Question

complete the proofs below by giving the missing statements and reasons.
1 given: \\( \angle p q r \\) is a right angle
prove: \\( \angle p q s \\) and \\( \angle s q r \\) are complementary
2 given: \\( \angle 2 \cong \angle 3 ; \angle 1 \\) and \\( \angle 2 \\) form a linear pair
prove: \\( \angle 1 \\) and \\( \angle 3 \\) are supplementary
3 given: \\( \angle 1 \\) and \\( \angle 2 \\) form a right angle; \\( m \angle 1 + m \angle 3 = 90 ^ { \circ } \\)
prove: \\( \angle 2 \cong \angle 3 \\)

Explanation:

1.

Step1: Given

Statement 1 is given in the problem.

Step2: Definition of a right angle

A right angle has a measure of \(90^{\circ}\).

Step3: Angle - addition postulate

If \(S\) is a point in the interior of \(\angle PQR\), then \(m\angle PQS + m\angle SQR=m\angle PQR\)

Step4: Substitution property of equality

Since \(m\angle PQR = 90^{\circ}\) (from step 2) and \(m\angle PQS + m\angle SQR=m\angle PQR\) (from step 3), we substitute \(m\angle PQR\) with \(90^{\circ}\)

Step5: Definition of complementary angles

Two angles are complementary if the sum of their measures is \(90^{\circ}\)

2.

Step1: Given

Statement 1 is given in the problem.

Step2: Definition of congruent angles

If \(\angle2\cong\angle3\), then \(m\angle2 = m\angle3\)

Step3: Given

Statement 3 is given in the problem.

Step4: Definition of a linear pair

If two angles form a linear pair, then they are supplementary

Step5: Definition of supplementary angles

If \(\angle1\) and \(\angle2\) are supplementary, then \(m\angle1 + m\angle2=180^{\circ}\)

Step6: Substitution property of equality

Since \(m\angle2 = m\angle3\) (from step 2) and \(m\angle1 + m\angle2 = 180^{\circ}\) (from step 5), we substitute \(m\angle2\) with \(m\angle3\)

Step7: Definition of supplementary angles

If \(m\angle1 + m\angle3=180^{\circ}\), then \(\angle1\) and \(\angle3\) are supplementary

3.

Step1: Given

Statement 1 is given in the problem.

Step2: Definition of complementary angles

If two angles form a right - angle, then they are complementary

Step3: Given

Statement 3 is given in the problem.

Step4: Definition of complementary angles

If \(m\angle1 + m\angle3 = 90^{\circ}\), then \(\angle1\) and \(\angle3\) are complementary

Step5: Congruent complements theorem

If \(\angle1\) and \(\angle2\) are complementary (\(m\angle1 + m\angle2=90^{\circ}\)) and \(\angle1\) and \(\angle3\) are complementary (\(m\angle1 + m\angle3 = 90^{\circ}\)), then \(\angle2\cong\angle3\)

Answer:

1.

1. Given
2. Definition of a right angle
3. Angle - addition postulate
4. Substitution property of equality
5. Definition of complementary angles

2.

1. Given
2. Definition of congruent angles
3. Given
4. Definition of a linear pair
5. Definition of supplementary angles
6. Substitution property of equality
7. Definition of supplementary angles

3.

1. Given
2. Definition of complementary angles
3. Given
4. Definition of complementary angles
5. Congruent complements theorem