Question

given: ∠4 ≅ ∠6
prove: ∠4 and ∠7 are right angles

statement
∠4 and ∠7 form a linear pair
∠4 and ∠7 are supplementary
∠4 ≅ ∠6
∠4 ≅ ∠7
∠0 ≅ ∠0
∠4 and ∠7 are right angles

reason
definition of linear pair
linear pair property
given
transitive property
transitive property
select

Explanation:

Step1: Use the property of linear pair

Since \(\angle4\) and \(\angle7\) form a linear pair, by the Linear Pair Property, \(\angle4+\angle7 = 180^{\circ}\), so \(\angle4\) and \(\angle7\) are supplementary.

Step2: Use the transitive property

Given \(\angle4\cong\angle6\) and \(\angle6\cong\angle7\) (because \(\angle4\cong\angle7\) from the previous transitive - like step, assume there is a relation with \(\angle6\) which is congruent to both). By the Transitive Property of Congruence, if \(\angle4\cong\angle6\) and \(\angle6\cong\angle7\), then \(\angle4\cong\angle7\).

Step3: Find the measure of the angles

Let \(m\angle4 = m\angle7=x\). Since \(\angle4+\angle7 = 180^{\circ}\), then \(x + x=180^{\circ}\), \(2x = 180^{\circ}\), \(x = 90^{\circ}\).

Answer:

In statement 5, \(\angle6\cong\angle7\) (because \(\angle4\cong\angle6\) (given) and \(\angle4\cong\angle7\) (from step 2)). In statement 6, the reason is "Congruent Supplementary Angles Theorem" (If two angles are congruent and supplementary, then each angle is a right angle).