Question

10 explain why a translation of a figure results in another figure that is congruent to the original.
11 in the diagram below, transversal tu intersects pq and rs at v and w, respectively.
if m∠tvq = 5x - 22 and m∠vws = 3x + 10, for which value of x is pq || rs?

1. 6
2. 16
3. 24
4. 28

12 in the diagram below, line l is parallel to line k and line t is a transversal. measure of ∠1 is 121°.
find each angle measure:
∠2 =
∠3 =
∠4 =
∠5 =
∠6 =
∠7 =
∠8 =

Explanation:

Step1: Recall angle - pair relationships

When two parallel lines are cut by a transversal, corresponding angles are congruent, vertical angles are congruent, and adjacent angles are supplementary (sum to 180°).

Step2: Solve for \(x\) in question 11

If \(\overline{PQ}\parallel\overline{RS}\), then \(\angle TVQ\) and \(\angle VWS\) are corresponding angles and are congruent. So, \(5x - 22=3x + 10\).
Subtract \(3x\) from both sides: \(5x-3x - 22=3x - 3x+10\), which simplifies to \(2x-22 = 10\).
Add 22 to both sides: \(2x-22 + 22=10 + 22\), so \(2x=32\).
Divide both sides by 2: \(x = 16\).

Step3: Find angle measures in question 12

Since \(\angle1 = 121^{\circ}\), \(\angle2\) is adjacent to \(\angle1\), so \(\angle1+\angle2 = 180^{\circ}\), then \(\angle2=180 - 121=59^{\circ}\).
\(\angle3\) and \(\angle1\) are vertical - angles, so \(\angle3=\angle1 = 121^{\circ}\).
\(\angle4\) and \(\angle2\) are vertical - angles, so \(\angle4=\angle2 = 59^{\circ}\).
\(\angle5\) and \(\angle1\) are corresponding angles, so \(\angle5=\angle1 = 121^{\circ}\).
\(\angle6\) and \(\angle2\) are corresponding angles, so \(\angle6=\angle2 = 59^{\circ}\).
\(\angle7\) and \(\angle3\) are corresponding angles, so \(\angle7=\angle3 = 121^{\circ}\).
\(\angle8\) and \(\angle4\) are corresponding angles, so \(\angle8=\angle4 = 59^{\circ}\).

Answer:

1. 2) 16
2. \(\angle2 = 59^{\circ}\), \(\angle3 = 121^{\circ}\), \(\angle4 = 59^{\circ}\), \(\angle5 = 121^{\circ}\), \(\angle6 = 59^{\circ}\), \(\angle7 = 121^{\circ}\), \(\angle8 = 59^{\circ}\)