Angle and parallel-line proofs: fill in the missing reason

Grade 10 Math Worksheet - Building Confidence

Circle the best answer.

1. Read this first

   Use this

   Given: lines CG and DH cross at Q. Prove: ∠CQH ≅ ∠GQD. Ladder: 1. Lines CG and DH cross at Q -- Given; 2. ∠CQH ≅ ∠GQD -- ?.

2. 1.

   Ari opens a pair of scissors, and the blades lie along lines CG and DH. Which reason fills the blank?

   • Linear pair postulate
   • Vertical angles theorem
   • Angle addition postulate
   • Given
3. 2.

   Bo builds a kite whose two sticks lie along lines CG and DH. Which reason fills the blank?

   • Given
   • Linear pair postulate
   • Vertical angles theorem
   • Definition of complementary angles
4. 3.

   Jun sees two stripes UV and WY on a rug, crossed by a seam HD. Which reason fills the blank?

   • Converse of the corresponding angles postulate
   • Given
   • Corresponding angles postulate
   • Alternate exterior angles theorem

5. Sketch, not to scale

   Line UV above line WY (U and W on the left), crossed by line HD at S on UV and at Q on WY (H at the top, D at the bottom). Marked: ∠HSU and ∠SQW with one arc each (no parallel arrows). Given: ∠HSU ≅ ∠SQW. Prove: ∠VSQ ≅ ∠SQW. Ladder: 1. ∠HSU ≅ ∠SQW — Given; 2. ∠VSQ ≅ ∠HSU — Vertical angles theorem; 3. ∠VSQ ≅ ∠SQW — ?.

6. 4.

   Kira sketches window bars UV and WY, crossed by a brace HD. Which reason fills the blank?

   • Given
   • Symmetric property
   • Alternate interior angles theorem
   • Transitive property

7. Sketch, not to scale

   Line UV above line WY (U and W on the left), crossed by line HD at S on UV and at Q on WY (H at the top, D at the bottom). Marked: ∠USQ and ∠WQD with one arc each (no parallel arrows). Given: ∠USQ ≅ ∠WQD. Prove: ∠HSV ≅ ∠WQD. Ladder: 1. ∠USQ ≅ ∠WQD — Given; 2. ∠HSV ≅ ∠USQ — Vertical angles theorem; 3. ∠HSV ≅ ∠WQD — ?.

8. 5.

   Luz traces streets UV and WY on a map, crossed by avenue HD. Which reason fills the blank?

   • Given
   • Alternate exterior angles theorem
   • Reflexive property
   • Transitive property

9. Use this

   Given: ∠SQY ≅ ∠USQ. Prove: UV ∥ WY. Ladder: 1. ∠SQY ≅ ∠USQ -- Given; 2. ∠WQD ≅ ∠SQY -- Vertical angles theorem; 3. ∠WQD ≅ ∠USQ -- Transitive property; 4. UV ∥ WY -- ?.

10. 6.

    Ren rules lines UV and WY on paper, then crosses them with line HD. Which reason fills the blank?

    • Converse of the corresponding angles postulate
    • Given
    • Converse of the same-side interior angles theorem
    • Corresponding angles postulate

11. Use this

    Given: ∠SQY ≅ ∠HSV. Prove: ∠WQD ≅ ∠HSV. Ladder: 1. ∠SQY ≅ ∠HSV -- Given; 2. ? -- Vertical angles theorem; 3. ∠WQD ≅ ∠HSV -- Transitive property.

12. 7.

    Milo draws a door's edges UV and WY, crossed by a ribbon HD. Which statement fills the blank?

    • ∠HSV ≅ ∠SQY
    • ∠WQD ≅ ∠SQY
    • ∠WQD ≅ ∠HSV
    • ∠WQD ≅ ∠YQD

13. Use this

    Given: ∠YQD ≅ ∠VSQ. Prove: ∠SQW ≅ ∠VSQ. Ladder: 1. ∠YQD ≅ ∠VSQ -- Given; 2. ? -- Vertical angles theorem; 3. ∠SQW ≅ ∠VSQ -- Transitive property.

14. 8.

    Noah draws a bike rack with bars UV and WY, crossed by a pole HD. Which statement fills the blank?

    • ∠SQW ≅ ∠SQY
    • ∠SQW ≅ ∠VSQ
    • ∠VSQ ≅ ∠YQD
    • ∠SQW ≅ ∠YQD
15. 9.

    Tala draws tent poles UV and WY, joined by a cord HD. Which statement fills the blank?

    • UV ∥ WY
    • ∠USQ ≅ ∠WQD
    • ∠USQ ≅ ∠SQY
    • ∠WQD ≅ ∠HSV

16. Use this

    Given: ∠YQD ≅ ∠HSU. Prove: UV ∥ WY. Ladder: 1. ∠YQD ≅ ∠HSU -- Given; 2. ∠SQW ≅ ∠YQD -- Vertical angles theorem; 3. ? -- Transitive property; 4. UV ∥ WY -- Converse of the corresponding angles postulate.

17. 10.

    Zee sews quilt stripes UV and WY, crossed by a stitch line HD. Which statement fills the blank?

    • UV ∥ WY
    • ∠SQW ≅ ∠SQY
    • ∠HSU ≅ ∠YQD
    • ∠SQW ≅ ∠HSU