Answer:
182
Step-by-step explanation:
First, we can say that ∠ACB and ∠BCF are opposite angles because they are not next to each other and are formed by intersecting lines. Therefore, ∠ACB = ∠BCF
Next, angles correspond with the side opposite of it. This means that, for example, ∠ACB corresponds with line AB and ∠BCF corresponds with line BF. Because ∠ACB and ∠BCF are equal and the triangles are similar, we can say that their corresponding sides have a ratio with each other. Similarly, ∠CAB corresponds to CB, ∠ABC corresponds to AC, ∠CWV corresponds to CV, and ∠CVW corresponds to CB.
Because the triangle CVW is isosceles, we can say that the angles whose corresponding sides are equal are equal as well, so ∠CAB = ∠ABC and ∠CVW = ∠CVW. Next, because ∠ACB and ∠BCF are equal and correspond to each other, and the other two angles in each triangle are equal to each other, we can say that the other two angles must be equal. Another way of putting this is like this:
Say that x = ∠CAB. Then,
∠CAB + ∠ABC + ∠ACB = 180
x + x + ∠ACB = 180
2x + ∠ACB = 180
subtract ∠ACB from both sides
180 - ∠ACB = 2x
Similarly, if y = ∠CWV,
180 - ∠WCV = 2y. Therefore, 2y=2x and y=x, so ∠CAB = ∠ABC = ∠CWV = ∠CVW.
Given this, we can say that sides CV and CW correspond with BC and AC. This means that the ratio between, for example, CV and BC is equal to the ratio between CW and AC. Therefore,
CV/BC = CW/AC
84/182 = 84/?
Filling in the blank, 84/182 is equal to 84/182, so ? = 182
