Question

∠p and ∠q are supplementary and ∠r and ∠p are supplementary. complete the proof that ∠r≅∠q.
statement reason
1 ∠p and ∠q are supplementary given
2 ∠r and ∠p are supplementary given
3 m∠p + m∠q = 180° definition of supplementary angles
4 m∠r + m∠p = 180° definition of supplementary angles
5 m∠p + m∠q = m∠r + m∠p transitive property of equality
6 m∠r = m∠q properties of addition, subtraction, multiplication, and division
7 definition of congruence

Explanation:

Step1: Recall given information

$\angle P$ and $\angle Q$ are supplementary, $\angle R$ and $\angle P$ are supplementary.

Step2: Apply definition of supplementary angles

$m\angle P + m\angle Q=180^{\circ}$, $m\angle R + m\angle P = 180^{\circ}$

Step3: Use transitive property of equality

Since $m\angle P + m\angle Q=180^{\circ}$ and $m\angle R + m\angle P = 180^{\circ}$, then $m\angle P + m\angle Q=m\angle R + m\angle P$.

Step4: Subtract $m\angle P$ from both sides

Subtracting $m\angle P$ from both sides of $m\angle P + m\angle Q=m\angle R + m\angle P$ gives $m\angle R=m\angle Q$.

Step5: Apply definition of congruence

If $m\angle R = m\angle Q$, then $\angle R\cong\angle Q$ by the definition of congruence.

Answer:

$\angle R\cong\angle Q$