Question

given that ∠5 ≅ ∠6, which of the following would complete the proof that ∠4 ≅ ∠7?
diagram with angles 4,5 and 6,7
statement | reason

1. ∠5 ≅ ∠6 | 1. given
2. ∠5 ≅ ∠4 | 2. ______ theorem
3. ∠6 ≅ ∠7 | 3. ______ theorem
4. ∠_ ≅ ∠_ | 4. substitution property

options:
○ 2) vertical angle, 3) vertical angle, 4) ∠6 ≅ ∠4
○ 2) vertical angle, 3) substitution property, 4) ∠6 ≅ ∠7
○ 2) vertical angle, 3) substitution property, 4) ∠4 ≅ ∠7
○ 2) vertical angle, 3) vertical angle, 4) ∠4 ≅ ∠7

Explanation:

Step1: Analyze ∠5 and ∠4

∠5 and ∠4 are vertical angles, so by Vertical Angle Theorem, ∠5 ≅ ∠4. So step 2 reason is Vertical Angle Theorem.

Step2: Analyze ∠6 and ∠7

∠6 and ∠7 are vertical angles, so by Vertical Angle Theorem, ∠6 ≅ ∠7. So step 3 reason is Vertical Angle Theorem.

Step3: Apply Substitution Property

We know ∠5 ≅ ∠6 (given), ∠5 ≅ ∠4 (vertical angles), ∠6 ≅ ∠7 (vertical angles). By substitution, ∠4 ≅ ∠7. So step 4 is ∠4 ≅ ∠7.

Answer:

1. Vertical Angle, 3) Vertical Angle, 4) $\angle 4 \cong \angle 7$ (the last option: 2) Vertical Angle, 3) Vertical Angle, 4) $\angle 4 \cong \angle 7$)