Question

complete the statement.

1. two angles are complementary if the sum of their measures is?°.
2. two angles are supplementary if the sum of their measures is?°.
3. if two angles share a common vertex and side, but have no common interior points, then the two angles are? angles.
4. a true statement that follows from other true statements is called a?

determine whether the angles whose measures are given are complementary, supplementary, or neither. also tell whether the angles are adjacent or nonadjacent.

1. image with angles 138° and 42°
2. image with angles 49° and 31°
3. image with angles 35° and 35°

find the measure of a complement of the angle.

1. (mangle y = 40^{circ})
2. (mangle k = 12^{circ})
3. (mangle p = 64^{circ})
4. (mangle t = 85^{circ})

find the measure of a supplement of the angle.

1. (mangle a = 54^{circ})
2. (mangle r = 115^{circ})
3. (mangle z = 22^{circ})
4. (mangle f = 90^{circ})

(angle pqs) and (angle sqr) are complementary angles. find the value of the variable.

1. image with angles (2x^{circ}) and (x^{circ})
2. image with angles ((4x + 10)^{circ}) and (4x^{circ})
3. image with angles ((3x + 10)^{circ}) and (x^{circ})
4. (angle abc) and (angle dbe) are right angles. name an angle that is congruent to (angle3). explain.

in exercises 20 and 21, use the drawing of a teeter - totter.

1. the marked angles are supplementary. find the value of (x).
2. by how many degrees would the angle of the teeter - totter have to change so that it forms a right angle with its vertical support bar? (hint: find the measure of a complement of a (74^{circ}) angle.)

Explanation:

Step1: Recall angle - related definitions

Complementary angles have a sum of 90°. Supplementary angles have a sum of 180°. Adjacent angles share a common vertex and side with no common interior points. A theorem is a true statement that follows from other true statements.

1. The sum of complementary angles is 90°.
2. The sum of supplementary angles is 180°.
3. If two angles share a common vertex and side, but have no common interior points, they are adjacent angles.
4. A true statement that follows from other true statements is called a theorem.

Step2: Analyze angle - pair relationships

1. For 138° and 42°, \(138 + 42=180\), so they are supplementary. They share a common vertex and side, so they are adjacent.
2. For 49° and 31°, \(49+31 = 80

eq90\) and \(80
eq180\), so they are neither complementary nor supplementary. They share a common vertex and side, so they are adjacent.

1. The two 35° angles are vertical angles (opposite each other when two lines intersect). They are neither complementary (\(35 + 35=70

eq90\)) nor supplementary (\(35+35 = 70
eq180\)), and they are non - adjacent.

Step3: Calculate complementary and supplementary angles

1. Complement of \(\angle Y\) with \(m\angle Y = 40^{\circ}\): \(90 - 40=50^{\circ}\)
2. Complement of \(\angle K\) with \(m\angle K = 12^{\circ}\): \(90 - 12 = 78^{\circ}\)
3. Complement of \(\angle P\) with \(m\angle P = 64^{\circ}\): \(90 - 64=26^{\circ}\)
4. Complement of \(\angle T\) with \(m\angle T = 85^{\circ}\): \(90 - 85 = 5^{\circ}\)
5. Supplement of \(\angle A\) with \(m\angle A = 54^{\circ}\): \(180 - 54=126^{\circ}\)
6. Supplement of \(\angle R\) with \(m\angle R = 115^{\circ}\): \(180 - 115 = 65^{\circ}\)
7. Supplement of \(\angle Z\) with \(m\angle Z = 22^{\circ}\): \(180 - 22=158^{\circ}\)
8. Supplement of \(\angle F\) with \(m\angle F = 90^{\circ}\): \(180 - 90 = 90^{\circ}\)

Step4: Solve for variables in complementary - angle problems

1. Since \(\angle PQS\) and \(\angle SQR\) are complementary, \(2x+x=90\), \(3x = 90\), \(x = 30\)
2. Since \(\angle PQS\) and \(\angle SQR\) are complementary, \(4x+(4x + 10)=90\), \(8x+10 = 90\), \(8x=80\), \(x = 10\)
3. Since \(\angle PQS\) and \(\angle SQR\) are complementary, \((3x + 10)+x=90\), \(4x+10 = 90\), \(4x=80\), \(x = 20\)

Step5: Identify congruent angles

1. Since \(\angle ABC\) and \(\angle DBE\) are right angles (\(90^{\circ}\)), \(\angle 1+\angle 2=90^{\circ}\) and \(\angle 2+\angle 3=90^{\circ}\). By the angle - addition postulate and the subtraction property of equality, \(\angle 1\) is congruent to \(\angle 3\) (because \(\angle 1=90^{\circ}-\angle 2\) and \(\angle 3=90^{\circ}-\angle 2\))

Step6: Solve for \(x\) in supplementary - angle problem

1. The marked angles are supplementary, so \(x + 74=180\), \(x=180 - 74 = 106\)

Step7: Calculate angle - change for right - angle formation

1. The complement of a \(74^{\circ}\) angle is \(90 - 74 = 16^{\circ}\), so the angle of the teeter - totter has to change by \(16^{\circ}\) to form a right angle with its vertical support bar.

Answer:

1. 90
2. 180
3. adjacent
4. theorem
5. Supplementary, adjacent
6. Neither, adjacent
7. Neither, non - adjacent
8. \(50^{\circ}\)
9. \(78^{\circ}\)
10. \(26^{\circ}\)
11. \(5^{\circ}\)
12. \(126^{\circ}\)
13. \(65^{\circ}\)
14. \(158^{\circ}\)
15. \(90^{\circ}\)
16. \(x = 30\)
17. \(x = 10\)
18. \(x = 20\)
19. \(\angle 1\), because \(\angle 1\) and \(\angle 3\) are both complementary to \(\angle 2\)
20. \(x = 106\)
21. \(16^{\circ}\)