congruent triangles: review for test - answers · determine if you can use sss, sas, asa, aas, and...

Name__________________________________ Date___________ Per. ________ Mr. Lambert/Ms. Williams

Congruent Triangles: Review for Test - Answers

a. b.

c. d.

e.

f.

g. Find the measures of angle 1, angle 2, angle 3 and angle 4.

h.

I.

J. In ABC, is extended to D, m B = 2y, m BCA = 6y, and m ACD = 3y. What is m A?

K.

L.

M-N.

O.

Given: 𝐴𝐶̅̅ ̅̅ ≅ 𝐸𝐶̅̅̅̅̅

𝐷𝐶̅̅ ̅̅ ≅ 𝐵𝐶̅̅ ̅̅

P.

Q. Given: ∡𝐽 ≅ ∡𝐿, ∡𝐽𝐾𝑀 ≅ ∡𝐿𝑀𝐾

∆JKM ≅ ∆ _______ because of ______.

R.

∆XWZ ≅ ∆ _______ because of ______.

S.

Determine if you can use SSS, SAS, ASA, AAS, and HL to prove triangles congruent. If not, say no.

Identifying Additional Congruent Parts

A.

B.

C.

D.

a.

b.

c.

d.

TODA

MODA

OMAN

TOND

TODA

MN

OA

MA

a.

b.

c.

d.

a.

b.

c.

d.

What additional information is needed for a HL congruence correspondence?

a.

b.

c.

d.

OA

MA

NO

MONA

JC

KE

KAEL

AL

CADBAD

CB

ADAD

ACAB

3.

7. In the accompanying diagram of BCD, ABC is an equilateral triangle and AD = AB. What is the value of x,

in degrees?

Proofs

1.

Statements Reasons

1. Quadrilateral ABCD, AFEC 1. Given

2. 𝐴𝐵̅̅ ̅̅ 𝐶𝐷̅̅ ̅̅ , 𝐴𝐷̅̅ ̅̅ 𝐶𝐵̅̅ ̅̅ 2. Given

3. 𝐷𝐹̅̅ ̅̅ 𝐴𝐶̅̅ ̅̅ , 𝐵𝐸̅̅ ̅̅ 𝐴𝐶̅̅ ̅̅ 3. Given

4. 𝐴𝐶̅̅ ̅̅ 𝐴𝐶̅̅ ̅̅ 4. Reflexive Property

5. ADC CBA 5. SSS (2, 2, 4)

6. ∡DAC ∡BCA 6. CPCTC

7. ∡DFA and ∡BEC are right angles 7. Def of perpendicular lines

8. ∡DFA ∡BEC 8. Right angles are congruent

9. AFD CEB 9. AAS (6, 8, 2)

10. 𝐷𝐹̅̅ ̅̅ 𝐵𝐸̅̅ ̅̅ 10. CPCTC

2.

Statements Reasons

1. ABC, 𝐵𝐷̅̅ ̅̅ is median and altitude to 𝐴𝐶̅̅ ̅̅ 1. Given

2. D is the midpoint of 𝐴𝐶̅̅ ̅̅ 2. Def of median

3. 𝐴𝐷̅̅ ̅̅ 𝐶𝐷̅̅ ̅̅ 3. Def of midpoint

4. 𝐵𝐷̅̅ ̅̅ 𝐴𝐶̅̅ ̅̅ 4. Def of altitude

5. ∡ADB and ∡CDB are right angles 5. Def of Perpendicular Lines

6. ∡ADB ∡CDB 6. Right angles are congruent

7. 𝐵𝐷̅̅ ̅̅ 𝐵𝐷̅̅ ̅̅ 7. Reflexive Property

8. ADB CDB 8. SAS (3, 6, 7)

9. 𝐵𝐴̅̅ ̅̅ 𝐵𝐶 ̅̅ ̅̅ ̅ 9. CPCTC

3.

Statements Reasons

1. ∡1 ∡5, ∡2 ∡6 1. Given

2. ∡1 + ∡2 ∡5 + ∡6 2. Addition Postulate (1)

3. ∡DBA ∡ECA 3. Substitution Property

5. ∡DBA supp ∡3; ∡ECA supp ∡4 5. Linear Pair Theorem

6. ∡3 ∡4 6. Congruent Supplements Theorem

7. 𝐴𝐵̅̅ ̅̅ 𝐴𝐶̅̅ ̅̅ 7.

8. ABC is isosceles 8. If the legs of a triangle are congruent, then triangle is isosceles.

4.

5.

6. Statements Reasons

1. ⨀O, 𝑂𝐷̅̅ ̅̅ 𝑂𝐸̅̅ ̅̅ 1. Given

2. 𝐴𝑂̅̅ ̅̅ 𝑂𝐵̅̅ ̅̅ 2. All radii of a circle are congruent.

3. ∡DOB ∡EOA 3. Given

4. ∡DOE ∡DOE 4. Reflexive Property

5. ∡DOB - ∡DOE ∡AOE - ∡DOE 5. Subtraction Postulate (3,4)

6. ∡AOD ∡EOB 6. Substitution Property

7. AOD BOE 7. SAS (1, 6, 2)

8. ∡A ∡B 8. CPCTC (7)

9. 𝐴𝐶 ̅̅ ̅̅̅ 𝐶𝐵̅̅ ̅̅

9.

10. 𝐴𝐷̅̅ ̅̅ 𝐸𝐵̅̅ ̅̅ 10. CPCTC (7)

11. 𝐴𝐶̅̅ ̅̅ – 𝐴𝐷̅̅ ̅̅ 𝐶𝐵̅̅ ̅̅ - 𝐸𝐵̅̅ ̅̅ 11. Subtraction Postulate (9,10)

12. 𝐶𝐷̅̅ ̅̅ 𝐶𝐸̅̅̅̅ 12. Substitution Property

7.

