Examkin

Question

Difficulty: MediumTriangle Properties and Angle Theorems

An exterior angle of a triangle measures 135∘135^\circ. The two nonadjacent interior angles of the triangle have measures in the ratio 2:32:3. What is the measure, in degrees, of the largest interior angle of the triangle?

Answer: 81 degrees

Answer

The measure of the largest interior angle of the triangle is 81 degrees.
According to the Exterior Angle Theorem, the sum of the two nonadjacent interior angles equals the measure of the exterior angle: 2x+3x=1352x + 3x = 135, which simplifies to 5x=1355x = 135 and gives x=27x = 27. The nonadjacent interior angles are 2(27)=54∘2(27) = 54^\circ and 3(27)=81∘3(27) = 81^\circ. The adjacent interior angle is 180∘−135∘=45∘180^\circ - 135^\circ = 45^\circ. Comparing the three angles (45∘45^\circ, 54∘54^\circ, 81∘81^\circ), the largest is 81∘81^\circ.

Step-by-Step Solution

1
Use the Exterior Angle Theorem to relate the two nonadjacent interior angles to the given exterior angle.
Equation: 2x+3x=1352x + 3x = 135
The theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of its two nonadjacent interior angles.
2
Solve the linear equation for xx.
x=27x = 27
Combine like terms to get 5x=1355x = 135, then divide both sides by 5.
3
Calculate the measures of the two nonadjacent interior angles.
First angle: 2(27)=54∘2(27) = 54^\circ; Second angle: 3(27)=81∘3(27) = 81^\circ
Multiply each part of the ratio by the scale factor x=27x = 27.
4
Calculate the measure of the remaining interior angle.
Adjacent angle: 180∘−135∘=45∘180^\circ - 135^\circ = 45^\circ
An interior angle and its adjacent exterior angle lie on a straight line and are supplementary (they sum to 180∘180^\circ).
5
Compare the three interior angle measures to find the largest.
The largest angle is 81∘81^\circ.
Comparing 45∘45^\circ, 54∘54^\circ, and 81∘81^\circ shows that 81∘81^\circ is the maximum value.

Key Concept

Exterior Angle Theorem and interior angle relationships in a triangle

Alternative Method

Find the adjacent interior angle first: 180∘−135∘=45∘180^\circ - 135^\circ = 45^\circ. Since the sum of all interior angles in a triangle is 180∘180^\circ, the sum of the remaining two interior angles must be 180∘−45∘=135∘180^\circ - 45^\circ = 135^\circ. Set up the ratio equation 2x+3x=1352x + 3x = 135 and solve for xx to find the other two angles.
Estimated Time:1m 15s
Rate this question

Topics

Triangle Properties and Angle Theorems

More questions from this topic