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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. 1
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Chapter 4 – Congruent Triangles4.2 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.

1

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!

2

3

Find the value of x.1. 2. 3.

4. 5.

4

50°

x°3x + 20

5x60°

3x

21

100°

x°72°

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.

and . Find mL, mM, and mLNM. _____________________

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

5

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

6

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.

4.4 Congruent Triangles

P olygons 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

7

425

3 1

9

11

107

8

6

120

(5x)°

x° 140°

35°

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.

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.

8

A B

O

CD

B

L A

C

K

H

O

R

S

A

4 cm

E

3. Name the opposite angle to BC .

4. Name the included angle between AB and AC .

4.5-4.7 Proving Triangles CongruentWays to Prove ’s :

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

Given: AS bisects PW ; PA≃AW

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; AX≃XE

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

Given:

MA // THAT //MH

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: UZ bisec ts CA UZ⊥CU ; UZ⊥ZA

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

Given: AT⊥FC

Isosceles FAC with legs FA , AC9

BC

A

P WS

A

X

PE

A

M

T

H

C

RU Z

A

CTF

A

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.

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.

10

C

D

B

A I

F

H

GE

R

S

Q

T

1212

10

C D

BA

10

SQP

R

U

R

V

TS

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 postulate1. 2.

∆ABC _____ by ________ ∆ABC ________ by __________

3. 4.

∆ABC ________ by __________ ∆ABC ________ by _________5. 6.

∆ABC ________ by _________ ∆ABC _______ by __________

7. 8.

11

D

E

F

CB

A

A

B

C

D

FE

A

B C

D

E

F D

CB

A

A

D

C

B

ER

QP

C

B

A

BA

DC

60

50

70

60

D

C

BA

6030

∆ABC ________ by ___________ ∆ABC ________ by ____________

9. 10.

∆ABC ________ by _________ ∆ABC _______ by ___________

11. 12.

∆ABC _________ by ___________ ∆ABC _________ by __________

13. 14.

∆ABC _________ by ___________ ∆ABC __________ by ___________Proofs!

#1 Given: SR≃UT ; SR //UT ; ∠ S≃∠U Prove: ST //UV

1. SR≃UT ; SR //UT ; ∠ S≃∠U 1. _____________________________

2. 1 4 2. __________________________________________

3. ∆RST ∆TUV 3. __________________________________________

4. 3 2 4. __________________________________________

5. ST //UV 5. __________________________________________

12

D C

BA

D

CB

A

D

B

A

C

N

U

C

A

B

D

C

B

A

C

B

D

A

#2 Given: D is the midpoint of AB; CA≃CB Prove: CD bisects ACB.

1. D is the midpoint of AB; CA≃CB 1. _________________________________________

2. AD≃DB 2. __________________________________________

3. CD≃CD 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. __________________________________________

#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. _________________________________________________

13

CA1 2

3 4

B

D

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. _________________________________________________

14

CA

D

1 2

3 4

B

Congruent Triangles Proofs

1. Given: ∠P≃∠ S ; O is the midpoint of PSProve: O is the midpoint of RQ

2. Given: CD⊥AB ; D is the midpoint of

Prove: CA≃CB

3. Given: SK //NR ; SN // KRProve: SK≃NR ;SN≃KR

4. Given: AD //ME; AD≃MEM is the midpoint AB

Prove: DM // EB

5. Given: AB⊥MKB is the midpoint of MK

Prove: ∠ x≃∠ y

6. Given:

CD⊥FM∠1≃∠2

Prove: CD bisects ∠MCF15

D

C

BA

4

32

1

R

N

K

S

D E

BMA

yx

KM B

A

2

1

F

M

DC

Q

SR

O

P

16


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