Question 376·Medium·Math

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In the figure shown, $\overline{AB} \parallel \overline{CD}$ . Which choice gives the measure of angle $x$ ?

With intersecting diagonals, check whether the angle you want sits opposite an angle in a triangle. Those vertical angles are equal, so you can place the unknown in a triangle with two known angles and use the $180^\circ$ total. A Desmos regression can find the remaining angle. Adding the two known angles alone gives their sum, not the missing angle.

Hints

Hint 1
At the intersection, vertical angles sit directly across from each other and have equal measures. Which angle inside triangle $DEC$ sits opposite $x$ ?
Hint 2
The three interior angles of a triangle add to $180^\circ$ . In triangle $DEC$ , count each given angle once alongside the angle at $E$ . What equation does that give?

Step-by-step

Approach 1: Use the triangle below the intersection
Step 1
Match the opposite angles
The top angle $x$ and $\angle DEC$ below it are vertical angles: they sit opposite each other where the diagonals cross. Vertical angles are equal, so $m\angle DEC=x^\circ$ .
Step 2
Write the triangle’s angle equation
Triangle $DEC$ contains the $25^\circ$ angle at $D$ , the $35^\circ$ angle at $C$ , and $x^\circ$ at $E$ . Its three interior angles add to $180^\circ$ , so $25+35+x=180$ . The two given angles don’t add up to $x$ ; all three angles add up to $180^\circ$ .
Step 3
Find the requested angle
Type $25+35+x_1\sim180$ in Desmos. The $\sim$ tells Desmos to find the value of $x_1$ that makes both sides equal. Desmos shows $x_1=120$ under PARAMETERS, so angle $x$ measures $120^\circ$ . Choice C.

Approach 2: Carry the angles to the top triangle
Step 1
Bring the 35-degree angle to A
Because $AB\parallel CD$ , diagonal $AC$ is a transversal, a line crossing both parallel bases. The angle at $C$ and $\angle BAE$ are alternate interior angles: they lie between the bases on opposite sides of $AC$ . So $m\angle BAE=35^\circ$ .
Step 2
Bring the 25-degree angle to B
Diagonal $BD$ crosses the same parallel bases. The angle at $D$ and $\angle ABE$ are alternate interior angles across $BD$ , so $m\angle ABE=25^\circ$ .
Step 3
Use the top triangle
The angles $35^\circ$ , $25^\circ$ , and $x^\circ$ are the three interior angles of triangle $AEB$ . A triangle’s angles total $180^\circ$ , so $35+25+x=180$ .
Step 4
Solve for x
Type $35+25+x_1\sim180$ in Desmos. It shows $x_1=120$ under PARAMETERS, so the angle at the top of the intersection measures $120^\circ$ . Choice C.

Question 376·Medium·Math

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