Answer:
Exterior angle = m∠160°
Step-by-step explanation:
Given the measures of two interior opposite angles, m75° and m∠85°, and that the prompt requires us to find the measure of the exterior angle of a triangle:
Exterior Angle Theorem
In order to find the measure of the exterior angle, we must apply the Exterior Angle Theorem which states that the measure of an exterior angle of a triangle is equal to the sum of the measures of two non-adjacent remote interior angles.
Solution:
Let x = exterior angle of a triangle
In order to solve for the measure of x, we can set up the following equation:
x = m∠75° + m∠85°
x = 160° ⇒ Measure of the exterior angle of a triangle.
Double-check:
We must verify whether we have the correct value for the exterior angle of a triangle. In reference to the Triangle Sum Theorem, which states that the sum of all interior angles of a triangle is equal to 180°.
Since we already have the measures of the two remote interior angles and the exterior angle of a triangle, we can find the measure of the third interior angle of a triangle by setting up the following equation:
m∠75° + m∠85° + m∠y° = 180°
Where ∠y is the unknown third interior angle of a triangle.
Solve for ∠y (Unknown third interior angle of a triangle)
m∠75° + m∠85° + m∠y° = 180°
160° + m∠y° = 180°
Subtract 160° from both sides of the equation:
160° - 160° + m∠y° = 180° - 160°
m∠y° = 20° ⇒ Third interior angle of a triangle.
Hence, the measures of the three interior angles of a triangle are: 75°, 85°, and 20°, for which the third angle (∠y) and the exterior angle of a triangle (∠x) are supplements:
m∠x° + m∠y° = 180°
160° + 20° = 180°
180° = 180° (True statement).
Therefore, the measure of the exterior angle of a triangle is 160°.
