This geometry parallel lines worksheet answer key solves all 24 problems of the parallel lines, transversals and proofs worksheet step by step.
This answer key gives a full worked solution for every problem on the parallel lines and transversals worksheet. Check each reason against the theorem you cited, and compare your algebra step by step.
Geometry Parallel Lines Worksheet Answer Key: problems 1–5
Problem 1: Name the angle pairs
- \(\angle 4\) and \(\angle 5\) lie between the lines on opposite sides of the transversal: alternate interior angles.
- \(\angle 2\) and \(\angle 6\) are in the same position at the two intersections: corresponding angles.
- \(\angle 1\) and \(\angle 8\) lie outside the lines on opposite sides of the transversal: alternate exterior angles.
- \(\angle 3\) and \(\angle 5\) lie between the lines on the same side of the transversal: same-side interior angles.
Problem 2: Corresponding angles
\(\angle 1\) and \(\angle 5\) are corresponding angles. By the Corresponding Angles Theorem they are equal.
\(m\angle 5\) = \(118^{\circ}\).
Problem 3: Alternate interior angles
\(\angle 3\) and \(\angle 6\) are alternate interior angles, so they are equal.
\(m\angle 6\) = \(72^{\circ}\).
Problem 4: Same-side interior angles
\(\angle 3\) and \(\angle 5\) are same-side interior angles, so they are supplementary.
\(180-53=127\). \(m\angle 5\) = \(127^{\circ}\).
Problem 5: Alternate exterior angles
\(\angle 2\) and \(\angle 7\) are alternate exterior angles. For parallel lines they are equal.
\(m\angle 7\) = \(109^{\circ}\).
Geometry Parallel Lines Worksheet Answer Key: problems 6–10
Problem 6: Find all eight angles
Vertical angles are equal: \(m\angle 4\) = \(124^{\circ}\). Corresponding angles are equal: \(m\angle 5\) = \(m\angle 8\) = \(124^{\circ}\). Each other angle forms a linear pair with a \(124^{\circ}\) angle: \(180-124=56\).
\(m\angle 1\) = \(m\angle 4\) = \(m\angle 5\) = \(m\angle 8\) = \(124^{\circ}\); \(m\angle 2\) = \(m\angle 3\) = \(m\angle 6\) = \(m\angle 7\) = \(56^{\circ}\).
Problem 7: Corresponding angles with algebra
Corresponding angles are equal: \(3x+15=5x-25\).
Subtract \(3x\) and add 25: \(40=2x\), so \(x=20\).
\(3(20)+15=75\) and \(5(20)-25=75\). Each angle measures \(75^{\circ}\).
Problem 8: Alternate interior with algebra
\(\angle 4\) and \(\angle 5\) are alternate interior angles, so \(4x-12=2x+30\).
Subtract \(2x\), add 12: \(2x=42\), so \(x=21\).
\(m\angle 4\) = \(4(21)-12=72\), so \(m\angle 4\) = \(72^{\circ}\). Check: \(2(21)+30=72\).
Problem 9: Same-side interior with algebra
Same-side interior angles are supplementary: \((2x+20)+(3x-10)=180\).
Combine: \(5x+10=180\), so \(5x=170\) and \(x=34\).
\(2(34)+20=88\) and \(3(34)-10=92\). The angles are \(88^{\circ}\) and \(92^{\circ}\); \(88+92=180\).
Problem 10: Same-side exterior with algebra
Same-side exterior angles between parallel lines are supplementary: \((4x+8)+(2x+22)=180\).
\(6x+30=180\), so \(6x=150\) and \(x=25\).
\(m\angle 1\) = \(4(25)+8=108\) and \(m\angle 7\) = \(2(25)+22=72\). So \(108^{\circ}\) and \(72^{\circ}\).
Geometry Parallel Lines Worksheet Answer Key: problems 11–15
Problem 11: Are the lines parallel? (alternate interior)
The alternate interior angles are equal (\(64^{\circ}\) = \(64^{\circ}\)).
Yes, \(a \parallel b\) by the Alternate Interior Angles Converse.
Problem 12: Are the lines parallel? (same-side interior)
Add the angles: \(112+78=190\). The sum is not \(180^{\circ}\).
No, the lines are not parallel, because same-side interior angles of parallel lines must be supplementary.
