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

Grade 10 Math Worksheet

Circle the best answer.

1. Read this first

   Use this

   Ladder: 1. Lines CG and DH cross at Q -- Given; 2. ∠CQH and ∠HQG are supplementary -- Linear pair postulate; 3. ∠GQD and ∠HQG are supplementary -- Linear pair postulate; 4. ∠CQH ≅ ∠GQD -- ?.

2. 1.

   Ari proves the vertical angles theorem on open scissors, blades along CG and DH. Which reason fills the blank?

   • Vertical angles theorem
   • Linear pair postulate
   • Congruent supplements theorem
   • Congruent complements theorem

3. Use this

   Given: lines CG and DH cross at Q. Prove: ∠HQG ≅ ∠DQC. Ladder: 1. Lines CG and DH cross at Q -- Given; 2. ∠HQG and ∠GQD are supplementary -- Linear pair postulate; 3. ∠DQC and ∠GQD are supplementary -- Linear pair postulate; 4. ∠HQG ≅ ∠DQC -- ?.

4. 2.

   Bo proves the vertical angles theorem on a kite, sticks along CG and DH. Which reason fills the blank?

   • Angle addition postulate
   • Congruent supplements theorem
   • Vertical angles theorem
   • Congruent complements theorem
5. 3.

   Kira proves the alternate exterior angles theorem on train rails UV and WY, crossed by road HD. Which reason fills the blank?

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

6. Use this

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

7. 4.

   Luz proves the alternate exterior angles theorem on map streets UV and WY, crossed by avenue HD. Which reason fills the blank?

   • Converse of the corresponding angles postulate
   • Same-side interior angles theorem
   • Vertical angles theorem
   • Corresponding angles postulate

8. Use this

   Given: m∠CQH = 115. Prove: m∠HQG = 65. Ladder: 1. m∠CQH = 115 -- Given; 2. ∠CQH and ∠HQG are supplementary -- Linear pair postulate; 3. m∠CQH + m∠HQG = 180 -- Definition of supplementary angles; 4. 115 + m∠HQG = 180 -- Substitution property; 5. m∠HQG = 65 -- ?.

9. 5.

   Pia opens scissors so one angle at the pivot Q measures 115°. Which reason fills the blank?

   • Addition property of equality
   • Given
   • Substitution property
   • Subtraction property of equality

10. Use this

    Sketch, not to scale: lines CG and DH cross at Q (C upper left, H upper right, G lower right, D lower left). Marked: ∠GQD labeled 128°. Given: m∠GQD = 128. Prove: m∠DQC = 52. Ladder: 1. m∠GQD = 128 -- Given; 2. ∠GQD and ∠DQC are supplementary -- Linear pair postulate; 3. m∠GQD + m∠DQC = 180 -- Definition of supplementary angles; 4. 128 + m∠DQC = 180 -- Substitution property; 5. m∠DQC = 52 -- ?.

11. 6.

    Ren sets kite sticks CG and DH so one angle at Q measures 128°. Which reason fills the blank?

    • Subtraction property of equality
    • Addition property of equality
    • Given
    • Definition of supplementary angles

12. 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 and ∠DQC are supplementary -- Linear pair postulate; 3. ? -- Linear pair postulate; 4. ∠CQH ≅ ∠GQD -- Congruent supplements theorem.

13. 7.

    Esme proves the vertical angles theorem with two crossed pencils along CG and DH. Which statement fills the blank?

    • ∠GQD ≅ ∠DQC
    • ∠GQD and ∠DQC are complementary
    • ∠GQD and ∠DQC are supplementary
    • ∠CQH ≅ ∠GQD

14. Use this

    Marked: none beyond the lines. Given: lines CG and DH cross at Q. Prove: ∠HQG ≅ ∠DQC. Ladder: 1. Lines CG and DH cross at Q -- Given; 2. ∠HQG and ∠CQH are supplementary -- Linear pair postulate; 3. ? -- Linear pair postulate; 4. ∠HQG ≅ ∠DQC -- Congruent supplements theorem.

15. 8.

    Jun proves the vertical angles theorem with scissors opened wide. Which statement fills the blank?

    • ∠HQG ≅ ∠DQC
    • ∠DQC and ∠CQH are complementary
    • ∠DQC ≅ ∠CQH
    • ∠DQC and ∠CQH are supplementary

16. Use this

    Given: UV ∥ WY. Prove: ∠VSQ ≅ ∠SQW. Ladder: 1. UV ∥ WY -- Given; 2. ∠VSQ ≅ ∠YQD -- Corresponding angles postulate; 3. ? -- Vertical angles theorem; 4. ∠VSQ ≅ ∠SQW -- Transitive property.

17. 9.

    Noah proves the alternate interior angles theorem on window bars UV and WY, crossed by brace HD. Which statement fills the blank?

    • ∠YQD ≅ ∠SQW
    • ∠VSQ ≅ ∠SQY
    • ∠VSQ ≅ ∠SQW
    • ∠YQD ≅ ∠SQY

18. Use this

    Given: UV ∥ WY and m∠HSV = 62. Prove: m∠SQY = 62. Ladder: 1. UV ∥ WY -- Given; 2. m∠HSV = 62 -- Given; 3. ? -- Corresponding angles postulate; 4. m∠SQY = m∠HSV -- Definition of congruent angles; 5. m∠SQY = 62 -- Substitution property.

19. 10.

    Tala measures 62° where road HD meets train rail UV. Which statement fills the blank?

    • UV ∥ WY
    • m∠SQY = 62
    • ∠HSV ≅ ∠YQD
    • ∠HSV ≅ ∠SQY