Question

complete the following proof by filling in the blanks. given: ∠1≅∠4 prove: ∠7≅∠6 statements reasons 1) ∠1≅∠4 1) given 2) ∠4≅∠6 2) ? 3) ? 3) transitive property of congruence 4) ∠7≅∠1 4) ? 5) ∠7≅∠6 5) ? statements reasons 1) ∠1≅∠4 1) given 2) ∠4≅∠6 2) 3) 3) transitive property of congruence 4) ∠7≅∠1 4) 5) ∠7≅∠6 5)

Explanation:

Step1: Identify vertical - angle property

Vertical angles are congruent. $\angle4$ and $\angle6$ are vertical angles, so $\angle4\cong\angle6$ because vertical angles are congruent.

Step2: Apply transitive property

Since $\angle1\cong\angle4$ (given) and $\angle4\cong\angle6$, by the transitive property of congruence, $\angle1\cong\angle6$.

Step3: Identify vertical - angle property again

$\angle7$ and $\angle1$ are vertical angles. So $\angle7\cong\angle1$ because vertical angles are congruent.

Step4: Apply transitive property again

Since $\angle1\cong\angle6$ and $\angle7\cong\angle1$, by the transitive property of congruence, $\angle7\cong\angle6$.

Answer:

1. Vertical angles are congruent
2. $\angle1\cong\angle6$
3. Vertical angles are congruent
4. Transitive property of congruence