8.

9. Given: APB with perpendicular bisector 𝑃𝑀̅̅̅̅̅

Prove: 𝐴𝑅̅̅ ̅̅ 𝐵𝑅̅̅ ̅̅

Statements Reasons

1. APB with perpendicular bisector 𝑃𝑀̅̅̅̅̅ 1. Given

2. M is the midpoint of 𝐴𝐵̅̅ ̅̅ 2. Def of segment bisector

3. 𝐴𝑀̅̅̅̅̅ 𝐵𝑀̅̅̅̅ ̅ 3. Def of midpoint

4. ∡AMP and ∡BMP are right angles 4. Def of perpendicular lines

5. ∡AMP ∡BMP 5. Right angles are congruent

6. 𝑅𝑀̅̅̅̅̅ 𝑅𝑀̅̅̅̅̅ 6. Reflexive Property

7. ARM BRM 7. SAS (3, 5, 6)

8. 𝐴𝑅̅̅ ̅̅ 𝑅𝐵̅̅ ̅̅ 8. CPCTC

P

A BM

R

10.

11.

12.

13.

14.

15.

Statements Reasons

1. ABC is isosceles with base 𝐶𝐵̅̅ ̅̅ 1. Given

2. 𝐴𝐶̅̅ ̅̅ 𝐴𝐵̅̅ ̅̅ 2. Definition of Isosceles Triangles

3. ∡ACB ∡ABC

3.

4. 𝐶𝑀̅̅̅̅̅ is a median and 𝐵𝐷̅̅ ̅̅ is a median 4. Given

5. D is the midpoint of 𝐴𝐶̅̅ ̅̅ and M is the midpoint of 𝐴𝐵̅̅ ̅̅ 5. Def of median

6. 𝐷𝐶̅̅ ̅̅ ½ 𝐴𝐶̅̅ ̅̅ , 𝐷𝐴̅̅ ̅̅ ½ 𝐴𝐶̅̅ ̅̅ ; 𝑀𝐵̅̅̅̅ ̅ = ½ 𝐴𝐵̅̅ ̅̅ , 𝑀𝐴̅̅̅̅̅ = ½ 𝐴𝐵̅̅ ̅̅ 6. Def of midpoint

7. 𝐷𝐶̅̅ ̅̅ 𝑀𝐵̅̅̅̅ ̅ 𝐷𝐴̅̅ ̅̅ 𝑀𝐴̅̅̅̅̅ 7. Division Postulate

8. 𝐶𝐵̅̅ ̅̅ 𝐶𝐵̅̅ ̅̅ 8. Reflexive Property

9. DCB MBC 9. SAS (7, 3, 8)

10. ∡CDB ∡BMC 10. CPCTC (9)

11. ∡DEC ∡MEB 11. Vertical angles are congruent

12. DCE MBE 12. AAS (10, 11, 7)

13. 𝐷𝐸̅̅ ̅̅ 𝑀𝐸̅̅̅̅̅ 13. CPCTC (12)

14. ∡CDB supp ∡ADE; ∡BMC sup ∡AME 14. Linear Pair Theorem

15. ∡ADE ∡AME 15. Congruent Supplements Theorem

16. ADE AME 16. SAS (7, 15, 13)

17. ∡DAE ∡MAE 17. CPCTC (16)

18. 𝐴𝑅̅̅ ̅̅ 𝐴𝑅̅̅ ̅̅ 18. Reflexive Property

19. CAR BAR 19. SAS (2, 17, 18)

20. ∡CRA ∡BRA 20. CPCTC (19)

21. ∡CRA supp ∡BRA 21. Linear Pair Theorem

22. ∡CRA and ∡BRA are right angles 22. If 2 angles are congruent and supplementary, then the angles are right.

23. 𝐴𝑅̅̅ ̅̅ ⊥ 𝐶𝐵̅̅ ̅̅ 23. If right angles are formed, then the lines that intersect are perpendicular.

16. Given: ABC is isosceles with base BC, X, Y, M are midpoints of 𝐴𝐵̅̅ ̅̅ , 𝐴𝐶̅̅ ̅̅ , and 𝐵𝐶̅̅ ̅̅ , respectively.

Prove: 𝑋𝑀̅̅̅̅̅ 𝑌𝑀̅̅̅̅̅

Statements Reasons

1. ABC is isosceles with base 𝐵𝐶̅̅ ̅̅ 1. Given

2. 𝐴𝐵̅̅ ̅̅ 𝐴𝐶̅̅ ̅̅ 2. Definition of Isosceles Triangles

3. X, Y, and M are midpoints of 𝐴𝐵̅̅ ̅̅ , 𝐴𝐶̅̅ ̅̅ , 𝐵𝐶̅̅ ̅̅ , respectively

3. Given

4. 𝐵𝑋̅̅ ̅̅ ½ 𝐴𝐵̅̅ ̅̅ ; 𝐶𝑌̅̅̅̅ ½ 𝐴𝐶̅̅ ̅̅ 4. Def of midpoint

5. 𝐵𝑋̅̅ ̅̅ 𝐶𝑌̅̅̅̅ 5. Division Postulate

6. ∡B ∡C

6.

7. 𝐵𝑀̅̅̅̅ ̅ 𝐶𝑀̅̅̅̅̅ 7. Def of midpoint

8. XBM YCM 8. SAS (5, 6, 7)

9. 𝑋𝑀̅̅̅̅̅ 𝑌𝑀̅̅̅̅̅ 9. CPCTC

17.

A

B CM

X Y