Question

1. find the value of x in the following: (a) (b) (c) 14. in the figure given alongside, op and oq are opposite rays: (a) if x = 32°, what is the value of y? (b) if y = 24°, what is the value of x? 15. determine all the marked angles in the following figure in which l || m.

Explanation:

13 (a):

Step1: Set up equation

Since the sum of the two - angles is 180° (linear - pair of angles), we have the equation \(x+(x - 44)=180\).

$$x+x - 44 = 180$$

Step2: Combine like - terms

$$2x-44 = 180$$

Step3: Add 44 to both sides

$$2x=180 + 44$$

$$2x=224$$

Step4: Divide both sides by 2

$$x=\frac{224}{2}=112$$

13 (b):

Step1: Set up equation

Since the sum of the two angles is 180° (linear - pair of angles), we have the equation \(x+(2x + 24)=180\).

$$x+2x+24 = 180$$

Step2: Combine like - terms

$$3x+24 = 180$$

Step3: Subtract 24 from both sides

$$3x=180 - 24$$

$$3x=156$$

Step4: Divide both sides by 3

$$x=\frac{156}{3}=52$$

13 (c):

Step1: Set up equation

Since the sum of the two angles is 90° (complementary angles), we have the equation \(x+2x=90\).

$$3x=90$$

Step2: Divide both sides by 3

$$x = 30$$

14 (a):

Step1: Use linear - pair property

Since OP and OQ are opposite rays, \(3x+(2y + 24)=180\). Given \(x = 32\), substitute \(x\) into the equation:

$$3\times32+(2y + 24)=180$$

$$96+2y+24 = 180$$

Step2: Combine like - terms

$$2y+120 = 180$$

Step3: Subtract 120 from both sides

$$2y=180 - 120$$

$$2y=60$$

Step4: Divide both sides by 2

$$y = 30$$

14 (b):

Step1: Use linear - pair property

Since \(3x+(2y + 24)=180\) and \(y = 24\), substitute \(y\) into the equation:

$$3x+2\times24 + 24=180$$

$$3x+48+24 = 180$$

$$3x+72 = 180$$

Step2: Subtract 72 from both sides

$$3x=180 - 72$$

$$3x=108$$

Step3: Divide both sides by 3

$$x = 36$$

15:

Step1: Use corresponding angles and alternate - interior angles properties

Since \(l\parallel m\), the corresponding angle to the 60° angle is also 60°.
The alternate - interior angle to the 50° angle is 50°.
For angle \(a\):
The angle adjacent to the 60° angle and \(a\) forms a straight - line, so \(a=180 - 60=120\).
For angle \(b\):
The angle adjacent to the 50° angle and \(b\) forms a straight - line, so \(b = 180-50 = 130\).

Answer:

13 (a): \(x = 112\)
13 (b): \(x = 52\)
13 (c): \(x = 30\)
14 (a): \(y = 30\)
14 (b): \(x = 36\)
15: \(a = 120^{\circ}\), \(b = 130^{\circ}\), the other marked angles are 60° and 50°