If the measure of ∠ABD is 112°, what are the measures of ∠ABC and ∠CBD?

To find the measures of ∠ABC and ∠CBD, we need to use the fact that the sum of the angles in a triangle is 180 degrees.

Since ∠ABD is 112°, we know that ∠ABC + ∠CBD + ∠ABD = 180°.

Let's substitute the given expressions for the angles into this equation:

(3x + 7) + (2x + 5) + 112 = 180

Simplifying the equation:

5x + 124 = 180

Next, let's solve for x:

5x = 180 - 124

5x = 56

x = 56 / 5

x = 11.2

Now, we can find the measures of ∠ABC and ∠CBD by substituting the value of x into their respective expressions:

∠ABC = 3x + 7 = 3(11.2) + 7 = 33.6 + 7 = 40.6 degrees

∠CBD = 2x + 5 = 2(11.2) + 5 = 22.4 + 5 = 27.4 degrees

Therefore, the measures of ∠ABC and ∠CBD are 40.6 degrees and 27.4 degrees, respectively.