Geometry Honors Notes – Chapter 4: Congruent Triangles – Solutions to Proof Practice Problems

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D

BA C

P O N

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A1 3 4 2B M C D

C

B

D

A

E

4.3 – Prove Triangles Congruent by SSS

PRACTICE #1 Given: ;AD CD≅ B is the midpoint of .AC Prove: ABD CBDΔ ≅ Δ Statements Reasons 1. AD CD≅ 1. Given 2. B is the midpoint of .AC 2. Given 3. AB CB≅ 3. Def. of midpoint. 4. BD BD≅ 4. Reflexive Property 5. ABD CBDΔ ≅ Δ 5. SSS (1, 3, 4) PRACTICE #2 Given: ,PR NT≅ ;NO SR≅ O is 1

3 of the way from N to P. S is 1

3 of the way from R to T. Prove: NRT RNPΔ ≅ Δ

Statements Reasons 1. PR NT≅ 1. Given 2. ,NO SR≅ 2. Given O is 1

3 of the way from N to P. S is 1

3 of the way from R to T.

3. NP RT≅ 3. Multiplication Prop. 4. NR NR≅ 4. Reflexive Prop. 5. NRT RNPΔ ≅ Δ 5. SSS (1, 3, 4)

4.4 – Prove Triangles Congruent by SAS and HL PRACTICE #1 Given: 1 2;∠ ≅∠ M is the midpt. of .BC BE CE≅ Prove: EMB EMCΔ ≅ Δ Statements Reasons 1. BE CE≅ 1. Given 2. 1 2∠ ≅ ∠ 2. Given 3. 1 is supp. to 3∠ ∠ 3. If 2 s form a st. ∠ ∠ 2 is supp. to 4∠ ∠ (assumed from diagram), then they are supp. 4. 3 4∠ ≅ ∠ 4. Congruent Supplements Th. 5. M is the midpt. of .BC 5. Given 6. MB MC≅ 6. Def. of midpt. 7. EMB EMCΔ ≅ Δ 7. SAS (1, 4, 6) PRACTICE #2 Given: ,BC AC⊥ ,BD AD⊥ AC AD≅ Prove: ACB ADBΔ ≅ Δ Statements Reasons 1. ,BC AC BD AD⊥ ⊥ 1. Given 2. is a rt. .ACB∠ ∠ 2. Def. of perpendicular lines. is a rt. .BDA∠ ∠ 3. ACBΔ and ADBΔ 3. Def. of right triangles. are right triangles. 4. AC AD≅ 4. Given 5. AB AB≅ 5. Reflexive Property 6. ACB ADBΔ ≅ Δ 6. HL (3, 4, 5)

Geometry Honors Notes – Chapter 4: Congruent Triangles – Solutions to Proof Practice Problems

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R

K3 4 5 6

M O P

3 45

612

A D

F

B EG

R S T V

B

GG

G

HH

HH

J

O M KC

4.5 – Prove Triangles Congruent by ASA & AAS PRACTICE #1 Given: 3 6,∠ ≅∠ ,KR PR≅ KRO PRM∠ ≅ ∠ Prove: KRM PROΔ ≅ Δ Statements Reasons 1. 3 6∠ ≅ ∠ 1. Given 2. 3 is supp. to 4.∠ ∠ 2. If 2 s form a st. ∠ ∠ 6 is supp. to 5.∠ ∠ (assumed from diagram), then they are supp. 3. 4 5∠ ≅ ∠ 3. Congruent Supp. Th. 4. KR PR≅ 4. Given 5. KRO PRM∠ ≅ ∠ 5. Given 6. MRO MRO∠ ≅ ∠ 6. Reflexive Property 7. KRM PRO∠ ≅ ∠ 7. Subtraction Property 8. KRM PROΔ ≅ Δ 8. ASA (3, 4, 7) PRACTICE #2 Given: 1 6,∠ ≅∠ BC EC≅ Prove: ABC DECΔ ≅ Δ Statements Reasons 1. 1 6∠ ≅ ∠ 1. Given 2. 1 is supp. to 2.∠ ∠ 2. If 2 s form a st. ∠ ∠ 6 is supp. to 5.∠ ∠ (assumed from diagram), then they are supp. 3. 2 5∠ ≅ ∠ 3. Congruent Supp. Th. 4. BC EC≅ 4. Given 5. 3 4∠ ≅ ∠ 5. Vertical angles are .≅ 6. ABC DECΔ ≅ Δ 6. ASA (3, 4, 5)

