD 4. __________________________________________

5. 1 2 5. __________________________________________

6. CD bisects ACB. 6. __________________________________________

#3 Given: AR≅ AQ; RS ≅ QT Prove: AS ≅ AT

1. AR≅ AQ; RS ≅ QT 1. ________________________

2. <R <Q 2. __________________________________________

3. ARS AQT 3. __________________________________________

4. AS ≅ AT 4. __________________________________________

14

#1

Given: AB CB

AC BD

Prove: Δ ADB Δ CDB

1. AB CB 1. _________________________________________________

2. AC BD 2. _________________________________________________

3. 1 & 2 are right ’s. 3. _________________________________________________

4. 1 2 4. _________________________________________________

5. BD BD 5. _________________________________________________

6. Δ ADB Δ CDB 6. _________________________________________________

#2

Given: AC BD

BD bisects ADC

Prove: AB CB

1. AC BD 1. _________________________________________________

2. 1 & 2 are right ’s 2. _________________________________________________

3. 1 2 3. _________________________________________________

4. BD BD 4. _________________________________________________

5. BD bisects ADC 5. _________________________________________________

6. 3 4 6. _________________________________________________

7. Δ ADB Δ CDB 7. _________________________________________________

8. AB CB 8. _________________________________________________

B

3 4

1 2

D

A C

D

B

3 4

1 2 A C

15

Congruent Triangles Proofs

1. Given: SP ; O is the midpoint of PS

Prove: O is the midpoint of RQ

2. Given: ABCD ; D is the midpoint of AB

Prove: CBCA

3. Given: KRSNNRSK //;//

Prove: KRSNNRSK ;

4. Given: MEADMEAD ;//

M is the midpoint AB

Prove: EBDM //

5. Given: MKAB

B is the midpoint of MK

Prove: yx

6. Given: 21

FMCD

Prove: CD bisects MCF

A B

C

D

S

K

N

R

1

2 3

4

A M B

E D

A

B M K

x y

C D

M

F

1

2

P

O

R S

Q

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