Question

for questions 18 - 20, if l || m, find the values of x and y. 18. (3x - 16)° (11y - 32)° (6x + 7)° 19. (8x - 14)° (5y + 16)° (5x + 34)° 20. (5y - 23)° (2x + 13)° 47° (3x)°

Explanation:

Step1: Identify angle - relationship for x in question 18

Since \(l\parallel m\), the angles \((3x - 16)^{\circ}\) and \((6x+7)^{\circ}\) are corresponding angles, so \(3x - 16=6x + 7\).
\[

$$\begin{align*} 3x-6x&=7 + 16\\ -3x&=23\\ x&=-\frac{23}{3} \end{align*}$$

\]
The angles \((11y - 32)^{\circ}\) and \((6x + 7)^{\circ}\) are also corresponding angles. Substitute \(x =-\frac{23}{3}\) into \(6x + 7\): \(6\times(-\frac{23}{3})+7=-46 + 7=-39\). Then \(11y-32=-39\), \(11y=-39 + 32=-7\), \(y =-\frac{7}{11}\)

Step2: Identify angle - relationship for x in question 19

Since \(l\parallel m\), the angles \((8x - 14)^{\circ}\) and \((5x+34)^{\circ}\) are corresponding angles. So \(8x-14 = 5x+34\).
\[

$$\begin{align*} 8x-5x&=34 + 14\\ 3x&=48\\ x&=16 \end{align*}$$

\]
The angles \((5y + 16)^{\circ}\) and \((5x + 34)^{\circ}\) are corresponding angles. Substitute \(x = 16\) into \(5x+34\): \(5\times16+34=80 + 34 = 114\). Then \(5y+16=114\), \(5y=114 - 16 = 98\), \(y=\frac{98}{5}\)

Step3: Identify angle - relationship for x in question 20

Since \(l\parallel m\), the angles \((2x + 13)^{\circ}\) and \((3x)^{\circ}\) are alternate - interior angles. So \(2x+13=3x\), \(3x-2x=13\), \(x = 13\)
The angles \((5y-23)^{\circ}\) and \((3x)^{\circ}\) are corresponding angles. Substitute \(x = 13\) into \(3x\): \(3\times13 = 39\). Then \(5y-23=39\), \(5y=39 + 23=62\), \(y=\frac{62}{5}\)

Answer:

1. \(x=-\frac{23}{3}\), \(y =-\frac{7}{11}\)
2. \(x = 16\), \(y=\frac{98}{5}\)
3. \(x = 13\), \(y=\frac{62}{5}\)