PRACTICE #3 Given: S and T trisect ,RV ,R V∠ ≅∠ BST BTS∠ ≅ ∠ Prove: BRS BVTΔ ≅ Δ Statements Reasons 1. R V∠ ≅ ∠ 1. Given 2. S and T trisect RV 2. Given 3. RS ST TV≅ ≅ 3. Def. of trisection pts. 4. BST BTS∠ ≅ ∠ 4. Given 5. is supp. to .BSR BST∠ ∠ 5. If 2 s form a st. ∠ ∠ is supp. to .BTV BTS∠ ∠ (assumed from diag.), then they are supp. 6. BSR BTV∠ ≅ ∠ 6. Congruent Supp. Th. 7. BRS BVTΔ ≅ Δ 7. ASA (1, 3, 6) 4.6.1 – Use Congruent Triangles (CPCTC) PRACTICE #1 Given: H is the midpt. of .GJ M is the midpt. of .OK , ,GO JK GJ OK≅ ≅ , 27,G K OK∠ ≅ ∠ = 24, 2 7,m GOH x m GHO y∠ = + ∠ = − 3 23, 4 105.m JMK y m MJK x∠ = − ∠ = − Find: , ,m GOH m GHO∠ ∠ and GH.

Answer: 6725

13.5

m GOHm GHOGH

∠ = °∠ = °=

4 105 24

43x x

x− = +=

2 7 3 23

16y y

y− = −=

27 / 2 13.5GH = =

Geometry Honors Notes – Chapter 4: Congruent Triangles – Solutions to Proof Practice Problems

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A B C D

E

F H

K J

G

34

12

D C

AE

F

B

BC

DP

AF

E

PRACTICE #2 Given: ,AEC DEB∠ ≅∠ ,BE CE≅ ABE DCE∠ ≅ ∠ Prove: AB CD≅ Statements Reasons 1. ABE DCE∠ ≅ ∠ 1. Given 2. BE CE≅ 2. Given 3. AEC DEB∠ ≅ ∠ 3. Given 4. BEC BEC∠ ≅ ∠ 4. Reflexive Property 5. AEB DEC∠ ≅ ∠ 5. Subtraction Property 6. AEB DECΔ ≅ Δ 6. ASA (1, 2, 5) 7. AB CD≅ 7. CPCTC PRACTICE #3 Given: ,KG GJ≅ 2 4,∠ ≅∠ 1∠ is comp. to 2.∠

3∠ is comp. to 4.∠ FGJ HGK∠ ≅ ∠ Prove: FG HG≅ Statements Reasons 1. 2 4∠ ≅∠ 1. Given 2. 1 is comp. to 2.∠ ∠ 2. Given 3 is comp. to 4.∠ ∠ 3. 1 3∠ ≅ ∠ 3. Congruent Comp. Th. 4. KG GJ≅ 4. Given 5. FGJ HGK∠ ≅ ∠ 5. Given 6. KGJ KGJ∠ ≅ ∠ 6. Reflexive Property 7. FGK HGJ∠ ≅ ∠ 7. Subtraction Property 8. FGK HGJΔ ≅ Δ 8. ASA (3, 4, 7) 9. FG HG≅ 9. CPCTC

PRACTICE #4 Given: ,AE FC≅ ,FB DE≅ CFB AED∠ ≅ ∠ Prove: FAB ECD∠ ≅ ∠ Statements Reasons 1. FB DE≅ 1. Given 2. CFB AED∠ ≅ ∠ 2. Given 3. is supp. to .CFB BFA∠ ∠ 3. If 2 s form a st. ∠ ∠ is supp. to .AED DEC∠ ∠ (assumed from diag.), then they are supp. 4. BFA DEC∠ ≅ ∠ 4. Congruent Supp. Th. 5. AE FC≅ 5. Given 6. EF EF≅ 6. Reflexive Property 7. AF EC≅ 7. Addition Property 8. FAB ECDΔ ≅ Δ 8. SAS (1, 4, 7) 9. FAB ECD∠ ≅ ∠ 9. CPCTC PRACTICE #5 Given: ,BC FE≅ ,PA PD≅ ,PF PC≅ ,AB FC⊥ .DE FC⊥ Prove: C F∠ ≅ ∠ Statements Reasons 1. ,AB FC DE FC⊥ ⊥ 1. Given 2. is a rt. .BAC∠ ∠ 2. Def. of ⊥ lines. is a rt. .EDF∠ ∠ 3. BACΔ and EDFΔ 3. Def. of right triangles. are right triangles. 4. BC FE≅ 4. Given 5. PA PD≅ 5. Given 6. PF PC≅ 6. Given 7. DF AC≅ 7. Addition Property 8. BAC EDFΔ ≅ Δ 8. HL (3, 4, 7) 9. C F∠ ≅ ∠ 9. CPCTC

