Question

1. given: ∠pqr is a right - angle. prove: ∠pqs and ∠sqr are complementary. statements: 1. ∠pqr is a right - angle. 2. m∠pqr = 90°. 3. m∠pqs + m∠sqr = m∠pqr. 4. m∠pqs + m∠sqr = 90°. 5. ∠pqs and ∠sqr are complementary. 2. given: ∠2 = ∠3, ∠1 and ∠2 form a linear pair. prove: ∠1 and ∠3 are supplementary. statements: 1. ∠2 = ∠3. 2. m∠2 = m∠3. 3. ∠1 and ∠2 form a linear pair. 4. ∠1 and ∠2 are supplementary. 5. m∠1 + m∠2 = 180°. 6. m∠1 + m∠3 = 180°. 7. ∠1 and ∠3 are supplementary. 3. given: ∠1 and ∠2 form a right - angle, m∠1 + m∠3 = 90°. prove: ∠2 ≅ ∠3. statements: 1. ∠1 and ∠2 form a right - angle. 2. ∠1 and ∠2 are complementary. 3. m∠1 + m∠3 = 90°. 4. ∠1 and ∠3 are complementary. 5. ∠2 ≅ ∠3.

Explanation:

Problem 1:

Step1: Define right - angle

Given $\angle PQR$ is a right - angle. Reason: Given.

Step2: Right - angle measure

$m\angle PQR = 90^{\circ}$. Reason: Definition of a right - angle (a right - angle measures $90^{\circ}$).

Step3: Angle addition postulate

$m\angle PQS+m\angle SQR=m\angle PQR$. Reason: Angle addition postulate (if point $S$ lies in the interior of $\angle PQR$, then the measure of $\angle PQR$ is the sum of the measures of $\angle PQS$ and $\angle SQR$).

Step4: Substitute angle measure

$m\angle PQS + m\angle SQR=90^{\circ}$. Reason: Substitution property (since $m\angle PQR = 90^{\circ}$).

Step5: Define complementary angles

$\angle PQS$ and $\angle SQR$ are complementary. Reason: Definition of complementary angles (two angles are complementary if the sum of their measures is $90^{\circ}$).

Problem 2:

Step1: Given equal angles

$\angle2=\angle3$. Reason: Given.

Step2: Equal angles have equal measures

$m\angle2 = m\angle3$. Reason: Definition of congruent angles (if two angles are congruent, then their measures are equal).

Step3: Define linear pair

$\angle1$ and $\angle2$ form a linear pair. Reason: Given.

Step4: Linear - pair property

$\angle1$ and $\angle2$ are supplementary. Reason: If two angles form a linear pair, then they are supplementary.

Step5: Supplementary - angle sum

$m\angle1 + m\angle2=180^{\circ}$. Reason: Definition of supplementary angles (the sum of the measures of supplementary angles is $180^{\circ}$).

Step6: Substitute angle measure

$m\angle1 + m\angle3=180^{\circ}$. Reason: Substitution property (since $m\angle2 = m\angle3$).

Step7: Define supplementary angles

$\angle1$ and $\angle3$ are supplementary. Reason: Definition of supplementary angles (if the sum of the measures of two angles is $180^{\circ}$, then the two angles are supplementary).

Problem 3:

Step1: Given right - angle formation

$\angle1$ and $\angle2$ form a right - angle. Reason: Given.

Step2: Define complementary angles

$\angle1$ and $\angle2$ are complementary. Reason: If two angles form a right - angle, then they are complementary.

Step3: Given angle - sum equation

$m\angle1 + m\angle3=90^{\circ}$. Reason: Given.

Step4: Define complementary angles

$\angle1$ and $\angle3$ are complementary. Reason: Definition of complementary angles (the sum of their measures is $90^{\circ}$).

Step5: Congruence of complementary angles

$\angle2\cong\angle3$. Reason: If two angles are complementary to the same angle, then they are congruent.

Answer:

For problem 1:

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

For problem 2:

1. Given
2. Definition of congruent angles
3. Given
4. Linear - pair property
5. Definition of supplementary angles
6. Substitution property
7. Definition of supplementary angles

For problem 3:

1. Given
2. If two angles form a right - angle, they are complementary
3. Given
4. Definition of complementary angles
5. If two angles are complementary to the same angle, they are congruent