Question

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

statement	reason
2 \\( \angle \square \\) and \\( \angle \square \\) are supplementary	select
3 \\( \angle 4 \cong \angle 6 \\)	given
4 \\( \angle \square \cong \angle \square \\)	select
5 \\( \angle \square \cong \angle \square \\)	transitive property
6 \\( \angle 4 \\) and \\( \angle 7 \\) are right angles	select

Explanation:

Step1: Fill in the blanks for statement 2

Since \(\angle4\) and \(\angle7\) form a linear pair (statement 1), by the Linear Pair Postulate, \(\angle4\) and \(\angle7\) are supplementary.

Step2: Fill in the blanks for statement 4

\(\angle6\) and \(\angle7\) are supplementary (because \(\angle4\) and \(\angle7\) are supplementary from step 2 and \(\angle4\cong\angle6\) (given)).

Step3: Fill in the blanks for statement 5

From step 2 (\(\angle4+\angle7 = 180^{\circ}\)) and step 4 (\(\angle6+\angle7=180^{\circ}\)), we can say \(\angle4\cong\angle7\) (if two angles are supplementary to the same angle, then they are congruent).

Step4: Use the transitive property

Since \(\angle4\cong\angle6\) (given) and \(\angle4\cong\angle7\) (from step 3), by the transitive property \(\angle6\cong\angle7\)

Step5: Conclude that \(\angle4\) and \(\angle7\) are right angles

Since \(\angle4\) and \(\angle7\) are congruent and supplementary (\(\angle4+\angle7 = 180^{\circ}\)), each angle is \(90^{\circ}\) (if two congruent angles are supplementary, then each is a right angle)

Answer:

1. \(\angle4\) and \(\angle7\) are supplementary (Reason: Linear Pair Postulate)
2. \(\angle6\cong\angle7\) (Reason: If two angles are supplementary to the same angle, then they are congruent)
3. \(\angle4\cong\angle7\) (using the transitive property with \(\angle4\cong\angle6\) (given) and \(\angle6\cong\angle7\) (from step 4))
4. Reason: If two congruent angles are supplementary, then each is a right angle