Geometry Honors Notes – Chapter 4: Congruent Triangles – Solutions to Proof Practice Problems

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H1 2

G F

E

A B C D

E

A

W

XZ

Y

B

C

D

F

GE

4.6.2 – Steps Beyond CPCTC PRACTICE #1 Given: G is the midpt. of FH EF EH≅ Prove: 1 2∠ ≅ ∠ Statements Reasons 1. G is the midpt. of FH 1. Given 2. FG HG≅ 2. Def. of midpt. 3. EF EH≅ 3. Given 4. Draw EG 4. Two pts. det. a segment. 5. EG EG≅ 5. Reflexive Property 6. EFG EHGΔ ≅ Δ 6. SSS (2, 3, 5) 7. EFG EHG∠ ≅ ∠ 7. CPCTC 8. 2 is supp. to .EFG∠ ∠ 8. If 2 s form a st. ∠ ∠ 1 is supp. to .EHG∠ ∠ (assumed from diagram), then they are supp. 9. 1 2∠ ≅ ∠ 9. Congruent Supp. Th. PRACTICE #2 Given: ,AEB DEC∠ ≅∠ ,AE DE≅ A D∠ ≅ ∠ Prove: AC BD≅ Statements Reasons 1. AEB DEC∠ ≅ ∠ 1. Given 2. AE DE≅ 2. Given 3. A D∠ ≅ ∠ 3. Given 4. AEB DECΔ ≅ Δ 4. ASA (1, 2, 3) 5. AB CD≅ 5. CPCTC 6. BC BC≅ 6. Reflexive Property 7. AC BD≅ 7. Addition Property

PRACTICE #3 Given: ;AZ ZB≅ Z is the midpt. of .XY ,AZX BZY∠ ≅∠ XW YW≅ Prove: AW BW≅ Statements Reasons 1. AZ ZB≅ 1. Given 2. AZX BZY∠ ≅ ∠ 2. Given 3. Z is the midpt. of .XY 3. Given 4. XZ ZY≅ 4. Def. of midpt. 5. AZX BZYΔ ≅ Δ 5. SAS (1, 2, 4) 6. AX BY≅ 6. CPCTC 7. XW YW≅ 7. Given 8. AW BW≅ 8. Subtraction Property PRACTICE #4 Given: DF bisects .CDE∠ EF bisects .CED∠ G is the midpt. of .DE DF EF≅ Prove: CDE CED∠ ≅ ∠ Statements Reasons 1. Draw .FG 1. Two pts. det. a segment. 2. DF EF≅ 2. Given 3. G is the midpt. of .DE 3. Given 4. DG EG≅ 4. Def. of midpt. 5. FG FG≅ 5. Reflexive Property 6. FDG FEGΔ ≅ Δ 6. SSS (2, 4, 5) 7. FDG FEG∠ ≅ ∠ 7. CPCTC 8. DF bisects .CDE∠ 8. Given EF bisects .CED∠ 9. CDE CED∠ ≅ ∠ 9. Multiplication Property

Geometry Honors Notes – Chapter 4: Congruent Triangles – Solutions to Proof Practice Problems

