Question

tell whether the angles are only adjacent, adjacent and form a linear pair, or not adjacent.

1. ∠1 and ∠2
2. ∠1 and ∠3
3. ∠2 and ∠4
4. ∠2 and ∠3

find the measure of each of the following.

1. supplement of ∠a
2. complement of ∠a
3. supplement of ∠b
4. complement of ∠b
5. multi - step an angles measure is 6 degrees more than 3 times the measure of its complement. find the measure of the angle.
6. landscaping a sprinkler swings back and forth between a and b in such a way that ∠1≅∠2. ∠1 and ∠3 are complementary, and ∠2 and ∠4 are complementary. if m∠1 = 47.5°, find m∠2, m∠3, and m∠4.
7. name each pair of vertical angles.

Explanation:

Step1: Recall angle - related definitions

Adjacent angles share a common side and a common vertex. A linear - pair of adjacent angles sum to 180 degrees. Complementary angles sum to 90 degrees and supplementary angles sum to 180 degrees. Vertical angles are opposite each other when two lines intersect.

Step2: Solve for question 7

The measure of angle \(A = 81.2^{\circ}\). The supplement of an angle \(\theta\) is \(180^{\circ}-\theta\). So the supplement of \(\angle A\) is \(180 - 81.2=98.8^{\circ}\).

Step3: Solve for question 8

The complement of an angle \(\theta\) is \(90^{\circ}-\theta\). So the complement of \(\angle A\) is \(90 - 81.2 = 8.8^{\circ}\).

Step4: Solve for question 9

Let the measure of \(\angle B=(6x - 5)^{\circ}\). First, we need to find \(x\) (not given in enough context here, assuming we just work with the supplement formula). The supplement of \(\angle B\) is \(180-(6x - 5)=180 - 6x+5=185 - 6x\) degrees.

Step5: Solve for question 10

The complement of \(\angle B\) is \(90-(6x - 5)=90 - 6x + 5=95 - 6x\) degrees.

Step6: Solve for question 11

Let the angle be \(y\) and its complement be \(90 - y\). We know that \(y=3(90 - y)+6\).
Expand the right - hand side: \(y = 270-3y + 6\).
Add \(3y\) to both sides: \(y+3y=270 + 6\).
\(4y=276\).
Divide both sides by 4: \(y = 69^{\circ}\).

Step7: Solve for question 12

Since \(\angle1\cong\angle2\) and \(m\angle1 = 47.5^{\circ}\), then \(m\angle2=47.5^{\circ}\).
Since \(\angle1\) and \(\angle3\) are complementary, and \(m\angle1 = 47.5^{\circ}\), then \(m\angle3=90 - 47.5 = 42.5^{\circ}\).
Since \(\angle2\) and \(\angle4\) are complementary, and \(m\angle2 = 47.5^{\circ}\), then \(m\angle4=90 - 47.5 = 42.5^{\circ}\).

Step8: Solve for question 13

When two lines \(AC\) and \(ED\) intersect at point \(B\), the vertical - angle pairs are \(\angle ABE\) and \(\angle CBD\), and \(\angle ABC\) and \(\angle EBD\).

Answer:

1. \(98.8^{\circ}\)
2. \(8.8^{\circ}\)
3. \(185 - 6x\) degrees
4. \(95 - 6x\) degrees
5. \(69^{\circ}\)
6. \(m\angle2 = 47.5^{\circ}\), \(m\angle3 = 42.5^{\circ}\), \(m\angle4 = 42.5^{\circ}\)
7. \(\angle ABE\) and \(\angle CBD\), \(\angle ABC\) and \(\angle EBD\)