This geometry parallel lines proofs worksheet gives 24 problems on parallel lines, transversals and proofs, ordered from easy to hard, with a separate answer key.
These 24 problems practice standard G-CO.C.9 with parallel lines cut by a transversal. The set covers naming angle pairs, finding missing measures, solving for x, and deciding whether lines are parallel.
The set also includes two-column proofs, perpendicular bisector problems, and six real-world word problems. Write every step so a classmate can follow your reasoning.
Geometry Parallel Lines Proofs Worksheet: problems 1–5
Problem 1: Name the angle pairs
Lines \(a\) and \(b\) are cut by transversal \(t\). Name the relationship for each pair of angles.
- \(\angle 4\) and \(\angle 5\)
- \(\angle 2\) and \(\angle 6\)
- \(\angle 1\) and \(\angle 8\)
- \(\angle 3\) and \(\angle 5\)
Problem 2: Corresponding angles
Lines \(a\) and \(b\) are parallel and \(m\angle 1\) = \(118^{\circ}\). Find \(m\angle 5\) and name the theorem you used.
Problem 3: Alternate interior angles
Lines \(a\) and \(b\) are parallel and \(m\angle 3\) = \(72^{\circ}\). Find \(m\angle 6\).
Problem 4: Same-side interior angles
Lines \(a\) and \(b\) are parallel and \(m\angle 3\) = \(53^{\circ}\). Find \(m\angle 5\).
Problem 5: Alternate exterior angles
Lines \(a\) and \(b\) are parallel and \(m\angle 2\) = \(109^{\circ}\). Find \(m\angle 7\) and explain.
Geometry Parallel Lines Proofs Worksheet: problems 6–10
Problem 6: Find all eight angles
Lines \(a\) and \(b\) are parallel and \(m\angle 1\) = \(124^{\circ}\). Find the measure of the other seven angles.
Problem 7: Corresponding angles with algebra
Lines \(a\) and \(b\) are parallel. \(m\angle 2\) = \((3x+15)^{\circ}\) and \(m\angle 6\) = \((5x-25)^{\circ}\). Find \(x\) and the measure of each angle.
Problem 8: Alternate interior with algebra
Lines \(a\) and \(b\) are parallel. \(m\angle 4\) = \((4x-12)^{\circ}\) and \(m\angle 5\) = \((2x+30)^{\circ}\). Find \(x\) and \(m\angle 4\).
Problem 9: Same-side interior with algebra
Lines \(a\) and \(b\) are parallel. Find \(x\) and both angle measures.
Problem 10: Same-side exterior with algebra
Lines \(a\) and \(b\) are parallel. \(m\angle 1\) = \((4x+8)^{\circ}\) and \(m\angle 7\) = \((2x+22)^{\circ}\). These are same-side exterior angles. Find \(x\) and each measure.
Geometry Parallel Lines Proofs Worksheet: problems 11–15
Problem 11: Are the lines parallel? (alternate interior)
A transversal crosses lines \(a\) and \(b\). Alternate interior angles \(\angle 3\) and \(\angle 6\) both measure \(64^{\circ}\). Is \(a \parallel b\)? Justify.
Problem 12: Are the lines parallel? (same-side interior)
Same-side interior angles \(\angle 3\) and \(\angle 5\) measure \(112^{\circ}\) and \(78^{\circ}\). Is \(a \parallel b\)? Justify.
Problem 13: Value of x that makes lines parallel (corresponding)
Corresponding angles measure \((5x+4)^{\circ}\) and \((7x-20)^{\circ}\). Find \(x\) so that \(a \parallel b\), then find the angle measure.
Problem 14: Value of x that makes lines parallel (alternate exterior)
Alternate exterior angles measure \((9x-7)^{\circ}\) and \((6x+23)^{\circ}\). Find \(x\) so that the lines are parallel, then find the angle measure.
Problem 15: Complete the proof
Given \(a \parallel b\), prove \(m\angle 4\) = \(m\angle 5\). Write the missing reason in each row.
| Statements | Reasons |
|---|---|
| \(a \parallel b\) | Given |
| \(m\angle 4\) = \(m\angle 8\) | ? |
| \(m\angle 8\) = \(m\angle 5\) | ? |
| \(m\angle 4\) = \(m\angle 5\) | ? |
Geometry Parallel Lines Proofs Worksheet: problems 16–20
Problem 16: Write a proof of the converse
Given \(m\angle 1\) = \(m\angle 8\), write a two-column proof that \(a \parallel b\).
Problem 17: Perpendicular bisector with algebra
Line \(\ell\) is the perpendicular bisector of \(\overline{AB}\), and \(P\) is on \(\ell\). Find \(x\) and \(PA\).
Problem 18: Distance to endpoints
Line \(\ell\) is the perpendicular bisector of \(\overline{AB}\) at \(M\). Point \(P\) is on \(\ell\), \(AM=12\) and \(PM=9\). Find \(PA\) and \(PB\).
Problem 19: Streets problem
Maple Street and Oak Street are parallel. Pine Street crosses both. At Maple Street the northeast angle between Pine and Maple measures \(71^{\circ}\). Find (a) the northeast angle at Oak Street and (b) the northwest angle at Oak Street.
Problem 20: Railroad crossing
Two rails of a railroad are parallel. A service road crosses both rails. Alternate interior angles measure \((3x+2)^{\circ}\) and \((2x+37)^{\circ}\). (a) Find \(x\). (b) Find the measure of each angle. (c) Find the measure of a same-side interior angle to one of them.
Geometry Parallel Lines Proofs Worksheet: problems 21–24
Problem 21: Bookshelf brace
Aisha wants to check whether two shelves in a bookcase are parallel. A diagonal brace crosses both, and the corresponding angles measure \(83^{\circ}\) and \(85^{\circ}\). Are the shelves parallel? Explain.
Problem 22: Bus stop
Two houses, A and B, are 16 blocks apart on a straight road. A bus stop \(P\) is placed on the perpendicular bisector of \(\overline{AB}\), 6 blocks away from the road. How far is the stop from each house?
Problem 23: Multi-step angle problem
Lines \(a\) and \(b\) are parallel. \(m\angle 1\) = \((6x-12)^{\circ}\) and \(m\angle 5\) = \((4x+18)^{\circ}\). Priya needs \(m\angle 4\) and \(m\angle 6\). Find \(x\), then both measures.
Problem 24: Notebook paper
The printed lines on a sheet of notebook paper are parallel. Ethan draws a line across two of them. Two same-side interior angles measure \((6y+4)^{\circ}\) and \((3y+14)^{\circ}\). Find \(y\) and both angle measures.
How to use this geometry parallel lines proofs worksheet
Print this geometry parallel lines proofs worksheet as a PDF, solve every problem on paper, then compare with the answer key.
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