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A W X

Z Y

5 6

A

B

Y

C

Z

D

E

Z P Y

W Q X

4.6.3 – Overlapping Triangles PRACTICE #1 Given: YW bisects .AX A X∠ ≅ ∠ , 5 6∠ ≅ ∠ Prove: ZW YW≅ Statements Reasons 1. A X∠ ≅ ∠ 1. Given 2. YW bisects .AX 2. Given 3. AW XW≅ 3. Def. of segment bisector. 4. 5 6∠ ≅ ∠ 4. Given 5. ZWY ZWY∠ ≅ ∠ 5. Reflexive Property 6. AWY XWZ∠ ≅ ∠ 6. Addition Property 7. AWY XWZΔ ≅ Δ 7. ASA (1, 3, 6) 8. ZW YW≅ 8. CPCTC PRACTICE #2 Given: ,YD ZD≅ ;BD CD≅ E is the midpt. of .YZ Prove: BYZ CZY∠ ≅ ∠ Statements Reasons 1. Draw .DE 1. Two pts det. a segment. 2. E is the midpt. of .YZ 2. Given 3. YE ZE≅ 3. Def. of midpt. 4. DE DE≅ 4. Reflexive Property 5. YD ZD≅ 5. Given 6. EYD EZDΔ ≅ Δ 6. SSS (3, 4, 5) 7. EYD EZD∠ ≅ ∠ 7. CPCTC 8. YD ZD≅ 8. Given 9. BDY CDZ∠ ≅∠ 9. Vertical angles are .≅ 10. BD CD≅ 10. Given 11. BYD CZDΔ ≅ Δ 11. SAS (8, 9, 10) 12. BYZ CZY∠ ≅ ∠ 12. Addition Property

4.6.4 – Detour Proofs PRACTICE #1 Given: bisects .PQ YZ

Q is the midpt. of .WX ,Y Z WZ XY∠ ≅ ∠ ≅ Prove: WQP XQP∠ ≅ ∠ To reach the required conclusion, we must prove that ___ WQP XQPΔ ≅ Δ ______________, but the given information is not sufficient to prove these triangles congruent. Therefore, we must detour through another pair of triangles _ ZWP YXPΔ ≅ Δ __________. Statements Reasons 1. bisects .PQ YZ 1. Given 2. ZP PY≅ 2. Def. of segment bis. 3. Z Y∠ ≅ ∠ 3. Given 4. WZ XY≅ 4. Given 5. ZWP YXPΔ ≅ Δ 5. SAS (2, 3, 4) 6. WP PX≅ 6. CPCTC 7. Q is the midpt. of .WX 7. Given 8. WQ QX≅ 8. Def. of midpt. 9. PQ PQ≅ 9. Reflexive Property 10. WQP XQPΔ ≅ Δ 10. SSS (6, 8, 9) 11. WQP XQP∠ ≅ ∠ 11. CPCTC

Geometry Honors Notes – Chapter 4: Congruent Triangles – Solutions to Proof Practice Problems

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1 2

P

QR S T U

R

P

S

T

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U

B A

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W X Y Z

3 4

PRACTICE #2 Given: ,PR PU≅ ,QR QU≅

RS UT≅ Prove: 1 2∠ ≅ ∠ Statements Reasons 1. Draw .PQ 1. Two pts det. a segment. 2. PQ PQ≅ 2. Reflexive Property

3. PR PU≅ 3. Given 4. QR QU≅ 4. Given 5. PQR PQUΔ ≅ Δ 5. SSS (2, 3, 4) 6. R U∠ ≅ ∠ 6. CPCTC 7. RS UT≅ 7. Given 8. PRS PUTΔ ≅ Δ 8. SAS (3, 6, 7) 9. 1 2∠ ≅ ∠ 9. CPCTC PRACTICE #3 Given: ,PT PU≅ PR PS≅ Prove: bisects .PQ RPS∠ Statements Reasons 1. PT PU≅ 1. Given 2. SPT RPU∠ ≅∠ 2. Reflexive Property 3. PR PS≅ 3. Given 4. SPT RPUΔ ≅ Δ 4. SAS (1, 2, 3) 5. T U∠ ≅∠ 5. CPCTC 6. RQT SQU∠ ≅∠ 6. Vertical angles are .≅

7. RT SU≅ 7. Subtraction Property 8. RTQ SUQΔ ≅ Δ 8. AAS (6, 5, 7)

9. RQ SQ≅ 9. CPCTC

10. PQ PQ≅ 10. Reflexive Property 12. RPQ SPQΔ ≅ Δ 12. SSS (3, 9, 10) 13. RPQ SPQ∠ ≅∠ 13. CPCTC 14. bisects .PQ RPS∠ 14. Def. of angle bisector.

