Question

Point D is in the interior of ∠ABC, m∠ABC=12x−110, m∠ABD=3x+40, and m∠DBC=2x−10. What is the m∠ABC?

a. 130°
b. 30°
c. 20°
d. 100°

Answer 1

By setting up and solving the equation 12x - 110 = (3x + 40) + (2x - 10), we find that x = 20 and the measure of ∠ABC is 130°, which is option (a).

Step-by-step explanation:

The question asks us to find the measure of ∠ABC when given the measures of ∠ABC, ∠ABD, and ∠DBC in terms of x. From the angles provided, we can write the following equation because point D is in the interior of ∠ABC:

m∠ABC = m∠ABD + m∠DBC

Substituting the given expressions for these angles gives us:

12x - 110 = (3x + 40) + (2x - 10)

Now, we combine like terms and solve for x:

12x - 110 = 5x + 30

x = 20

Finally, we find m∠ABC by inserting x back into the expression for m∠ABC:

m∠ABC = 12(20) - 110 = 240 - 110 = 130°

So the measure of ∠ABC is 130°, which corresponds to option (a).