Question

hi i was just wondering how to do this question (attached below) - ive been trying to figure it out for ages but have had no luck - could use some help! thanks

Answer 1

Answer: 88°

Step-by-step explanation:

We know that the ratio of ∠DCB : ∠ACD is 3:1. In other words, ∠DCB is $\frac{3}{3+1}$, or $\frac{3}{4}$ of the whole angle (i.e., ∠ACB), while ∠ACD is $\frac{1}{4}$ of the whole angle.

To easily find ∠ACB, which is the sum of both angles, we can add up all the angles of $\triangle ABC$ and set it equal to 180°.

$m\angle A + m\angle B + m\angle ACB = 180\\75+53+m\angle ACB=180\\128+m\angle ACB=180\\m\angle ACB=52$

From here, we can calculate ∠BCD by multiplying the value of ∠ACB by three-fourths.

$m\angle BCD = \frac{3}{4}(m\angle ACB)\\m\angle BCD = \frac{3}{4}(52)\\m\angle BCD = 39$

Similar to what we did to get the measure of ∠ACB, we can add up all the angles measures of $\triangle DBC$ to get the measure of ∠BDC.

$m\angle B + m\angle BDC + m\angle BCD = 180\\53+m\angle BDC + 39= 180\\92+ m\angle BDC=180\\m\angle BDC = 88$

The measure of ∠BDC is 88°.