Solving for Triangle Angles with Given Exterior and Interior Angle Ratios

In this article, we will tackle a classic geometry problem: finding the angles of a triangle when given one exterior angle and the ratio of its opposite interior angles. This problem involves understanding basic properties of triangles and applying algebraic methods to solve for unknown angles. Let's dive into the solution step by step.

Problem Statement

Given that one of the exterior angles of a triangle is 135° and the ratio of its opposite interior angles is 1:2, we need to find the angles of the triangle.

Step 1: Finding the Corresponding Interior Angle

The first step is to find the measure of the corresponding interior angle. The exterior angle of a triangle is equal to the sum of the two opposite interior angles. Therefore, if one exterior angle is 135°, the sum of the two opposite interior angles can be found as follows:

The formula is:

(A B) 135°

where A and B are the measures of the two opposite interior angles.

Step 2: Using the Ratio of the Interior Angles

According to the problem, the two opposite interior angles, A and B, are in the ratio 1:2. This means:

A x and B 2x

Step 3: Substituting into the Equation

Substituting the expressions for A and B into the sum of the angles gives:

A B 135°

Substituting A x and B 2x into the equation:

x 2x 135°

This simplifies to:

3x 135°

Step 4: Solving for x

x frac{135°}{3} 45°

Step 5: Finding the Angles A and B

Now that we know x 45°, we can find the values of A and B:

A x 45°

and

B 2x 2 times 45° 90°

Step 6: Finding the Third Angle

In any triangle, the sum of all interior angles is 180°. If we denote the third angle as C, we can find C as follows:

A B C 180°

Substituting A 45° and B 90° into the equation:

45° 90° C 180°

This simplifies to:

C 180° - 135° 45°

Conclusion

The angles of the triangle are:

A 45°, B 90°, C 45°

Thus, the angles of the triangle are 45°, 90°, and 45°. This problem demonstrates the application of basic triangle properties and algebraic methods in solving for unknown angles.

Alternative Problem and Solution

Consider another problem where the exterior angle is 120° and the interior angles are in the ratio 2:3.

Alternative Steps:

1. The corresponding interior angle to the exterior angle of 120° is 60°.

2. Let the unknown angles be 2x and 3x.

3. The sum of the interior angles is 180°, and we know one interior angle is 60°. Therefore:

2x 3x 60° 180°

4. Simplifying, we get:

5x 60° 180°

5. Solving for x:

x frac{120°}{5} 24°

6. The angles are:

2x 48° and 3x 72°

Summary

Both problems demonstrate the importance of understanding the relationship between exterior and interior angles in a triangle. By applying simple algebra, we can solve for the unknown angles. The key steps involve identifying the corresponding interior angle, using the given ratios, and summing the angles appropriately.