1) Given: AB ∥ CD; AB ≅ CD
Prove: ΔABC ≅ ΔDCB
Statement Reason
1. AB ∥ CD 1. Given
2. AB ≅ CD 2. Given
3. ∠ABC ≅ ∠DCB 3. Alternate Interior Angles Theorem (from 1)
4. BC ≅ CB 4. Reflexive Property of Congruence
5. ΔABC ≅ ΔDCB 5. SAS Congruence Postulate (from 2, 3, 4)
2) Given: ∠J ≅ ∠M, K is midpoint of JM
Prove: ΔJKN ≅ ΔMKL
Statement Reason
1. ∠J ≅ ∠M 1. Given
2. K is midpoint of JM 2. Given
3. JK ≅ MK 3. Definition of Midpoint (from 2)
4. ∠JKN ≅ ∠MKL 4. Vertical Angles Theorem
5. ΔJKN ≅ ΔMKL 5. ASA Congruence Postulate (from 1, 3, 4)
3) Given: ∠X ≅ ∠W, ZT ⊥ XW
Prove: ΔXTZ ≅ ΔWTZ
Statement Reason
1. ∠X ≅ ∠W 1. Given
2. ZT ⊥ XW 2. Given
3. ∠ZTX ≅ ∠ZTW 3. Definition of Perpendicular Lines (both are right angles)
4. ZT ≅ TZ 4. Reflexive Property of Congruence
5. ΔXTZ ≅ ΔWTZ 5. AAS Congruence Theorem (from 1, 3, 4)
4) Given: RM ⊥ LP; RL ≅ RP
Prove: ΔRML ≅ ΔRMP
Statement Reason
1. RM ⊥ LP 1. Given
2. RL ≅ RP 2. Given
3. ∠RML ≅ ∠RMP 3. Definition of Perpendicular Lines (both are right angles)
4. RM ≅ MR 4. Reflexive Property of Congruence
5. ΔRML ≅ ΔRMP 5. HL Congruence Theorem (for right triangles, from 2, 3, 4)
5) Given: AC bisects ∠BCD; BC ≅ DC
Prove: ΔABC ≅ ΔADC
Statement Reason
1. AC bisects ∠BCD 1. Given
2. BC ≅ DC 2. Given
3. ∠BCA ≅ ∠DCA 3. Definition of Angle Bisector (from 1)
4. AC ≅ CA 4. Reflexive Property of Congruence
5. ΔABC ≅ ΔADC 5. SAS Congruence Postulate (from 2, 3, 4)
6) Given: AB ≅ DC; AD ≅ BC
Prove: ΔABD ≅ ΔCDB
Statement Reason
1. AB ≅ DC 1. Given
2. AD ≅ BC 2. Given
3. BD ≅ DB 3. Reflexive Property of Congruence
4. ΔABD ≅ ΔCDB 4. SSS Congruence Postulate (from 1, 2, 3)
7) Given: ∠R ≅ ∠U; ST bisects ∠RSU
Prove: ΔRST ≅ ΔUST
Statement Reason
1. ∠R ≅ ∠U 1. Given
2. ST bisects ∠RSU 2. Given
3. ∠RST ≅ ∠UST 3. Definition of Angle Bisector (from 2)
4. ST ≅ TS 4. Reflexive Property of Congruence
5. ΔRST ≅ ΔUST 5. AAS Congruence Theorem (from 1, 3, 4)
8) Given: ∠BDA and ∠BDC are right angles; BA ≅ BC
Prove: ΔBDA ≅ ΔBDC
Statement Reason
1. ∠BDA and ∠BDC are right angles 1. Given
2. BA ≅ BC 2. Given
3. ∠BDA ≅ ∠BDC 3. All right angles are congruent
4. BD ≅ DB 4. Reflexive Property of Congruence
5. ΔBDA ≅ ΔBDC 5. HL Congruence Theorem (for right triangles, from 2, 3, 4)
9) Given: RN ⊥ MP; N is midpoint of MP
Prove: ΔRNM ≅ ΔRNP
Statement Reason
1. RN ⊥ MP 1. Given
2. N is midpoint of MP 2. Given
3. MN ≅ PN 3. Definition of Midpoint (from 2)
4. ∠RNM ≅ ∠RNP 4. Definition of Perpendicular Lines (both are right angles)
5. RN ≅ NR 5. Reflexive Property of Congruence
6. ΔRNM ≅ ΔRNP 6. SAS Congruence Postulate (from 3, 4, 5)
10) Given: ∠C ≅ ∠D; AB ⊥ CD
Prove: ΔABC ≅ ΔABD
Statement Reason
1. ∠C ≅ ∠D 1. Given
2. AB ⊥ CD 2. Given
3. ∠ABC ≅ ∠ABD 3. Definition of Perpendicular Lines (both are right angles)
4. AB ≅ BA 4. Reflexive Property of Congruence
5. ΔABC ≅ ΔABD 5. AAS Congruence Theorem (from 1, 3, 4)
Parent Tip: Review the logic above to help your child master the concept of geometry triangle proofs worksheet.
