Question

∠p and ∠r are complementary and ∠p and ∠q are complementary. complete the proof that ∠q≅∠r.
statement\treason
1 ∠p and ∠r are complementary\tgiven
2 ∠p and ∠q are complementary\tgiven
3 m∠p + m∠r = 90°\tdefinition of complementary angles
4 m∠p + m∠q = 90°\tdefinition of complementary angles
5 m∠p + m∠r = m∠p + m∠q\ttransitive property of equality
6 m∠q = m∠r\tpaste...
7 ∠q≅∠r\tdefinition of congruence
angles forming a linear pair sum to 180°
definition of angle bisector
definition of complementary angles
definition of congruence
definition of perpendicular lines

Explanation:

Step1: State given information

Given $\angle P$ and $\angle R$ are complementary, $\angle P$ and $\angle Q$ are complementary.

Step2: Apply definition of complementary angles

By definition, $m\angle P + m\angle R=90^{\circ}$ and $m\angle P + m\angle Q = 90^{\circ}$.

Step3: Use transitive property of equality

Since both sums equal $90^{\circ}$, $m\angle P + m\angle R=m\angle P + m\angle Q$.

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

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

Step5: Use definition of congruence

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

Answer:

The reason for statement 6 is "Subtraction Property of Equality" and the reason for statement 7 is "Definition of congruence".