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

Grade 10 Math Worksheet - Challenge Me

Circle the best answer.

1. Read this first

   Use this

   Given: UV ∥ WY. Prove: ∠VSQ and ∠SQY are supplementary. Ladder: 1. UV ∥ WY -- Given; 2. ∠VSQ ≅ ∠SQW -- Alternate interior angles theorem; 3. m∠VSQ = m∠SQW -- Definition of congruent angles; 4. ∠SQY and ∠SQW are supplementary -- Linear pair postulate; 5. m∠SQY + m∠SQW = 180 -- Definition of supplementary angles; 6. m∠SQY + m∠VSQ = 180 -- ?; 7. ∠VSQ and ∠SQY are supplementary -- Definition of supplementary angles.

2. 1.

   Bo proves the same-side interior angles theorem on rug stripes UV and WY. Which reason fills the blank?

   • Substitution property
   • Given
   • Vertical angles theorem
   • Addition property of equality

3. Use this

   Given: ∠USQ and ∠SQW are supplementary. Prove: UV ∥ WY. Ladder: 1. ∠USQ and ∠SQW are supplementary -- Given; 2. ∠SQW and ∠SQY are supplementary -- Linear pair postulate; 3. ∠USQ ≅ ∠SQY -- ?; 4. UV ∥ WY -- Converse of the alternate interior angles theorem.

4. 2.

   Kira checks that tent poles UV and WY are parallel, using a cord HD. Which reason fills the blank?

   • Congruent supplements theorem
   • Alternate interior angles theorem
   • Congruent complements theorem
   • Vertical angles theorem

5. Use this

   Given: UV ∥ WY. Prove: ∠USQ and ∠SQW are supplementary. Ladder: 1. UV ∥ WY -- Given; 2. ∠USQ ≅ ∠WQD -- Corresponding angles postulate; 3. m∠USQ = m∠WQD -- Definition of congruent angles; 4. ? -- Linear pair postulate; 5. m∠SQW + m∠WQD = 180 -- Definition of supplementary angles; 6. m∠SQW + m∠USQ = 180 -- Substitution property; 7. ∠USQ and ∠SQW are supplementary -- Definition of supplementary angles.

6. 3.

   Esme proves the same-side interior angles theorem a second way, on map streets UV and WY. Which statement fills the blank?

   • ∠SQW and ∠WQD are complementary
   • ∠SQW ≅ ∠WQD
   • ∠USQ and ∠SQW are supplementary
   • ∠SQW and ∠WQD are supplementary

7. Use this

   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: parallel arrows on UV and WY. Given: UV ∥ WY. Prove: ∠VSQ and ∠SQY are supplementary. Ladder: 1. UV ∥ WY -- Given; 2. ∠VSQ ≅ ∠YQD -- Corresponding angles postulate; 3. m∠VSQ = m∠YQD -- Definition of congruent angles; 4. ? -- Linear pair postulate; 5. m∠SQY + m∠YQD = 180 -- Definition of supplementary angles; 6. m∠SQY + m∠VSQ = 180 -- Substitution property; 7. ∠VSQ and ∠SQY are supplementary -- Definition of supplementary angles.

8. 4.

   Jun proves the same-side interior angles theorem a second way, on window bars UV and WY. Which statement fills the blank?

   • ∠VSQ and ∠SQY are supplementary
   • ∠SQY ≅ ∠YQD
   • ∠SQY and ∠YQD are supplementary
   • ∠SQY and ∠YQD are complementary

9. Use this

   Sketch, not to scale: rays QC, QD, QF and QG come out of point Q in that order, turning one way. Marked: right-angle boxes at ∠CQF and ∠DQG. Given: ∠CQF and ∠DQG are right angles. Prove: ∠CQD ≅ ∠FQG. Ladder: 1. ∠CQF and ∠DQG are right angles -- Given; 2. m∠CQF = 90 and m∠DQG = 90 -- Definition of right angle; 3. ? -- Angle addition postulate; 4. m∠DQF + m∠FQG = m∠DQG -- Angle addition postulate; 5. m∠CQD + m∠DQF = 90 and m∠DQF + m∠FQG = 90 -- Substitution property; 6. ∠CQD and ∠DQF are complementary, and so are ∠DQF and ∠FQG -- Definition of complementary angles; 7. ∠CQD ≅ ∠FQG -- Congruent complements theorem.

10. 5.

    Milo folds a paper fan whose folds are rays QC, QD, QF and QG. Which statement fills the blank?

