Question

write the letter of the property, definition, or postulate that justifies each statement.

1. if (mangle abc = 90^{circ}), then (angle abc) is a right angle.
2. if (mangle3 + mangle4=180^{circ}), then (angle3) and (angle4) are supplementary angles.
3. if (mangle pqr = mangle rst), then (angle pqrcongangle rst)
4. if (angle1) and (angle2) form a right angle, then (angle1) and (angle2) are complementary angles.
5. if (angle x) and (angle y) are supplementary and (angle x) and (angle z) are supplementary, then (angle ycongangle z)
6. if (angle j) and (angle k) are vertical angles, then (angle jcongangle k)
7. if (angle pqr) and (angle stu) are complementary angles, then (mangle pqr + mangle stu = 90^{circ}).
8. if (angle5) and (angle6) form a linear, then (angle5) and (angle6) are supplementary angles.

complete the proofs below by filling in the missing statements and reasons.

1. given: (angle1) and (angle2) form a linear pair, (angle1congangle3) prove: (angle2) and (angle3) are supplementary

statements reasons

1. (angle1) and (angle2) form a linear pair 1.
2. (angle1) and (angle2) are supplementary 2.
3. (mangle1 + mangle2 = 180^{circ}) 3.
4. (angle1congangle3) 4.
5. (mangle1=mangle3) 5.
6. (mangle3 + mangle2 = 180^{circ}) 6.
7. (angle2) and (angle3) are supplementary 7.
8. given: (angle pqr) is a right angle, (angle tqp) and (angle sqr) are complementary angles prove: (angle pqscongangle tqp)

statements reasons

1. (angle pqr) is a right angle 1.
2. (mangle pqr = 90^{circ}) 2.
3. (mangle pqs + mangle sqr=angle pqr) 3.
4. (mangle pqs + mangle sqr = 90^{circ}) 4.
5. (angle pqs) and (angle sqr) are complementary angles 5.
6. (angle tqp) and (angle sqr) are complementary angles 6.
7. (angle pqscongangle tqp) 7.

Explanation:

Step1: Identify the property for 11

If \(m\angle ABC = 90^{\circ}\), then \(\angle ABC\) is a right - angle by the definition of a right - angle. So the answer for 11 is E.

Step2: Identify the property for 12

If \(m\angle3 + m\angle4=180^{\circ}\), then \(\angle3\) and \(\angle4\) are supplementary angles by the definition of supplementary angles. So the answer for 12 is C.

Step3: Identify the property for 13

If \(m\angle PQR=m\angle RST\), then \(\angle PQR\cong\angle RST\) by the definition of congruence. So the answer for 13 is A.

Step4: Identify the property for 14

If \(\angle1\) and \(\angle2\) form a right - angle, then \(\angle1\) and \(\angle2\) are complementary angles by the definition of complementary angles. So the answer for 14 is B.

Step5: Identify the property for 15

If \(\angle X\) and \(\angle Y\) are supplementary and \(\angle X\) and \(\angle Z\) are supplementary, then \(\angle Y\cong\angle Z\) by the congruent supplements theorem. So the answer for 15 is L.

Step6: Identify the property for 16

If \(\angle J\) and \(\angle K\) are vertical angles, then \(\angle J\cong\angle K\) by the vertical angles theorem. So the answer for 16 is H.

Step7: Identify the property for 17

If \(\angle PQR\) and \(\angle STU\) are complementary angles, then \(m\angle PQR + m\angle STU = 90^{\circ}\) by the definition of complementary angles. So the answer for 17 is B.

Step8: Identify the property for 18

If \(\angle5\) and \(\angle6\) form a linear pair, then \(\angle5\) and \(\angle6\) are supplementary angles by the definition of a linear pair (supplement). So the answer for 18 is J.

Step9: Complete the proof for 19

1. Given
2. Definition of a linear pair
3. Definition of supplementary angles (\(m\angle1 + m\angle2=180^{\circ}\) since \(\angle1\) and \(\angle2\) are supplementary)
4. Given
5. Definition of congruent angles (\(\angle1\cong\angle3\) implies \(m\angle1 = m\angle3\))
6. Substitution property of equality (\(m\angle1\) replaced with \(m\angle3\) in \(m\angle1 + m\angle2 = 180^{\circ}\))
7. Definition of supplementary angles (\(m\angle3 + m\angle2=180^{\circ}\) implies \(\angle2\) and \(\angle3\) are supplementary)

Step10: Complete the proof for 20

1. Given
2. Definition of a right - angle (\(\angle PQR\) is a right - angle implies \(m\angle PQR = 90^{\circ}\))
3. Angle addition postulate (\(m\angle PQS+m\angle SQR=\angle PQR\))
4. Substitution property of equality (\(m\angle PQR = 90^{\circ}\) into \(m\angle PQS + m\angle SQR=\angle PQR\))
5. Definition of complementary angles (\(m\angle PQS + m\angle SQR = 90^{\circ}\) implies \(\angle PQS\) and \(\angle SQR\) are complementary)
6. Given
7. Congruent complements theorem (since \(\angle PQS\) and \(\angle SQR\) are complementary and \(\angle TQP\) and \(\angle SQR\) are complementary, then \(\angle PQS\cong\angle TQP\))

Answer:

1. E
2. C
3. A
4. B
5. L
6. H
7. B
8. J
9. Statements: 1. Given; 2. Definition of a linear pair; 3. Definition of supplementary angles; 4. Given; 5. Definition of congruent angles; 6. Substitution property of equality; 7. Definition of supplementary angles
10. Statements: 1. Given; 2. Definition of a right - angle; 3. Angle addition postulate; 4. Substitution property of equality; 5. Definition of complementary angles; 6. Given; 7. Congruent complements theorem