Question

Given this triangle, solve for x. Then find the measure of angle C.

a) x = 20°, angle C = 100°
b) x = 24°, angle C = 72°
c) x = 28°, angle C = 92°
d) x = 16°, angle C = 84°

Answer 1

x = 28°, angle C = 92° is the measure of angle C. Thus the correct option is C. x = 28°, angle C = 92°.

Step-by-step explanation:

In the given triangle, let's denote the three angles as A, B, and C, and the side lengths opposite to these angles as a, b, and c, respectively. According to the triangle angle sum property, the sum of the angles in any triangle is always 180°.

Now, the given values are x = 28° and angle C = 92°. We know that angle C is opposite side c. So, angle A = 180° - (x + angle C) = 180° - (28° + 92°) = 180° - 120° = 60°.

Now, using the Law of Sines, we can find the side lengths a, b, and c in the triangle. The Law of Sines states: a/sin(A) = b/sin(B) = c/sin(C).

Let's denote the length of side c as
`\(c\).` We can set up the equation as:

`\[c/\sin(C) = a/\sin(A).\]`

`\[c/\sin(92\textdegree) = b/\sin(60\textdegree).\]`

`\[c = (b \cdot \sin(92\textdegree))/(\sin(60\textdegree)).\]`

Next, using the fact that the sum of angles in a triangle is 180°, we can find angle B:

`\[B = 180\textdegree - A - C = 180\textdegree - 60\textdegree - 92\textdegree = 28\textdegree.\]`

So, the final solution is
`\(x = 28\textdegree\)` and
`\(angle C = 92\textdegree\)`, as given in option c).

Thus the correct option is C. x = 28°, angle C = 92°.