In triangle , and . Point lies on such that . What is the measure, in degrees, of ?

Solution available with step-by-step explanation

SAT Lines, Angles & Triangles Question 42 | Free SAT Prep | Aniko

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Question 42·Hard·Lines, Angles, and Triangles

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In triangle $ABC$ , $\angle ABC = 46^\circ$ and $\angle ACB = 71^\circ$ . Point $D$ lies on $\overline{AC}$ such that $\angle BDC = 74^\circ$ . What is the measure, in degrees, of $\angle ABD$ ?

Your answer

(Express the answer as an integer)

For geometry problems with a point on a side of a triangle and several angles given, first use the triangle angle-sum rule to find any missing vertex angles. Then, use straight-line (linear pair) relationships where a new point lies on a segment: angles that share a side and lie on a straight line add to $180^\circ$ . After finding all needed interior angles, focus on the smaller triangle that contains the unknown angle and apply the triangle angle-sum again. Keeping a clear, labeled sketch and systematically writing angle equations helps avoid mixing up which angle is which and speeds up solving on the SAT.

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Start with triangle ABC

Use the fact that the sum of the interior angles of triangle $ABC$ is $180^\circ$ to find $\angle BAC$ .

Use that D is on segment AC

Because $D$ lies on $\overline{AC}$ , the segments $\overline{AD}$ and $\overline{DC}$ form a straight line. Think about how this relates $\angle BDC$ and $\angle BDA$ .

Look at triangle ABD

Once you know $\angle BAD$ and $\angle BDA$ , apply the triangle angle sum in triangle $ABD$ to solve for $\angle ABD$ .

Desmos Guide

Compute angle BAC

In Desmos, type 180 - (46 + 71) to find the measure of $\angle BAC$ (which equals $\angle BAD$ ).

Compute angle BDA

Next, type 180 - 74 in Desmos to find the measure of $\angle BDA$ , using the fact that $\angle BDA$ and $\angle BDC$ form a linear pair on line $AC$ .

Compute angle ABD

Finally, in Desmos type 180 - ([result for BAD] + [result for BDA]), or directly 180 - (63 + 106), to get the measure of $\angle ABD$ . The numerical output is the requested angle.

Step-by-step Explanation

Find angle A in triangle ABC

In any triangle, the three interior angles add up to $180^\circ$ .

For triangle $ABC$ :

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

Substitute the given values $\angle ABC = 46^\circ$ and $\angle ACB = 71^\circ$ :

\angle BAC + 46^\circ + 71^\circ = 180^\circ

\angle BAC = 180^\circ - (46^\circ + 71^\circ)

This gives the measure of $\angle BAC$ , which is the same as $\angle BAD$ because $D$ lies on segment $AC$ .

Relate angles BDC and BDA using the straight line AC

Points $A$ , $D$ , and $C$ lie on the same straight line $AC$ . That means $\overline{AD}$ and $\overline{DC}$ are parts of the same straight line but in opposite directions.

At point $D$ , the angles $\angle BDA$ and $\angle BDC$ share side $\overline{BD}$ , and their other sides, $\overline{DA}$ and $\overline{DC}$ , are opposite rays on a straight line.

Therefore, $\angle BDA$ and $\angle BDC$ form a linear pair, so they add up to $180^\circ$ :

\angle BDA + \angle BDC = 180^\circ

Given $\angle BDC = 74^\circ$ ,

\angle BDA = 180^\circ - 74^\circ

Now you know $\angle BDA$ as well.

Use triangle ABD to find ∠ABD

Now look at triangle $ABD$ . Its three interior angles are $\angle ABD$ , $\angle BAD$ , and $\angle BDA$ .

We already found:

$\angle BAD = \angle BAC = 180^\circ - (46^\circ + 71^\circ) = 63^\circ$
$\angle BDA = 180^\circ - 74^\circ = 106^\circ$

In triangle $ABD$ :

\angle ABD + \angle BAD + \angle BDA = 180^\circ

Substitute the known angles:

\angle ABD + 63^\circ + 106^\circ = 180^\circ

Combine the known angles:

\angle ABD + 169^\circ = 180^\circ

Subtract $169^\circ$ from both sides:

\angle ABD = 180^\circ - 169^\circ = 11^\circ

So the measure of $\angle ABD$ is $11$ degrees.

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Desmos Guide

Step-by-step Explanation

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Desmos Guide

Step-by-step Explanation