One exterior angle of an isosceles triangle measures 14°. What are the possible measures for the other two exterior angles? Calculate the mean measure of the exterior angles of a t... One exterior angle of an isosceles triangle measures 14°. What are the possible measures for the other two exterior angles? Calculate the mean measure of the exterior angles of a triangle. What types of triangles have some exterior angles equal? Read more

Steps to Solve

1. Understand Exterior Angles of a Triangle

The exterior angle of a triangle is equal to the sum of the two opposite interior angles. If one exterior angle measures 14°, the corresponding interior angle is given by:

$$ \text{Interior Angle} = 180° - 14° = 166° $$

Since it is an isosceles triangle, two interior angles are equal. Let these angles be denoted as $a$:

$$ a + a + 14° = 180° $$

2. Set Up the Equation for Angles

Using the equation for the sum of the angles in the triangle, we have:

$$ 2a + 14° = 180° $$

3. Solve for the Interior Angles

Rearranging the equation gives:

$$ 2a = 180° - 14° $$

Calculate $180° - 14°$:

$$ 2a = 166° $$

Dividing both sides by 2 yields:

$$ a = 83° $$

4. Determine the Two Remaining Exterior Angles

The remaining exterior angles for the isosceles triangle would be the exterior angles corresponding to the two equal interior angles:

$$ \text{Exterior Angle} = 180° - a = 180° - 83° = 97° $$

So both the remaining exterior angles measure 97°.

5. Calculate the Mean Measurement of the Exterior Angles

The mean measurement of the exterior angles of a triangle can be calculated by taking the average of the three exterior angles:

Mean =

$$ \text{Mean} = \frac{\text{Angle 1} + \text{Angle 2} + \text{Angle 3}}{3} $$

Using our angles:

$$ \text{Mean} = \frac{14° + 97° + 97°}{3} $$

Calculating the total:

$$ \text{Total} = 14° + 97° + 97° = 208° $$

Now divide by 3:

$$ \text{Mean} = \frac{208°}{3} \approx 69.33° $$

6. Types of Triangles with Exterior Angles

Every triangle, whether scalene, isosceles, or equilateral, has exterior angles. The relevant exterior angles can be calculated from their respective interior angles.