For triangle XYZ, m∠X = (5g + 18)° and the exterior angle to ∠X measures (8g + 32)°. Find the measure of ∠X and its exterior angle.
a Interior angle = 68°; exterior angle = 112°
b Interior angle = 112°; exterior angle = 68°
c Interior angle = 73°; exterior angle = 105°
d Interior angle = 105°; exterior angle = 73°
To find the measure of ∠X, we need to set up an equation using the fact that the sum of the angles in a triangle is 180°.
∠X + ∠Y + ∠Z = 180°
Substituting the given values, we have:
(5g + 18)° + ∠Y + ∠Z = 180°
To find the measure of the exterior angle to ∠X, we know that it is equal to the sum of the other two interior angles of the triangle. So we have:
(8g + 32)° = ∠Y + ∠Z
Now we have a system of equations with two variables (∠Y and ∠Z) and can solve for ∠X, ∠Y, and ∠Z.
From equation 2, we can solve for ∠Y:
∠Y = (8g + 32)° - ∠Z
Substituting this into equation 1:
(5g + 18)° + ((8g + 32)° - ∠Z) + ∠Z = 180°
Simplifying:
5g + 18 + 8g + 32 = 180
13g + 50 = 180
13g = 130
g = 10
Now that we know g = 10, we can find the measure of ∠X and its exterior angle by substituting g back into the equations:
∠X = (5(10) + 18)° = 68°
Exterior angle = (8(10) + 32)° = 112°
Therefore, the answer is option a) Interior angle = 68°; exterior angle = 112°.