According to the diagram, exterior angle BCD is the sum of angles ABC and ACD. Since ACD is an exterior angle of triangle ABC, it measures 130 degrees. Therefore, angle BCD measures 130 degrees.
The exterior angle theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles.
In other words, if we have an exterior angle at point A of triangle ABC, then the measure of that angle is equal to the sum of the measures of angles B and C.
This theorem can be proven using a variety of methods, but one of the simplest proofs is to use the fact that the sum of the measures of the angles in a triangle is 180 degrees.
In the diagram, angle ACD is an exterior angle of triangle ABC.
We want to show that the measure of angle ACD is equal to the sum of the measures of angles B and C.
To do this, we first draw a line segment from point A to point C, as shown in the diagram.
This line segment creates two new triangles, ABD and ADC.
We know that the sum of the measures of the angles in a triangle is 180 degrees, so we have the following equations:
∠A + ∠B + ∠C = 180 degrees
∠A + ∠ABD + ∠ADC = 180 degrees
Subtracting these two equations, we get the following:
∠B + ∠C = ∠ABD + ∠ADC
But angle ACD is an exterior angle of triangle ABD, so it measures ∠ABD + ∠B.
Therefore, we can substitute ∠ABD + ∠B for ∠ACD in the above equation, which gives us the following:
∠B + ∠C = (∠ABD + ∠B) + ∠ADC
Simplifying both sides of the equation, we get the following:
∠B + ∠C = ∠B + ∠C
This tells us that ∠B + ∠C = 180 degrees, which means that the measure of angle ACD is equal to the sum of the measures of angles B and C.
This is the exterior angle theorem.
Since ACD is an exterior angle of triangle ABC, it measures 180 - ∠CAB = 180 - (14x + 16) = 166 - 14x degrees.
Therefore, the measure of exterior angle BCD is:
∠BCD = ∠ABC + ∠ACD = (22x - 2) + (166 - 14x) = 130 degrees