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Grade 10 · Math · Geometry

10th grade Angle and parallel-line proofs: fill in the missing reason worksheet

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

Grade 10 Math Worksheet - Challenge Me

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Circle the best answer.

  1. Read this first

    Use this

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

  2. Ari proves the same-side interior angles theorem on train rails UV and WY. Which reason fills the blank?

    • Substitution property
    • Definition of supplementary angles
    • Transitive property
    • Linear pair postulate

  3. 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.

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

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

  5. 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.

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

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

  7. Luz checks that a door's edges UV and WY are parallel, using a ribbon HD. Which reason fills the blank?

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

  8. 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.

  9. 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 and ∠WQD are supplementary
    • ∠USQ and ∠SQW are supplementary
    • ∠SQW ≅ ∠WQD

  10. 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.

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

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

  12. 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.

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

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

  14. 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.

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

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

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

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

  17. 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.

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

    • ∠UQV ≅ ∠WQY — Congruent complements theorem
    • m∠UQV = m∠WQY — Subtraction property of equality
    • m∠UQV = m∠VQW — Subtraction property of equality
    • m∠UQV = m∠WQY — Addition property of equality
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10th grade Angle and parallel-line proofs: fill in the missing reason worksheet – printable
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About this worksheet

Angle and parallel-line proofs: fill in the missing reason. Every version has 10 questions, and the answer key prints on the second page.

What your child will learn

  • I can name the reason for each step of a proof about angles and parallel lines.
  • I can tell a theorem from its converse: parallel lines give angles, and angles give parallel lines.
  • I can choose the statement that a reason gives, and spot a rung that skips ahead.

Words to know

rung: One line of the proof: a statement with its reason beside it.

Given / Prove: Given: the facts we start with. Prove: the statement the last rung must reach.

converse: The if-then turned around; it is a different statement, true or not.

linear pair: Two angles that share a side and together make a straight line.

vertical angles: Opposite angles where two lines cross.

supplementary / complementary: Two angles adding to 180° / adding to 90°.

transitive property: If a matches b and b matches c, then a matches c.

substitution property: Swap a value for an equal one inside a statement.

Common mistakes

Your child gave the rule for the wrong kind of angle pair. Point to each angle on the drawing and say which corner of each crossing it sits in.

Your child wrote "Given" for the statement the proof is trying to reach. Only the facts on the Given line are given. Cover the Prove line and ask what is known.

Your child named the converse of a theorem when the theorem itself was needed. The theorem goes from parallel lines to equal angles; its converse goes from equal angles to parallel lines.

What to say

Ask: which fact do we already have?

Questions parents ask

What is the most common mistake with angle and parallel-line proofs: fill in the missing reason?

Your child gave the rule for the wrong kind of angle pair. Point to each angle on the drawing and say which corner of each crossing it sits in.

Are these 10th grade math worksheets free to print?

Yes. All 6 printable versions at each level are free to print, with the answer key, and no account is needed.

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