Which angle pairs prove that two lines are parallel?
Grade 10 Math Worksheet - Challenge Me
Circle the best answer, or write your answer on the line.
Luz draws two lines on paper. m∠DGH = (3x + 5)° and m∠GHQ = (2x + 25)°. What value of x makes DF ∥ QS?
Milo checks two bike lanes. m∠UGF = (4x − 10)° and m∠SHV = (x + 35)°. What value of x makes DF ∥ QS?
Use the picture in question 1.
Noah checks two stripes on a rug. m∠FGH = (6x + 4)° and m∠GHS = (2x + 16)°. What value of x makes DF ∥ QS?
Use the picture in question 1.
Zee's window is quadrilateral PQRS with PQ ∥ SR and PS ∥ QR. Diagonal PR has m∠QPR = 38° and m∠PRQ = 41°. What is m∠P?
Ari's rug is quadrilateral PQRS with PQ ∥ SR and PS ∥ QR. Diagonal PR has m∠QPR = 25° and m∠PRQ = 52°. What is m∠P?
Use the picture in question 4.
Bo's map tile is quadrilateral PQRS with PQ ∥ SR and PS ∥ QR. Diagonal PR has m∠QPR = 33° and m∠PRQ = 49°. What is m∠S?
Use the picture in question 4.
Jun measures corresponding angles where line t crosses lines l, m and n: 68° at l, 68° at m and 71° at n. Which statement is justified?
- m ∥ n, since their angles, 68° and 71°, differ by just 3°, which is close enough to be equal
- No two lines are parallel: corresponding angles must add to 180°, and 68° + 68° does not equal 180°
- l ∥ m, since their corresponding angles are both 68°, while n is parallel to neither
- l ∥ n, since 71° at n and its supplement 109° add to 180°, which is the test for parallel lines
Kira can use one given to prove DF ∥ QS. Which given is enough?
- m∠DGH = 68° and m∠GHS = 112°, alternate interior angles that add to 180°
- m∠UGF = 72° and m∠FGH = 108°, a linear pair that adds to 180°
- m∠FGH = 112° and m∠GHQ = 112°, alternate interior angles that are congruent
- m∠FGH = 112° and m∠GHS = 112°, same-side interior angles that are congruent
Luz writes: 1. Assume DF ∥ QS. 2. Then alternate interior angles are congruent. 3. So m∠DGH = m∠GHS. 4. So DF ∥ QS. What is wrong?
Use the picture in question 8.
- Step 1 assumes the conclusion; the proof must start from the congruent angles and use the converse
- Nothing is wrong: Step 2 is a correct theorem, and Step 3 follows from it, so the proof holds
- Step 2 should name corresponding angles, since ∠DGH and ∠GHS are corresponding angles here
- Step 3 is wrong: alternate interior angles between parallel lines are supplementary, not congruent
Milo writes: m∠UGF + m∠GHS = 180°, so by the converse of the Same-Side Interior Angles Theorem, DF ∥ QS. What is wrong?
Use the picture in question 8.
- The name is wrong: the proof should cite the converse of the Alternate Interior Angles Theorem
- ∠UGF and ∠GHS are corresponding angles, and the converse for them needs them to be congruent, not supplementary
- Nothing is wrong: angles at the two crossings that add to 180° make the lines parallel
- The sum is wrong: for parallel lines these two angles add to 360°, so the test fails