Which angle pairs prove that two lines are parallel?

Answer Key - Grade 10 Math Worksheet

Circle the best answer, or write your answer on the line.

1. Read this first

   Sketch, not to scale

   Lines DF and QS (no arrows given), crossed by line UV at G on DF and at H on QS; U is above, V below; D and Q are on the left. Proof. Step 1: Given m∠DGH = 64° and m∠SHV = 116°. Step 2: m∠GHQ = 116°, because ∠SHV and ∠GHQ are vertical angles, so they are congruent. Step 3: m∠DGH + m∠GHQ = 180°, so DF ∥ QS. Reason for Step 3: ?

2. 1.

   Kira is writing a two-column proof that DF ∥ QS. Which reason justifies Step 3?

   • Vertical angles are congruent (the reason for Step 2)
   • If corresponding angles are congruent, then the lines are parallel (converse of the Corresponding Angles Theorem)
   • If lines are parallel, then same-side interior angles are supplementary (Same-Side Interior Angles Theorem)
   • If same-side interior angles are supplementary, then the lines are parallel (converse of the Same-Side Interior Angles Theorem)

3. Use this

   Proof. Step 1: Given m∠UGF = 125° and m∠GHQ = 55°. Step 2: m∠FGH = 55°, because ∠UGF and ∠FGH are a linear pair, so they add to 180°. Step 3: m∠FGH = m∠GHQ, so DF ∥ QS. Reason for Step 3: ?

4. 2.

   Luz is writing a two-column proof that DF ∥ QS. Which reason justifies Step 3?

   • If lines are parallel, then alternate interior angles are congruent (Alternate Interior Angles Theorem)
   • If alternate interior angles are congruent, then the lines are parallel (converse of the Alternate Interior Angles Theorem)
   • If same-side interior angles are supplementary, then the lines are parallel (converse of the Same-Side Interior Angles Theorem)
   • Linear pair angles add to 180° (the reason for Step 2)

5. Sketch, not to scale

   Lines DF and QS (no arrows given), crossed by line UV at G on DF and at H on QS; U is above, V below; D and Q are on the left. Proof. Step 1: Given m∠UGF = 100° and m∠SHV = 80°. Step 2: m∠UGD = 80°, because ∠UGF and ∠UGD are a linear pair, so they add to 180°. Step 3: m∠UGD = m∠SHV, so DF ∥ QS. Reason for Step 3: ?

6. 3.

   Milo is writing a two-column proof that DF ∥ QS. Which reason justifies Step 3?

   • If alternate interior angles are congruent, then the lines are parallel (converse of the Alternate Interior Angles Theorem)
   • If alternate exterior angles are congruent, then the lines are parallel (converse of the Alternate Exterior Angles Theorem)
   • Linear pair angles add to 180° (the reason for Step 2)
   • If lines are parallel, then alternate exterior angles are congruent (Alternate Exterior Angles Theorem)
7. 4.

   Noah is proving DF ∥ QS from an equation. Step 3 is missing. Which statement belongs there?

   • x = 14, from the given
   • ∠UGF and ∠GHS are alternate interior angles, and they are congruent
   • ∠UGF and the angle next to it on line UV are a linear pair
   • ∠UGF and ∠GHS are corresponding angles, and they are congruent
8. 5.

   Pia is proving DF ∥ QS from an equation. Step 3 is missing. Which statement belongs there?

   • ∠DGH and the angle next to it on line UV are a linear pair
   • x = 19, from the given
   • ∠DGH and ∠GHS are alternate interior angles, and they are congruent
   • ∠DGH and ∠GHS are corresponding angles, and they are congruent
9. 6.

   Ren is proving DF ∥ QS from an equation. Step 3 is missing. Which statement belongs there?

   • x = 20, from the given
   • ∠UGD and ∠SHV are same-side exterior angles, and they are congruent
   • ∠UGD and ∠SHV are alternate exterior angles, and they are congruent
   • ∠UGD and the angle next to it on line DF are a linear pair
10. 7.

    Tala is proving DF ∥ QS from an equation. Step 3 is missing. Which statement belongs there?

    • ∠DGH and the angle next to it on line UV are a linear pair
    • x = 13, from the given
    • ∠DGH and ∠QHV are corresponding angles, and they are congruent
    • ∠DGH and ∠QHV are alternate interior angles, and they are congruent
12. 8.

    Esme checks two guitar strings. ∠DGH and ∠GHS are alternate interior angles, m∠DGH = (4x + 2)° and m∠GHS = (2x + 40)°. What value of x makes DF ∥ QS?

    19
13. 9.

    Jun checks two streets on a map. ∠UGD and ∠SHV are alternate exterior angles, m∠UGD = (6x − 15)° and m∠SHV = (4x + 25)°. What value of x makes DF ∥ QS?

    Use the picture in question 8.

    20
14. 10.

    Kira checks the two rails of a ladder. ∠DGH and ∠QHV are corresponding angles, m∠DGH = (5x − 6)° and m∠QHV = (3x + 20)°. What value of x makes DF ∥ QS?

    Use the picture in question 8.

    13