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

   Use this

   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∠UGD = 108° and m∠GHS = 72°. Step 2: m∠UGF = 72°, because ∠UGD and ∠UGF are a linear pair, so they add to 180°. Step 3: m∠UGF = m∠GHS, so DF ∥ QS. Reason for Step 3: ?

2. 1.

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

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

3. 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: ?

4. 2.

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

   • 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)
   • 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)

5. 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: ?

6. 3.

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

   • 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)
   • If alternate interior angles are congruent, then the lines are parallel (converse of the Alternate Interior Angles Theorem)
   • If lines are parallel, then alternate interior angles are congruent (Alternate Interior Angles Theorem)

7. 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: ?

8. 4.

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

   • Linear pair angles add to 180° (the reason for Step 2)
   • If alternate interior angles are congruent, then the lines are parallel (converse of the Alternate Interior Angles Theorem)
   • If lines are parallel, then alternate exterior angles are congruent (Alternate Exterior Angles Theorem)
   • If alternate exterior angles are congruent, then the lines are parallel (converse of the Alternate Exterior Angles Theorem)
9. 5.

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

   • ∠UGF and ∠GHS are corresponding angles, and they are congruent
   • 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
10. 6.

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

    • x = 20, from the given
    • ∠UGD and the angle next to it on line DF are a linear pair
    • ∠UGD and ∠SHV are same-side exterior angles, and they are congruent
    • ∠UGD and ∠SHV are alternate exterior angles, and they are congruent
12. 7.

    Bo checks two toy-train rails. ∠UGF and ∠GHS are corresponding angles, m∠UGF = (3x + 16)° and m∠GHS = (5x − 12)°. What value of x makes DF ∥ QS?

    14
13. 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?

    Use the picture in question 7.

    19
14. 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 7.

    20
15. 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 7.

    13