Question

1. write a proof for this form of theorem 2 - 2. proof if two angles are supplements of congruent angles, then the two angles are congruent. given: ∠1 and ∠2 are supplementary. ∠3 and ∠4 are supplementary. ∠2≅∠4 prove: ∠1≅∠3

Explanation:

Step1: Recall the definition of supplementary angles

If $\angle1$ and $\angle2$ are supplementary, then $\angle1+\angle2 = 180^{\circ}$, so $\angle1=180^{\circ}-\angle2$. If $\angle3$ and $\angle4$ are supplementary, then $\angle3+\angle4 = 180^{\circ}$, so $\angle3=180^{\circ}-\angle4$.

Step2: Use the given congruence

Since $\angle2\cong\angle4$, then $\angle2=\angle4$ (congruent angles have equal measures).

Step3: Substitute

Substitute $\angle2$ for $\angle4$ in the equation for $\angle3$. We get $\angle3 = 180^{\circ}-\angle2$.

Step4: Compare the expressions for $\angle1$ and $\angle3$

We have $\angle1=180^{\circ}-\angle2$ and $\angle3 = 180^{\circ}-\angle2$. So, $\angle1=\angle3$. Since angles with equal measures are congruent, $\angle1\cong\angle3$.

Answer:

The proof is shown above, demonstrating that if two angles are supplements of congruent angles, then the two angles are congruent.