Alternative Proof: Statements Reasons 1. PT PU≅ 1. Given 2. SPT RPU∠ ≅∠ 2. Reflexive Property 3. PR PS≅ 3. Given 4. SPT RPUΔ ≅ Δ 4. SAS (1, 2, 3) 5. PRU PST∠ ≅∠ 5. CPCTC 6. T U∠ ≅∠ 6. CPCTC 7. RT SU≅ 7. Subtraction Property (1-3) 8. is supp. to .TRQ PRU∠ ∠ 8. If 2 s form a st. ∠ ∠ is supp. to .USQ PST∠ ∠ (assumed from diag.), then they are supp. 9. TRQ USQ∠ ≅∠ 9. Congruent Supp. Th. 10. RTQ SUQΔ ≅ Δ 10. ASA (6, 7, 9)

11. RQ SQ≅ 11. CPCTC 12. RPQ SPQΔ ≅ Δ 12. SAS (3, 5, 11) 13. RPQ SPQ∠ ≅∠ 13. CPCTC 14. bisects .PQ RPS∠ 14. Def. of angle bisector. 4.7 – Use Isosceles and Equilateral Triangles PRACTICE #1 Given: 3 4,∠ ≅∠ ,BX AY≅ BW AZ≅ Prove: WTZΔ is isosceles. Statements Reasons 1. 3 4∠ ≅∠ 1. Given 2. 3 is supp. to .WBX∠ ∠ 2. If 2 s form a st. ∠ ∠ 4 is supp. to .YAZ∠ ∠ (assumed from diag.), then they are supp. 3. WBX ZAY∠ ≅ ∠ 3. Congruent Supp. Th 4. BX AY≅ 4. Given 5. BW AZ≅ 5. Given 6. WBX ZAYΔ ≅ Δ 6. SAS (4, 3, 5) 7. W Z∠ ≅∠ 7. CPCTC 8. WTZΔ is isosceles. 8. If at least two s∠ of a Δ are ,≅ the triangle is isosceles.

Geometry Honors Notes – Chapter 4: Congruent Triangles – Solutions to Proof Practice Problems

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C

D G

E F

3 45

A

D EP

B C

PRACTICE #2 Given: ,CE CF≅ 3;F∠ ≅∠ is supp. to 5.E∠ ∠ Prove: CDGΔ is isosceles. Statements Reasons 1. CE CF≅ 1. Given 2. E F∠ ≅ ∠ 2. Base Angles Theorem 3. 3F∠ ≅∠ 3. Given 4. 3E∠ ≅∠ 4. Transitive Property 5. 4 is supp. to 5.∠ ∠ 5. If 2 s form a st. ∠ ∠ (assumed from diag.), then they are supp. 6. is supp. to 5.E∠ ∠ 6. Given 7. 4E∠ ≅∠ 7. Congruent Supp. Th. 8. 3 4∠ ≅∠ 8. Transitive Property 9. CDGΔ is isosceles. 9. If at least two s∠ of a Δ are ,≅ the triangle is isosceles.

PRACTICE #3 Given: ABCΔ is isosceles with .AB AC≅ D is the midpt. of .AB E is the midpt. of .AC Prove: PBCΔ is isosceles. Statements Reasons 1. ABCΔ is isosceles 1. Given with .AB AC≅ 2. D is the midpt. of .AB 2. Given 3. E is the midpt. of .AC 3. Given 4. DB EC≅ 4. Division Property 5. ABC ACB∠ ≅ ∠ 5. Base Angles Theorem 6. BC BC≅ 6. Reflexive Property 7. DBC ECBΔ ≅ Δ 7. SAS (4, 5, 6) 8. DCB EBC∠ ≅ ∠ 8. CPCTC 9. PBCΔ is isosceles. 9. If at least two s∠ of a Δ are ,≅ the triangle is isosceles. Alternative Proof: Statements Reasons 1. ABCΔ is isosceles 1. Given with .AB AC≅ 2. A A∠ ≅ ∠ 2. Reflexive Property 3. D is the midpt. of .AB 3. Given 4. E is the midpt. of .AC 4. Given 5. AD AE≅ 5. Division Property 6. ABE ACDΔ ≅ Δ 6. SAS (1, 2, 5) 7. ABE ACD∠ ≅ ∠ 7. CPCTC 8. ABC ACB∠ ≅ ∠ 8. Base Angles Theorem 9. PBC PCB∠ ≅ ∠ 9. Subtraction Property 10. PBCΔ is isosceles. 10.If at least two s∠ of a Δ are ,≅ the triangle is isosceles.