In triangle ABC, m∠B = (14x − 22)° and the measure of the exterior angle to ∠B is (6x − 18)°. Find m∠B.
154°
132°
48°
11°
Let's first determine what kind of angle the exterior angle to ∠B is.
In a triangle, the sum of the measures of the interior angles is 180°.
So, in triangle ABC, the sum of the measures of angles ∠A, ∠B, and ∠C is 180°.
The measure of the exterior angle to ∠B is equal to the sum of the measures of angles ∠B and ∠C.
Given that the measure of the exterior angle to ∠B is (6x − 18)°, we can set up the equation:
6x - 18 = (14x - 22) + ∠C
Now let's simplify the equation:
6x - 18 = 14x - 22 + ∠C
Rearranging the terms:
∠C = 6x - 18 - 14x + 22
∠C = -8x + 4
We know that the sum of the measures of angles ∠A, ∠B, and ∠C is 180°, so:
∠A + ∠B + ∠C = 180
Substituting the given expressions:
∠A + (14x - 22) + (-8x + 4) = 180
Now we can simplify:
∠A + 14x - 22 - 8x + 4 = 180
Combining like terms:
6x - 18 = 180
Adding 18 to both sides:
6x = 198
Dividing both sides by 6:
x = 33
Now we can substitute the value of x back into the original expression for ∠B:
∠B = 14x - 22
∠B = 14(33) - 22
∠B = 462 - 22
∠B = 440
Therefore, m∠B = 440°.