Answer: 88°
Step-by-step explanation:
We know that the ratio of ∠DCB : ∠ACD is 3:1. In other words, ∠DCB is [tex]\frac{3}{3+1}[/tex], or [tex]\frac{3}{4}[/tex] of the whole angle (i.e., ∠ACB), while ∠ACD is [tex]\frac{1}{4}[/tex] of the whole angle.
To easily find ∠ACB, which is the sum of both angles, we can add up all the angles of [tex]\triangle ABC[/tex] and set it equal to 180°.
[tex]m\angle A + m\angle B + m\angle ACB = 180\\75+53+m\angle ACB=180\\128+m\angle ACB=180\\m\angle ACB=52[/tex]
From here, we can calculate ∠BCD by multiplying the value of ∠ACB by three-fourths.
[tex]m\angle BCD = \frac{3}{4}(m\angle ACB)\\m\angle BCD = \frac{3}{4}(52)\\m\angle BCD = 39[/tex]
Similar to what we did to get the measure of ∠ACB, we can add up all the angles measures of [tex]\triangle DBC[/tex] to get the measure of ∠BDC.
[tex]m\angle B + m\angle BDC + m\angle BCD = 180\\53+m\angle BDC + 39= 180\\92+ m\angle BDC=180\\m\angle BDC = 88[/tex]
The measure of ∠BDC is 88°.