    • m∠CQD + m∠DQF = m∠CQF
    • m∠CQD + m∠DQF = 180
    • ∠CQD ≅ ∠FQG
    • m∠CQD + m∠FQG = m∠CQG

11. Use this

    Given: ∠UQW and ∠VQY are right angles. Prove: ∠UQV ≅ ∠WQY. Ladder: 1. ∠UQW and ∠VQY are right angles -- Given; 2. m∠UQW = 90 and m∠VQY = 90 -- Definition of right angle; 3. m∠UQV + m∠VQW = m∠UQW -- Angle addition postulate; 4. ? -- Angle addition postulate; 5. m∠UQV + m∠VQW = 90 and m∠VQW + m∠WQY = 90 -- Substitution property; 6. ∠UQV and ∠VQW are complementary, and so are ∠VQW and ∠WQY -- Definition of complementary angles; 7. ∠UQV ≅ ∠WQY -- Congruent complements theorem.

12. 6.

    Noah draws four sun rays QU, QV, QW and QY from one center Q. Which statement fills the blank?

    • m∠UQV + m∠WQY = m∠UQY
    • ∠UQV ≅ ∠WQY
    • m∠VQW + m∠WQY = m∠VQY
    • m∠VQW + m∠WQY = 180
13. 7.

    Tala proves the converse of the alternate exterior angles theorem on quilt stripes UV and WY. Which statement and reason fill the blanks?

    • ∠HSU ≅ ∠VSQ — Reflexive property
    • ∠VSQ ≅ ∠YQD — Corresponding angles postulate
    • ∠VSQ ≅ ∠YQD — Symmetric property
    • ∠VSQ ≅ ∠YQD — Transitive property

14. Use this

    Given: ∠HSV ≅ ∠WQD. Prove: UV ∥ WY. Ladder: 1. ∠HSV ≅ ∠WQD -- Given; 2. ∠USQ ≅ ∠HSV -- Vertical angles theorem; 3. ? -- ?; 4. UV ∥ WY -- Converse of the corresponding angles postulate.

15. 8.

    Zee proves the converse of the alternate exterior angles theorem on train rails UV and WY. Which statement and reason fill the blanks?

    • ∠USQ ≅ ∠WQD — Symmetric property
    • ∠USQ ≅ ∠WQD — Corresponding angles postulate
    • ∠HSV ≅ ∠USQ — Reflexive property
    • ∠USQ ≅ ∠WQD — Transitive property

16. Use this

    Given: ∠CQD and ∠DQF are complementary, and so are ∠FQG and ∠DQF. Prove: ∠CQD ≅ ∠FQG. Ladder: 1. ∠CQD and ∠DQF are complementary, and so are ∠FQG and ∠DQF -- Given; 2. m∠CQD + m∠DQF = 90 -- Definition of complementary angles; 3. m∠FQG + m∠DQF = 90 -- Definition of complementary angles; 4. m∠CQD + m∠DQF = m∠FQG + m∠DQF -- Substitution property; 5. ? -- ?; 6. ∠CQD ≅ ∠FQG -- Definition of congruent angles.

17. 9.

    Pia proves the congruent complements theorem on a paper fan with folds QC, QD, QF and QG. Which statement and reason fill the blanks?

    • m∠CQD = m∠FQG — Subtraction property of equality
    • m∠CQD = m∠DQF — Subtraction property of equality
    • m∠CQD = m∠FQG — Addition property of equality
    • ∠CQD ≅ ∠FQG — Congruent complements theorem

18. Use this

    Given: ∠UQV and ∠VQW are complementary, and so are ∠WQY and ∠VQW. Prove: ∠UQV ≅ ∠WQY. Ladder: 1. ∠UQV and ∠VQW are complementary, and so are ∠WQY and ∠VQW -- Given; 2. m∠UQV + m∠VQW = 90 -- Definition of complementary angles; 3. m∠WQY + m∠VQW = 90 -- Definition of complementary angles; 4. m∠UQV + m∠VQW = m∠WQY + m∠VQW -- Substitution property; 5. ? -- ?; 6. ∠UQV ≅ ∠WQY -- Definition of congruent angles.

19. 10.

    Ren proves the congruent complements theorem on sun rays QU, QV, QW and QY. Which statement and reason fill the blanks?

    • m∠UQV = m∠WQY — Addition property of equality
    • m∠UQV = m∠VQW — Subtraction property of equality
    • m∠UQV = m∠WQY — Subtraction property of equality
    • ∠UQV ≅ ∠WQY — Congruent complements theorem