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°