Question

1.6 linear pairs & vertical angles homework
page 50&51 (14,16,20,22,24,26,28,32,34,37 - 42)
for questions 14&16, use the diagram.

1. identify all the linear pairs that include ∠7.
2. are ∠2 and ∠5 vertical angles? explain your reasoning.

in exercises 20 - 24, find the measure of each ∠.

1. two angles form a linear pair. the measure of one angle is twice the measure of the other angle.
2. the measure of an angle is 9 times the measure of its complement.
3. the ratio of the measure of an angle to the measure of its complement is 2:7.
4. m∠2 = 24°. find the measure of the supplement of ∠2.
5. the arm of a crossing - gate moves 42° from a vertical position. how many more degrees does the arm have to move so that it is horizontal?
6. the measure of an angle is 6° less than the measure of its complement.
7. two angles form a linear pair. the measure of one angle is 15° less than ⅔ the measure of the other angle.

Explanation:

Step1: Recall linear - pair and vertical - angle definitions

A linear pair of angles are adjacent angles that form a straight line (sum to 180°). Vertical angles are non - adjacent angles formed by two intersecting lines and are congruent.

Step2: Solve question 14

From the diagram, the linear pairs that include ∠7 are ∠6 and ∠7, ∠7 and ∠8.

Step3: Solve question 16

∠2 and ∠5 are not vertical angles. Vertical angles are formed by two intersecting lines. ∠2 and ∠5 are not formed by the intersection of two lines in a way that they are non - adjacent and opposite each other.

Step4: Solve question 20

Let one angle be \(x\) and the other be \(2x\). Since they form a linear pair, \(x + 2x=180^{\circ}\). Combining like terms gives \(3x = 180^{\circ}\), so \(x = 60^{\circ}\) and \(2x=120^{\circ}\).

Step5: Solve question 22

Let the angle be \(x\) and its complement be \(90 - x\). Given \(x = 9(90 - x)\). Expand: \(x=810 - 9x\). Add \(9x\) to both sides: \(10x = 810\), so \(x = 81^{\circ}\) and its complement is \(9^{\circ}\).

Step6: Solve question 24

Let the angle be \(2x\) and its complement be \(7x\). Then \(2x+7x = 90^{\circ}\), \(9x = 90^{\circ}\), \(x = 10^{\circ}\), so the angle is \(20^{\circ}\).

Step7: Solve question 26

The supplement of an angle \(\theta\) is \(180^{\circ}-\theta\). Given \(\theta = 24^{\circ}\), the supplement is \(180 - 24=156^{\circ}\).

Step8: Solve question 28

A vertical position is \(90^{\circ}\) from a horizontal position. If the arm has moved \(42^{\circ}\) from the vertical, then it needs to move \(90 - 42=48^{\circ}\) more to be horizontal.

Step9: Solve question 32

Let the angle be \(x\) and its complement be \(90 - x\). Given \(x=(90 - x)-6\). Expand: \(x = 90 - x-6\). Add \(x\) to both sides: \(2x=84\), so \(x = 42^{\circ}\).

Step10: Solve question 34

Let one angle be \(x\) and the other be \(y\). Since they form a linear pair, \(x + y=180^{\circ}\), and \(x=\frac{2}{3}y-15^{\circ}\). Substitute \(x\) into the first equation: \(\frac{2}{3}y-15^{\circ}+y = 180^{\circ}\). Combine like terms: \(\frac{2y+3y}{3}=195^{\circ}\), \(\frac{5y}{3}=195^{\circ}\), \(y = 117^{\circ}\), then \(x = 63^{\circ}\).

Answer:

1. ∠6 and ∠7, ∠7 and ∠8
2. No. They are not formed by two intersecting lines as vertical - angle definition requires.
3. 60° and 120°
4. 81°
5. 20°
6. 156°
7. 48°
8. 42°
9. 63° and 117°