Problem 13: Value of x that makes lines parallel (corresponding)
The lines are parallel exactly when corresponding angles are equal: \(5x+4=7x-20\).
\(24=2x\), so \(x=12\).
\(5(12)+4=64\) and \(7(12)-20=64\). Each angle is \(64^{\circ}\).
Problem 14: Value of x that makes lines parallel (alternate exterior)
Set the alternate exterior angles equal: \(9x-7=6x+23\).
\(3x=30\), so \(x=10\).
\(9(10)-7=83\) and \(6(10)+23=83\). Each angle is \(83^{\circ}\).
Problem 15: Complete the proof
Each missing reason is shown in the table.
| Statements | Reasons |
|---|---|
| \(a \parallel b\) | Given |
| \(m\angle 4\) = \(m\angle 8\) | Corresponding Angles Theorem |
| \(m\angle 8\) = \(m\angle 5\) | Vertical Angles Theorem |
| \(m\angle 4\) = \(m\angle 5\) | Transitive Property of Equality |
Geometry Parallel Lines Worksheet Answer Key: problems 16–20
Problem 16: Write a proof of the converse
| Statements | Reasons |
|---|---|
| \(m\angle 1\) = \(m\angle 8\) | Given |
| \(m\angle 1\) = \(m\angle 4\) | Vertical Angles Theorem |
| \(m\angle 4\) = \(m\angle 8\) | Substitution Property of Equality |
| \(a \parallel b\) | Corresponding Angles Converse (\(\angle 4\) and \(\angle 8\) are corresponding) |
Problem 17: Perpendicular bisector with algebra
By the Perpendicular Bisector Theorem, \(PA=PB\): \(3x+4=5x-8\).
\(12=2x\), so \(x=6\).
\(PA=3(6)+4=22\) and \(PB=5(6)-8=22\). \(PA=22\).
Problem 18: Distance to endpoints
Right triangle \(\triangle PMA\) has legs 12 and 9: \(PA^{2}=12^{2}+9^{2}=144+81=225\), so \(PA=15\).
By the Perpendicular Bisector Theorem, \(PB=PA\). \(PA=PB=15\).
Problem 19: Streets problem
- The northeast angles are corresponding angles, so the angle at Oak Street is \(71^{\circ}\).
- The northwest and northeast angles at Oak Street form a linear pair: \(180-71=109\). The northwest angle is \(109^{\circ}\).
Problem 20: Railroad crossing
- Alternate interior angles are equal: \(3x+2=2x+37\), so \(x=35\).
- \(3(35)+2=107\) and \(2(35)+37=107\). Each is \(107^{\circ}\).
- Same-side interior angles are supplementary: \(180-107=73\). The angle is \(73^{\circ}\).
Geometry Parallel Lines Worksheet Answer Key: problems 21–24
Problem 21: Bookshelf brace
Parallel lines have equal corresponding angles. Here \(83 \neq 85\).
No, the shelves are not parallel. They are \(2^{\circ}\) off, so one shelf must be adjusted.
Problem 22: Bus stop
\(M\) is the midpoint, so \(AM=16 : 2=8\). Right triangle \(\triangle PMA\): \(PA^{2}=8^{2}+6^{2}=64+36=100\), so \(PA=10\).
By the Perpendicular Bisector Theorem, \(PB=PA\). The stop is 10 blocks from each house.
Problem 23: Multi-step angle problem
\(\angle 1\) and \(\angle 5\) are corresponding: \(6x-12=4x+18\), so \(2x=30\) and \(x=15\). Then \(m\angle 1\) = \(6(15)-12=78\) and \(m\angle 5\) = \(78^{\circ}\).
\(\angle 4\) and \(\angle 1\) are vertical angles: \(m\angle 4\) = \(78^{\circ}\).
\(\angle 5\) and \(\angle 6\) form a linear pair: \(180-78=102\), so \(m\angle 6\) = \(102^{\circ}\).
Problem 24: Notebook paper
Same-side interior angles are supplementary: \((6y+4)+(3y+14)=180\).
\(9y+18=180\), so \(9y=162\) and \(y=18\).
\(6(18)+4=112\) and \(3(18)+14=68\). The angles are \(112^{\circ}\) and \(68^{\circ}\).
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Geometry Parallel Lines, Transversals and Proofs worksheet answer key
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