Question 146·Easy·Geometry and Trigonometry

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In $\triangle ABC$ , $\angle A$ and $\angle B$ are congruent, and $\angle C$ measures $72^{\circ}$ . What is the measure, in degrees, of $\angle A$ ?

For triangle angle questions, first translate any congruent angle information into variables (for example, if two angles are congruent, call each one $x$ ). Then use the fact that the interior angles of a triangle always add to $180$ degrees to write an equation, combine like terms, and solve for the variable. Finally, match your result to the answer choices, watching for distractors that reflect common mistakes like forgetting to divide among congruent angles.

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Identify the type of triangle

If two angles in a triangle are congruent, what does that tell you about those angles and possibly the sides?

Represent the unknown angles with a variable

Let $\angle A$ and $\angle B$ both be $x$ degrees. How can you write each angle of the triangle in terms of $x$ ?

Use the triangle angle sum

The three interior angles of any triangle add up to $180$ degrees. Write an equation involving $x$ , $x$ , and $72$ .

Solve the equation

After you combine like terms, you should get an equation of the form $2x + 72 = 180$ . Solve this equation to find $x$ , the measure of $\angle A$ .

Desmos Guide

Set up the angle equation in Desmos

Enter the linear equation y = 2x + 72 to represent the sum of the two equal angles and the $72^\circ$ angle, and enter the horizontal line y = 180 to represent the total angle sum of a triangle.

Find the x-value where the lines meet

Use the intersection tool (or tap/click where the graphs intersect) to find the point where the two graphs cross; the x-coordinate of this intersection is the measure of $\angle A$ in degrees.

Step-by-step Explanation

Use the fact that angles A and B are congruent

In $\triangle ABC$ , we are told that $\angle A$ and $\angle B$ are congruent (equal in measure).

Let the common measure be $x$ degrees. Then:

$\angle A = x$
$\angle B = x$
$\angle C = 72$ degrees (given).

Apply the triangle angle sum

The sum of the interior angles in any triangle is $180$ degrees.

So we write the equation:

x + x + 72 = 180

Combine like terms:

2x + 72 = 180

Now isolate $2x$ :

2x = 180 - 72 = 108

Solve for x and match to the answer choices

From the previous step we have:

2x = 108

Divide both sides by $2$ :

x = \frac{108}{2} = 54

So $\angle A$ measures $54$ degrees, which corresponds to choice B.

Question 146·Easy·Geometry and Trigonometry

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation