Question 126·200 Super-Hard SAT Math Questions·Geometry and Trigonometry

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In convex pentagon $ABCDE$ , segment $AB$ is parallel to segment $CD$ , and segment $BC$ is parallel to segment $DE$ . The measure of $\angle A$ is $96^\circ$ , and the measure of $\angle E$ is $142^\circ$ . Which choice is the measure, in degrees, of $\angle C$ ?

Parallel sides in a convex pentagon cue same-side interior angles: angles inside the parallel lines on one side of a crossing segment. Each such pair adds to $180^\circ$ . Use both parallel relationships and the pentagon’s angle total, then let Desmos solve the resulting equation. The trap is treating those angles as equal rather than supplementary.

Hints

Hint 1
A transversal is a segment that crosses two lines. Segment $BC$ crosses the parallel lines containing $AB$ and $CD$ . Which two pentagon angles lie inside those lines on the same side of $BC$ ?
Hint 2
Now use $CD$ as the crossing segment for the other parallel pair, $BC$ and $DE$ . The same-side interior angles at its ends add to $180^\circ$ ; they aren’t necessarily equal.
Hint 3
A pentagon can be split into three triangles, so its interior angles add to $540^\circ$ . Include all five corners in that sum, then use the parallel-line relationships to write two of them in terms of $\angle C$ .

Step-by-step

Use both parallel pairs and the pentagon angle sum
Step 1
Relate the angles at B and C
Let $B$ and $C$ stand for the angle measures at those corners, in degrees. Segment $BC$ crosses the parallel lines containing $AB$ and $CD$ . The angles at $B$ and $C$ are same-side interior angles, so they add to $180^\circ$ :
$B+C=180.$
Step 2
Relate the angles at C and D
Let $D$ stand for the angle measure at $D$ . This time, $CD$ crosses the parallel lines containing $BC$ and $DE$ . The angles at $C$ and $D$ are also same-side interior angles, so they add rather than match:
$C+D=180.$
Step 3
Write the pentagon angle sum
Drawing diagonals from $A$ to $C$ and $D$ splits the convex pentagon into three triangles. So its interior angles total $3(180)=540^\circ$ . Include the given measures at $A$ and $E$ :
$96+B+C+D+142=540.$
Step 4
Express B and D using C
Subtract $C$ from each parallel-line equation to express both other unknown angles in terms of $C$ :
$B=180-C,\qquad D=180-C.$
Step 5
Substitute into the angle sum
Replace $B$ and $D$ in the pentagon equation. Each $180-C$ accounts for one different corner, while the middle $C$ is the angle you need:
$96+(180-C)+C+(180-C)+142=540.$
Step 6
Solve for the requested angle
Type $96+(180-x_1)+x_1+(180-x_1)+142\sim540$ in Desmos. Use $x_1$ for the unknown angle measure and $\sim$ to ask Desmos to fit the equality. Under PARAMETERS, it shows $x_1=58$ . So the interior angle $\angle C$ measures $58^\circ$ . Choice D.

Question 126·200 Super-Hard SAT Math Questions·Geometry and Trigonometry

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