Question

use the given information to prove that \\( \angle 3 \\) and \\( \angle 6 \\) are right angles.

given: \\( \angle 3 \cong \angle 5 \\)
prove: \\( \angle 3 \\) and \\( \angle 6 \\) are right angles

statement	reason
2	\\( \angle \square \\) and \\( \angle \square \\) are supplementary	select
3	\\( \angle 3 \cong \angle 5 \\)	given

Explanation:

Step1: <A Short Explanation of the Step>

Use the property of linear pair.
If two angles form a linear pair, then they are supplementary. Since \(∠3\) and \(∠6\) form a linear pair (from statement 1), by the Linear Pair Postulate, \(∠3\) and \(∠6\) are supplementary.

Step2: <A Short Explanation of the Step>

Use the property of vertical angles.
\(∠5\) and \(∠6\) are vertical angles. By the Vertical Angles Theorem, \(∠5\cong∠6\).

Step3: <A Short Explanation of the Step>

Use substitution and the definition of right - angle.
Given \(∠3\cong∠5\), and since \(∠5\cong∠6\), then \(∠3\cong∠6\) (Transitive Property of Congruence). Also, \(∠3 + ∠6=180^{\circ}\) (because they are supplementary). Let \(m∠3 = m∠6=x\). Then \(x + x=180^{\circ}\), \(2x = 180^{\circ}\), \(x = 90^{\circ}\).

For statement 2:
Since \(∠3\) and \(∠6\) form a linear pair (statement 1), by the Linear Pair Postulate, \(∠3\) and \(∠6\) are supplementary.

Answer:

1. \(∠3\) and \(∠6\) are supplementary (Reason: Linear Pair Postulate)