Question

given: ∠c ≅ ∠l
prove: ∠v ≅ ∠g

1. ∠c ≅ ∠l
2. pairs of vertical angles are congruent.

Explanation:

Step1: State given information

$\angle C\cong\angle L$

Step2: Identify vertical - angle pair

$\angle VBC$ and $\angle GBL$ are vertical angles, so $\angle VBC\cong\angle GBL$

Step3: Prove the target angles congruent

Use angle - sum property of triangles. Since $\angle V = 180^{\circ}-\angle C-\angle VBC$ and $\angle G=180^{\circ}-\angle L - \angle GBL$, and $\angle C\cong\angle L$, $\angle VBC\cong\angle GBL$, we get $\angle V\cong\angle G$

Answer:

1. $\angle C\cong\angle L$ (Given)
2. $\angle VBC\cong\angle GBL$ (Pairs of vertical angles are congruent)
3. $\angle V\cong\angle G$ (By the Angle - Angle (AA) similarity criterion or the fact that in two triangles, if two pairs of corresponding angles are congruent, the third pair is also congruent. Since the sum of angles in a triangle is 180 degrees, in $\triangle VBC$ and $\triangle GBL$, we have $\angle V = 180^{\circ}-\angle C-\angle VBC$ and $\angle G=180^{\circ}-\angle L - \angle GBL$. Because $\angle C\cong\angle L$ and $\angle VBC\cong\angle GBL$, then $\angle V\cong\angle G$)