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In triangle ABC, segment BD bisects angle ABC. If the measure of angle ABD is represented by (8x + 35) degrees, what are the measures of angles CBD and ABC? A) m/ABD = (8x + 35) degrees, m/CBD = (11x - 23) degrees, m/ABC = (19x + 12) degrees (Correct) B) m/ABD = (8x + 35) degrees, m/CBD = (11x + 23) degrees, m/ABC = (19x - 12) degrees C) m/ABD = (8x - 35) degrees, m/CBD = (11x - 23) degrees, m/ABC = (19x + 12) degrees D) m/ABD = (8x - 35) degrees, m/CBD = (11x + 23) degrees, m/ABC = (19x - 12) degrees

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Explanation:

To find the measures of angles CBD and ABC in triangle ABC where segment BD bisects angle ABC, you can use the angle bisector theorem. According to the theorem, if a segment bisects an angle in a triangle, it divides the opposite side in such a way that the ratio of the two line segments is equal to the ratio of the two adjacent sides.

In this case:

m/ABD = m/ACD (since BD bisects angle ABC)

Now, let's denote the measure of angle ABD as (8x + 35) degrees. Then, the measure of angle ACD is also (8x + 35) degrees.

Now, to find m/CBD and m/ABC:

m/CBD = 180 - m/ABD - m/ACD (because the angles in a triangle add up to 180 degrees)

m/CBD = 180 - (8x + 35) - (8x + 35)

m/CBD = 180 - 16x - 70

Now, let's simplify:

m/CBD = 110 - 16x

Next, for m/ABC:

m/ABC = m/ABD + m/CBD (because angle ABC is the sum of angles ABD and CBD)

m/ABC = (8x + 35) + (110 - 16x)

Now, simplify:

m/ABC = 8x + 35 + 110 - 16x

m/ABC = 144 - 8x

So, the measures of the angles are as follows:

m/ABD = (8x + 35) degrees

m/CBD = 110 - 16x degrees

m/ABC = 144 - 8x degrees

None of the answer choices provided matches these calculations. It seems there might be an error in the options given, or there could be a mistake in the problem statement.

User Mark Adelsberger
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