SAT Lines, Angles & Triangles Question 3 | Free SAT Prep

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Question 3·Hard·Lines, Angles, and Triangles

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In $\triangle ABC$ , point $D$ lies on side $AC$ and point $E$ lies on side $AB$ such that $DE \parallel BC$ . Ray $BF$ is the external bisector of the angle formed by $\overline{AB}$ and the extension of $\overline{BC}$ beyond $B$ .

If $\angle ADE = 28^{\circ}$ and $\angle EBF = 19^{\circ}$ , what is the measure, in degrees, of $\angle BAC$ ?

Your answer

For geometry angle-chasing problems with parallel lines and angle bisectors, first use similar triangles (formed by a line parallel to one side of a triangle) to transfer given angles to the main triangle's vertices. Next, carefully interpret whether a bisected angle is interior or exterior; exterior and interior angles at a vertex form a straight line, so they are supplementary (sum to $180^\circ$ ). Once you have the interior angles at two vertices, quickly apply the triangle angle sum $A + B + C = 180^\circ$ to solve for the remaining angle. Always label angles on a quick sketch to avoid mixing up interior vs. exterior angles or the vertex you are solving for.

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Use similar triangles from DE parallel to BC

Because $DE \parallel BC$ , triangles $ADE$ and $ABC$ are similar.

Which angle in triangle $ABC$ corresponds to $\angle ADE$ ?
Use this to identify the measure of $\angle ACB$ in the big triangle.

Use the external angle bisector at B

Focus on ray $BF$ .

What two rays form the exterior angle that $BF$ is bisecting?
If $\angle EBF = 19^\circ$ , how can you use that to find the full measure of this exterior angle at $B$ ?

Connect the exterior angle at B to the interior angle at B

The interior angle at $B$ (inside the triangle) and the exterior angle at $B$ lie on a straight line.

What is the relationship between angles that form a straight line?
Once you know the exterior angle measure, how do you find the interior angle $\angle ABC$ ?

Finish with the triangle angle sum

Now you know $\angle ABC$ and $\angle ACB$ .

Use the fact that the three interior angles of a triangle add up to $180^\circ$ .
Set up an equation involving $\angle BAC$ , $\angle ABC$ , and $\angle ACB$ and solve for $\angle BAC$ .

Desmos Guide

Check the final angle calculation

After you reason that the interior angle at $B$ is $180 - 2 \cdot 19$ degrees and the angle at $C$ is $28$ degrees, you can let Desmos handle the arithmetic.

In Desmos, type the expression

180 - (180 - 2\cdot 19) - 28

The value that Desmos outputs is the measure (in degrees) of $\angle BAC$ .

Step-by-step Explanation

Use similar triangles to identify angle C

We are told $DE \parallel BC$ , with $D$ on $AC$ and $E$ on $AB$ .

Because $DE \parallel BC$ , triangles $ADE$ and $ABC$ are similar (by AA similarity). Therefore, corresponding angles are equal:

\angle ADE = \angle ACB.

Given $\angle ADE = 28^\circ$ , we get

\angle ACB = 28^\circ.

Interpret what angle EBF represents

Ray $BF$ is the external bisector of the angle formed by $\overline{AB}$ and the extension of $\overline{BC}$ beyond $B$ .

Extend $\overline{BC}$ past $B$ to form an exterior angle with $\overline{BA}$ ; this is the exterior angle at $B$ .
Because $E$ lies on $\overline{AB}$ , segment $\overline{BE}$ lies along $\overline{BA}$ .
$BF$ bisects that exterior angle, so $\angle EBF$ is half of the exterior angle at $B$ .

Given $\angle EBF = 19^\circ$ , the measure of the exterior angle at $B$ is

2 \cdot 19^\circ = 38^\circ.

Relate the exterior and interior angles at B

At vertex $B$ , the interior angle $\angle ABC$ and the exterior angle (the one we just found as $38^\circ$ ) lie on a straight line, so they are supplementary (sum to $180^\circ$ ):

\angle ABC + 38^\circ = 180^\circ.

Thus the interior angle at $B$ is

\angle ABC = 180^\circ - 38^\circ = 142^\circ.

Apply the triangle angle sum to find angle A

In any triangle, the three interior angles sum to $180^\circ$ :

\angle BAC + \angle ABC + \angle ACB = 180^\circ.

We have already found

$\angle ABC = 142^\circ$ ,
$\angle ACB = 28^\circ$ .

Substitute these values:

\angle BAC + 142^\circ + 28^\circ = 180^\circ\\[0.7em] \angle BAC + 170^\circ = 180^\circ\\[0.7em] \angle BAC = 180^\circ - 170^\circ = 10^\circ.

So the measure of $\angle BAC$ is $10^\circ$ .

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Step-by-step Explanation

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Step-by-step Explanation