Question

given: ∠3 ≅ ∠5
prove: ∠3 and ∠6 are right angles
1 ∠3 and ∠6 form a linear pair
2 ∠3 and ∠6 are supplementary
3 ∠3 ≅ ∠5
4 ∠□ ≅ ∠□
5 ∠□ ≅ ∠□
6 ∠3 and ∠6 are right angles

Explanation:

Step1: Use the definition of supplementary angles

Since \(\angle3\) and \(\angle6\) form a linear pair, by the definition of supplementary angles, \(\angle3+\angle6 = 180^{\circ}\).

Step2: Use the property of congruent angles

Given \(\angle3\cong\angle5\), and \(\angle5\) and \(\angle6\) are vertical angles (assuming the geometric figure context), so \(\angle5\cong\angle6\) (vertical angles are congruent).

Step3: Apply the transitive property

Since \(\angle3\cong\angle5\) and \(\angle5\cong\angle6\), by the transitive property of congruence, \(\angle3\cong\angle6\).

Step4: Solve for the measure of the angles

Let \(m\angle3 = m\angle6=x\). Then from \(\angle3+\angle6 = 180^{\circ}\), we have \(x + x=180^{\circ}\), \(2x = 180^{\circ}\), \(x = 90^{\circ}\).

Answer:

1. For statement 2: Definition of supplementary angles.
2. For statement 4: \(\angle5\cong\angle6\) (vertical angles are congruent).
3. For statement 5: \(\angle3\cong\angle6\) (transitive property).
4. For statement 6: Since \(m\angle3=m\angle6 = 90^{\circ}\), \(\angle3\) and \(\angle6\) are right angles (definition of right angles: an angle with measure \(90^{\circ}\) is a right angle).