SAT Lines, Angles & Triangles Question 45 | Free SAT Prep | Aniko

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Question 45·Hard·Lines, Angles, and Triangles

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In $\triangle ABC$ , point $D$ lies on side $AC$ . The following angle measures are known:

$\angle CBD = 28^{\circ}$
$\angle BDC = 27^{\circ}$
$\angle BAD = 20^{\circ}$

What is the measure, in degrees, of $\angle ABC$ ?

Your answer

(Express the answer as an integer)

For geometry problems with points on the sides of a triangle, start by marking all given angles on a clear sketch. Look for straight lines (collinear points) to create linear pairs that sum to $180^\circ$ , and use the triangle angle-sum rule in smaller triangles formed by interior segments like $BD$ . Often, once you find one or two key interior angles, you can get the target angle by simple addition or subtraction, so keep track of how segments split the main angles of the triangle.

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Notice the straight line at D

Point $D$ lies on side $AC$ . What does that tell you about $\angle ADC$ , and how is $\angle BDC$ related to that straight angle?

Relate angles ADB and BDC

Since $A$ , $D$ , and $C$ are collinear and $BD$ comes off that line, angles $\angle ADB$ and $\angle BDC$ sit next to each other along a straight line. What must the sum of those two angles be?

Use triangle ABD, then angle addition at B

Once you know $\angle ADB$ , use the fact that the three angles in triangle $ABD$ add to $180^\circ$ to find $\angle ABD$ . Finally, remember that $BD$ splits angle $ABC$ into $\angle ABD$ and $\angle DBC$ —how can you use that to get $\angle ABC$ ?

Desmos Guide

Compute angle ADB from the linear pair

In Desmos, type 180 - 27 to calculate $\angle ADB$ using the fact that $\angle ADB$ and $\angle BDC$ form a straight angle. The result is the measure of $\angle ADB$ in degrees.

Use the triangle angle sum to find angle ABD

Next, type 180 - 20 - (result_of_step_1) to represent $180^\circ - \angle BAD - \angle ADB$ and find $\angle ABD$ from triangle $ABD$ . Desmos will show the value of $\angle ABD$ .

Add the two angles at B to get angle ABC

Finally, type (result_of_step_2) + 28 to add $\angle ABD$ to $\angle DBC = 28^\circ$ . The value Desmos displays is the measure of $\angle ABC$ in degrees; use that as your final answer.

Step-by-step Explanation

Use the straight line through A, D, and C

Because point $D$ lies on side $AC$ , points $A$ , $D$ , and $C$ are collinear. That means $\angle ADC$ is a straight angle, so it measures $180^\circ$ .

Ray $DB$ comes off this straight line, so the two angles $\angle ADB$ and $\angle BDC$ sit next to each other along the straight line $AC$ and form a linear pair. Therefore,

\angle ADB + \angle BDC = 180^\circ.

We are given $\angle BDC = 27^\circ$ , so

\angle ADB = 180^\circ - 27^\circ = 153^\circ.

Find angle ABD in triangle ABD

Now look at triangle $ABD$ . Its three interior angles are:

$\angle BAD = 20^\circ$ (given),
$\angle ADB = 153^\circ$ (from Step 1),
$\angle ABD$ (unknown for now).

The sum of the interior angles in any triangle is $180^\circ$ , so

\angle BAD + \angle ADB + \angle ABD = 180^\circ.

Substitute the known values:

20^\circ + 153^\circ + \angle ABD = 180^\circ.

\angle ABD = 180^\circ - 20^\circ - 153^\circ = 7^\circ.

Combine angles at B to get angle ABC

Segment $BD$ lies inside triangle $ABC$ , splitting angle $ABC$ into two adjacent angles:

$\angle ABD$ (between $BA$ and $BD$ ),
$\angle DBC$ (between $BD$ and $BC$ ).

\angle ABC = \angle ABD + \angle DBC.

We know $\angle ABD = 7^\circ$ (from Step 2) and $\angle DBC = \angle CBD = 28^\circ$ (given), so

\angle ABC = 7^\circ + 28^\circ = 35^\circ.

Thus, the measure of $\angle ABC$ is $35$ degrees.

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Desmos Guide

Step-by-step Explanation

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Desmos Guide

Step-by-step Explanation