Question 368·Hard·Math

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In triangle $ABC$ , point $D$ lies on segment $\overline{AB}$ and point $E$ lies on segment $\overline{AC}$ . Segment $\overline{DE}$ is drawn.

The measure of angle $BAC$ is $40^\circ$ . The measure of angle $ABC$ is $(3x-5)^\circ$ . The measure of angle $AED$ is $(2x+10)^\circ$ .

Which choice gives the value of $x$ for which $\overline{DE}$ is parallel to $\overline{BC}$ ?

Note: Figure not drawn to scale.

A segment inside a triangle may need to be parallel to one of its sides. Corresponding angles are angles in matching positions where a line crosses two parallel lines; use them to move the marked angle into the large triangle. Then add that triangle’s angles to get an equation and solve it in Desmos. Don’t match angles by how they look in a figure that isn’t drawn to scale.

Hints

Hint 1
Side $AC$ crosses $DE$ at $E$ and $BC$ at $C$ . If those segments are parallel, corresponding angles at the two crossings are equal. Which angle at $C$ is in the same position as $\angle AED$ ?
Hint 2
The interior angles of triangle $ABC$ add to $180^\circ$ . Once you know which angle at $C$ matches the marked angle at $E$ , include that measure with the angles at $A$ and $B$ in one sum.
Hint 3
Use a regression to solve your angle equation in Desmos: change $x$ to $x_1$ and $=$ to $\sim$ . Read the value under PARAMETERS, rather than estimating angles from the drawing.

Step-by-step

Match angles, then use the triangle sum
Step 1
Find the matching angle in the large triangle
For $DE$ to be parallel to $BC$ , side $AC$ crosses the two parallel segments at $E$ and $C$ . Corresponding angles occupy matching positions at those crossings. The angle at $E$ matches the angle at $C$ , not the angle at $B$ . So $m\angle ACB=m\angle AED=(2x+10)^\circ$ .
Step 2
Write the large triangle’s angle equation
The three interior angles of triangle $ABC$ add to $180^\circ$ . Use the marked angle at $E$ as the measure of angle $C$ , and count each corner once: $40+(3x-5)+(2x+10)=180$ .
Step 3
Solve for the value that makes the segments parallel
Type $40+(3x_1-5)+(2x_1+10)\sim180$ . Desmos uses $x_1$ as the unknown; a regression finds the value that makes the two sides equal. Under PARAMETERS, it shows $x_1=27$ . So $DE$ is parallel to $BC$ when $x=27$ . Choice C.

Question 368·Hard·Math

Strategy

Hints

Step-by-step

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Strategy

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Step-by-step

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