Question

1. choose a segment that is skew to ab
2. if g || f and m∠1 = x and m∠2 = 4x - 20. find the measure of ∠2?
3. if a || b and m∠1 = 4x - 1 and m∠2 = 6x - 11. find the measure of ∠1?

Explanation:

Step1: Use property of parallel - lines

When two parallel lines are cut by a transversal, corresponding angles (or alternate - interior/alternate - exterior angles depending on the orientation) are equal. In problem 16, since \(g\parallel f\), \(\angle1\) and \(\angle2\) are either corresponding or alternate - interior/exterior angles, so \(m\angle1=m\angle2\). We set up the equation \(x = 4x-20\).
\[

$$\begin{align*} 4x - x&=20\\ 3x&=20\\ x&=\frac{20}{3} \end{align*}$$

\]
Then \(m\angle2=4x - 20=4\times\frac{20}{3}-20=\frac{80 - 60}{3}=\frac{20}{3}\) (There is a mistake in the hand - written answer of 160. The correct way is as above).

In problem 17, since \(a\parallel b\), \(m\angle1=m\angle2\) (corresponding or alternate - interior/exterior angles). Set up the equation \(4x-1 = 6x - 11\).

Step2: Solve the equation for \(x\)

\[

$$\begin{align*} 6x-4x&=11 - 1\\ 2x&=10\\ x&=5 \end{align*}$$

\]

Step3: Find \(m\angle1\)

Substitute \(x = 5\) into the expression for \(m\angle1\). \(m\angle1=4x-1=4\times5-1=19\)

Answer:

For problem 16: \(\frac{20}{3}\)
For problem 17: \(19\)