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Difficulty: EasyTriangle Properties and Angle Theorems

In △PQR\triangle PQR, the measure of interior angle ∠P\angle P is 40∘40^\circ. If the measure of ∠Q\angle Q is twice the measure of ∠P\angle P, what is the measure, in degrees, of the interior angle ∠R\angle R?

Answer: 60 degrees

Answer

The measure of interior angle ∠R\angle R is 60 degrees.
By definition, the interior angles of any triangle sum to 180 degrees. Given that angle P is 40 degrees, and angle Q is twice angle P (80 degrees), the sum of angles P and Q is 120 degrees. Subtracting this from 180 degrees gives a remaining measure of 60 degrees for angle R.

Step-by-Step Solution

1
Calculate the measure of angle Q
m∠Q=80∘m\angle Q = 80^\circ
The problem states that the measure of angle Q is twice the measure of angle P, which is 40 degrees.
2
Apply the Triangle Angle Sum Theorem
m\angle P + m\angle Q + m\angle R = 180^\circ
The sum of the measures of the interior angles of any triangle is always 180 degrees.
3
Solve for the measure of angle R
m∠R=60∘m\angle R = 60^\circ
Substitute the values of angles P and Q into the equation: 40 + 80 + m\angle R = 180, which simplifies to 120 + m\angle R = 180, and solving for m\angle R gives 60.

Key Concept

The sum of the measures of the interior angles of a triangle is always 180 degrees.
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Topics

Triangle Properties and Angle Theorems

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