Question

given: \\( \angle 4 \cong \angle 6 \\) prove: \\( \angle 4 \\) and \\( \angle 7 \\) are right angles

Explanation:

Step1: Identify supplementary angles

Since \(\angle4\) and \(\angle7\) form a linear pair, by the Linear Pair Postulate, \(\angle4\) and \(\angle7\) are supplementary.

Step2: Use vertical - angle property

\(\angle6\) and \(\angle7\) are vertical angles. By the Vertical Angles Theorem, \(\angle6\cong\angle7\).

Step3: Apply transitive property

Given \(\angle4\cong\angle6\) (from step 3 in the problem) and \(\angle6\cong\angle7\) (from step 2), by the Transitive Property of Congruence, \(\angle4\cong\angle7\).

Answer:

1. \(\angle4\) and \(\angle7\) are supplementary (Reason: Linear Pair Postulate)
2. \(\angle6\cong\angle7\) (Reason: Vertical Angles Theorem)
3. \(\angle4\cong\angle7\) (Since \(\angle4\cong\angle6\) (given) and \(\angle6\cong\angle7\) (from step 4), by Transitive Property)