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Question 5·Medium·Lines, Angles, and Triangles

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In $\triangle ABC$ , side $BC$ is extended through $C$ to point $D$ , as shown.

If $m\angle A = 4x - 7^{\circ}$ , $m\angle B = 3x + 22^{\circ}$ , and $m\angle ACD = 9x - 5^{\circ}$ , what is $m\angle ACB$ ?

For triangle angle problems with an exterior angle, first identify which angle is exterior and recall that it equals the sum of the two non-adjacent interior angles. Translate the angle relationships into a simple linear equation in x, solve carefully, and then substitute back to get actual angle measures. Finally, check exactly which angle the question asks for (interior vs. exterior or a specific vertex) to avoid picking an intermediate result instead of the correct final angle.

Hints

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Identify the special angle at C

Look at $\angle ACD$ . How is it positioned relative to $\triangle ABC$ ? Is it inside the triangle or outside on an extension of a side?

Relate the exterior angle to interior angles

For a triangle, the measure of an exterior angle equals the sum of the two interior angles that are not adjacent to it. Which angles in the problem play those roles?

Set up and solve for x

Write an equation setting $9x - 5$ equal to the sum of $4x - 7$ and $3x + 22$ , then solve for $x$ carefully. After you have $x$ , plug it back into the expressions to get the actual angle measures.

Be careful which angle is being asked for

The problem asks for $m\angle ACB$ , the interior angle at C. How is this angle related to the exterior angle $m\angle ACD$ that you can find first?

Desmos Guide

Use Desmos to solve for x

In Desmos, enter two expressions as functions: y1 = 9x - 5 and y2 = (4x - 7) + (3x + 22). Find their intersection point; the x-coordinate of this intersection is the value of x that satisfies the equation.

Compute the exterior angle at C

In a new expression line, type 9*(x_value) - 5, replacing x_value with the x-coordinate you found from the intersection. The output is the measure of the exterior angle $m\angle ACD$ .

Find the interior angle ACB from the exterior angle

Since $\angle ACB$ and $\angle ACD$ form a straight line, type 180 - (exterior_angle_value) in a new expression, replacing exterior_angle_value with the result from the previous step. The number you see is the measure of $\angle ACB$ .

Step-by-step Explanation

Use the exterior angle theorem

At vertex C, side $BC$ is extended to $D$ , so $\angle ACD$ is an exterior angle of $\triangle ABC$ .

The exterior angle theorem says:

The measure of an exterior angle of a triangle equals the sum of the measures of the two non-adjacent interior angles.

So here:

m\angle ACD = m\angle A + m\angle B

In symbols with the given expressions:

9x - 5 = (4x - 7) + (3x + 22)

Set up and simplify the equation for x

Start with the equation from the exterior angle relationship:

9x - 5 = (4x - 7) + (3x + 22)

Combine like terms on the right side:

$4x + 3x = 7x$
$-7 + 22 = 15$

So the equation becomes:

9x - 5 = 7x + 15

Solve for x and find the exterior angle

Solve the equation:

9x - 5 = 7x + 15

Subtract $7x$ from both sides:

2x - 5 = 15

Add 5 to both sides:

2x = 20

Divide by 2:

x = 10

Now plug $x = 10$ into $m\angle ACD = 9x - 5$ to find the exterior angle at C:

m\angle ACD = 9(10) - 5 = 90 - 5 = 85^{\circ}

This is the exterior angle at C, not the one the question asks for.

Find the interior angle at C, ∠ACB

Angle $\angle ACD$ and $\angle ACB$ form a linear pair (they are adjacent and lie on a straight line), so they are supplementary:

m\angle ACB + m\angle ACD = 180^{\circ}

Substitute $m\angle ACD = 85^{\circ}$ :

m\angle ACB + 85 = 180

Subtract 85 from both sides:

m\angle ACB = 95^{\circ}

So $m\angle ACB = 95^{\circ}$ , which corresponds to answer choice D) 95.

Question 5·Medium·Lines, Angles, and Triangles

Strategy

Hints

Desmos Guide

Step-by-step Explanation

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Strategy

Hints

Desmos Guide

Step-by-step Explanation