This answer key shows a complete solution and a scoring guide for each question of the parallel lines and transversals test. Award partial credit using the point breakdown listed after each solution.
Question 1: Name the angle pairs (2 points)
- \(\angle 4\) and \(\angle 6\) are between the lines on the same side of the transversal: same-side interior angles.
- \(\angle 1\) and \(\angle 8\) are outside the lines on opposite sides: alternate exterior angles.
Scoring: 1 point for each correct name.
Question 2: Find missing angles (2 points)
- \(m\angle 6\) = \(54^{\circ}\) (alternate interior angles are equal).
- \(m\angle 5\) = \(180-54=126\) (same-side interior angles are supplementary). \(m\angle 8\) = \(126^{\circ}\) because \(\angle 8\) and \(\angle 5\) are vertical angles. Both are \(126^{\circ}\).
Scoring: 1 point for part (a). 1 point for part (b): 0.5 for each angle.
Question 3: Corresponding angles and x (2 points)
Corresponding angles are equal: \(7x-9=4x+24\). Then \(3x=33\), so \(x=11\).
\(7(11)-9=68\) and \(4(11)+24=68\). Each angle is \(68^{\circ}\).
Scoring: 1 point for the correct equation, 1 point for \(x=11\) and both angle measures.
Question 4: Same-side interior angles and x (2 points)
Same-side interior angles are supplementary: \((3x+12)+(5x-8)=180\). Then \(8x+4=180\) and \(8x=176\), so \(x=22\).
\(3(22)+12=78\) and \(5(22)-8=102\). The angles are \(78^{\circ}\) and \(102^{\circ}\).
Scoring: 1 point for the correct equation and \(x=22\), 1 point for both angle measures.
Question 5: Decide if lines are parallel (2 points)
- Yes. The alternate interior angles are equal, so \(a \parallel b\) by the Alternate Interior Angles Converse.
- No. \(100+70=170\), which is not \(180^{\circ}\), so the same-side interior angles are not supplementary.
Scoring: 1 point per part: 0.5 for the decision and 0.5 for the reason.
Question 6: Proof that lines are parallel (3 points)
| Statements | Reasons |
|---|---|
| \(m\angle 2\) = \(m\angle 7\) | Given |
| \(m\angle 2\) = \(m\angle 3\) | Vertical Angles Theorem |
| \(m\angle 3\) = \(m\angle 7\) | Substitution Property of Equality |
| \(a \parallel b\) | Corresponding Angles Converse |
Scoring: 1 point for \(m\angle 2\) = \(m\angle 3\) with its reason, 1 point for \(m\angle 3\) = \(m\angle 7\), 1 point for the conclusion with the correct converse. Other valid chains also earn full credit.
Question 7: Perpendicular Bisector Theorem proof (3 points)
| Statements | Reasons |
|---|---|
| \(AM=BM\) | Definition of bisector (M is the midpoint) |
| \(m\angle PMA\) = \(m\angle PMB\) = \(90^{\circ}\) | Definition of perpendicular lines |
| \(PM=PM\) | Reflexive Property |
| \(\triangle PMA\) is congruent to \(\triangle PMB\) | SAS (two sides and the included angle) |
| \(PA=PB\) | Corresponding parts of congruent triangles are congruent |
Scoring: 1 point for the two equal sides and the equal right angles, 1 point for triangle congruence by SAS, 1 point for the conclusion with the CPCTC reason.
Question 8: Equidistant point (2 points)
\(PA=PB\), so \(4x-3=2x+13\). Then \(2x=16\), so \(x=8\).
\(PA=4(8)-3=29\) (and \(PB=2(8)+13=29\)).
Scoring: 1 point for the equation and \(x=8\), 1 point for \(PA=29\).
Question 9: Parallel avenues (2 points)
The angles are alternate exterior angles, so they are equal: \(2x+15=3x-10\). Then \(x=25\).
Each angle is \(2(25)+15=65\) degrees. The northwest angle at Cedar forms a linear pair with the northeast angle: \(180-65=115\). It measures \(115^{\circ}\).
Scoring: 1 point for \(x=25\) with the correct equation, 1 point for \(115^{\circ}\).
20 points
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