Question

for questions 9 - 11, find the values of x and y if l || m.

1. (10x - 17)° (6y + 29)° (8x + 1)°
2. (3y + 11)° (7x - 30)° (5x + 14)°
3. (7y - 23)° (23x - 16)° (8x - 21)°

topic 4: parallel & perpendicular lines on the coordinate plane
know your formulas!
slope formula: m = \\(\frac{y_2 - y_1}{x_2 - x_1}\\) slope - intercept form: y = mx + b
standard form: ax + by = c point - slope formula: y - y_1 = m(x - x_1)
describe the slopes of parallel lines: same
describe the slopes of perpendicular lines: negative reciprocal

Explanation:

Question 9

Step1: Use corresponding - angles property

Since \(l\parallel m\), the corresponding angles are equal. So \(10x - 17=8x + 1\).
Solve for \(x\):
\[

$$\begin{align*} 10x-8x&=1 + 17\\ 2x&=18\\ x&=9 \end{align*}$$

\]

Step2: Use linear - pair property

The angles \((6y + 29)^{\circ}\) and \((8x + 1)^{\circ}\) form a linear - pair. Substitute \(x = 9\) into \(8x+1\), we get \(8\times9 + 1=73^{\circ}\).
Then \(6y+29 + 73=180\) (because linear - pair angles sum to \(180^{\circ}\)).
\[

$$\begin{align*} 6y+102&=180\\ 6y&=180 - 102\\ 6y&=78\\ y&=13 \end{align*}$$

\]

Question 10

Step1: Use corresponding - angles property

Since \(l\parallel m\), the corresponding angles are equal. So \(7x-30 = 5x+14\).
Solve for \(x\):
\[

$$\begin{align*} 7x-5x&=14 + 30\\ 2x&=44\\ x&=22 \end{align*}$$

\]

Step2: Use linear - pair property

The angles \((3y + 11)^{\circ}\) and \((7x-30)^{\circ}\) form a linear - pair. Substitute \(x = 22\) into \(7x-30\), we get \(7\times22-30=124^{\circ}\).
Then \(3y + 11+124 = 180\) (because linear - pair angles sum to \(180^{\circ}\)).
\[

$$\begin{align*} 3y+135&=180\\ 3y&=180 - 135\\ 3y&=45\\ y&=15 \end{align*}$$

\]

Question 11

Step1: Use same - side interior angles property

Since \(l\parallel m\), the same - side interior angles sum to \(180^{\circ}\). So \((23x-16)+(8x - 21)=180\).
\[

$$\begin{align*} 23x+8x-16 - 21&=180\\ 31x-37&=180\\ 31x&=180 + 37\\ 31x&=217\\ x&=7 \end{align*}$$

\]

Step2: Use vertical - angles property

The angle \((7y-23)^{\circ}\) and the angle vertical to \((23x - 16)^{\circ}\) are equal. Substitute \(x = 7\) into \(23x-16\), we get \(23\times7-16 = 145^{\circ}\).
Then \(7y-23=145\).
\[

$$\begin{align*} 7y&=145 + 23\\ 7y&=168\\ y&=24 \end{align*}$$

\]

Answer:

Question 9: \(x = 9\), \(y = 13\)
Question 10: \(x = 22\), \(y = 15\)
Question 11: \(x = 7\), \(y = 24\)