In \triangle ABC, point D lies on side BC. The segment AD divides the interior angle \angl… — ACT

In $\triangle ABC$ , point $D$ lies on side $BC$ . The segment $AD$ divides the interior angle $\angle BAC$ into two angles, $\angle BAD$ and $\angle DAC$ , whose measures are in the ratio $3:2$ , respectively. The measures of the interior angles $\angle B$ and $\angle C$ are in the ratio $5:4$ , respectively. If the measure of $\angle ADC$ is $104^\circ$ , what is the measure of $\angle BAC$ ?

A
$54^\circ$
B
$72^\circ$
C
$76^\circ$
$90^\circ$ Answer
E
$108^\circ$

Answer

90^\circ

The correct answer is

90^\circ

. By expressing the angles in terms of variables using their ratios, we set up two independent linear equations:

3x + 5y = 104

(from the exterior angle theorem on

\triangle ABD

) and

2x + 4y = 76

(from the sum of angles in

\triangle ADC

). Solving this system yields

x = 18

. Since

\angle BAC

is composed of

\angle BAD

and

\angle DAC

, its measure is

3x + 2x = 5x = 5(18^\circ) = 90^\circ

Step-by-Step Solution

Define variables for the partitioned angles and the base angles using the given ratios.

Let the measures of

\angle BAD

and

\angle DAC

3x

and

2x

respectively, so that

\angle BAC = 5x

. Let the measures of

\angle B

and

\angle C

5y

and

4y

respectively.

Ratios express quantities as multiples of a common variable, which simplifies setting up equations.

Apply the exterior angle theorem to

\triangle ABD

at vertex

D

The exterior angle

\angle ADC = \angle BAD + \angle B \Rightarrow 104^\circ = 3x + 5y

The measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles.

Apply the angle sum theorem to

\triangle ADC

\angle DAC + \angle C + \angle ADC = 180^\circ \Rightarrow 2x + 4y + 104^\circ = 180^\circ \Rightarrow 2x + 4y = 76^\circ

The sum of the measures of the interior angles of any triangle is always

180^\circ

Solve the system of linear equations: (1)

3x + 5y = 104

and (2)

2x + 4y = 76

Multiply equation (1) by 2 and equation (2) by 3 to align the coefficients of

x

6x + 10y = 208

6x + 12y = 228

Subtract the first aligned equation from the second:

2y = 20 \Rightarrow y = 10

.
Substitute

y = 10

back into equation (2):

2x + 4(10) = 76 \Rightarrow 2x = 36 \Rightarrow x = 18

Solving the system of linear equations determines the values of the variables

x

and

y

Calculate the measure of

\angle BAC

\angle BAC = 5x = 5(18^\circ) = 90^\circ

We defined the total measure of

\angle BAC

as the sum of its two partitioned parts,

3x + 2x = 5x

Key Concept

Using triangle angle properties and exterior angle theorems to set up and solve systems of linear equations.

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