Question

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Please help me with these proofs. Part 1

Answer 1

Problem 7

Given:

• C is the midpoint of AE,
• ∠BCA ≅ ∠DCE,
• ∠A ≅ ∠E

We have two pairs of congruent angles and the included sides are congruent. It gives us ASA congruence. AB ≅ DE as corresponding parts of congruent triangles.

Statement Reason

• C is the midpoint of AE Given
• ∠BCA ≅ ∠DCE Given
• ∠A ≅ ∠E Given
• AC ≅ EC Definition of a midpoint
• ΔABC ≅ ΔEDC ASA congruence
• AB ≅ DE CPCTC

• Proved

Problem 8

Given:

• AH bisects ∠BHT,
• AH ⊥ BT

We have two right triangles with common leg and the other pair of legs are congruent as bisected segment. We get SAS congruence with two sides and included right angle.

Statement Reason

• AH bisects ∠BHT Given
• AH ⊥ BT Given
• ∠BAH ≅ ∠TAH = 90° Definition of perpendicular
• AH ≅ AH Reflexive property (common side)
• BA ≅ TA Definition of segment bisector
• ΔBAH ≅ ΔTAH SAS congruence

• Proved

Problem 10

Given:

• AB ⊥ AC,
• CD ⊥ BD,
• AC ≅ BD

ΔABC and ΔCDB are right triangles with common hypotenuse and congruent pair of legs. It gives us HL congruence and it proves the corresponding angles are congruent.

Statement Reason

• AB ⊥ AC Given
• CD ⊥ BD Given
• AC ≅ BD Given
• ∠A & ∠D are right angles Definition of perpendicular
• BC ≅ BC Reflexive property (common side)
• ΔABC ≅ ΔDCB HL congruence
• ∠ABC ≅ ∠BCD CPCTC

• Proved