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Difficulty: EasyTriangle Properties and Angle Theorems

In △XYZ\triangle XYZ, the measure of ∠X\angle X is 40∘40^\circ. If the measure of ∠Y\angle Y is three times the measure of ∠X\angle X, what is the measure, in degrees, of ∠Z\angle Z?

Answer: 20 degrees

Answer

The measure of ∠Z\angle Z is 2020 degrees.
To find the measure of ∠Z\angle Z, we first determine the measure of ∠Y\angle Y. Since ∠Y\angle Y is three times the measure of ∠X\angle X (40∘40^\circ), we calculate 3×40∘=120∘3 \times 40^\circ = 120^\circ. Because the interior angles of a triangle must sum to 180∘180^\circ, the remaining angle ∠Z\angle Z is found by subtracting the measures of ∠X\angle X and ∠Y\angle Y from 180∘180^\circ: 180∘−40∘−120∘=20∘180^\circ - 40^\circ - 120^\circ = 20^\circ.

Step-by-Step Solution

1
Calculate the measure of ∠Y\angle Y.
The measure of ∠Y\angle Y is 120∘120^\circ.
The measure of ∠Y\angle Y is specified to be three times the measure of ∠X\angle X, which is given as 40∘40^\circ. Multiplying 40∘40^\circ by 33 gives 120∘120^\circ.
2
Calculate the measure of ∠Z\angle Z.
The measure of ∠Z\angle Z is 20∘20^\circ.
The interior angles of any triangle sum to 180∘180^\circ. Subtracting the sum of the measures of ∠X\angle X (40∘40^\circ) and ∠Y\angle Y (120∘120^\circ) from 180∘180^\circ yields the measure of ∠Z\angle Z.

Key Concept

The sum of the interior angles of a triangle is always 180∘180^\circ.
Estimated Time:45s
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Topics

Triangle Properties and Angle Theorems

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