Question

1. find the value of x in the figure below. (round your answer to the nearest whole number)

figure of triangle abc with angle at a: 80°, angle at b: 50°, side ab: 8 in, side bc: 10 in, side ac: x
options: 6 inches, 8 inches, 10 inches, 5 inches

Explanation:

Step1: Find the third angle

The sum of angles in a triangle is \(180^\circ\). Given \(\angle A = 80^\circ\) and \(\angle B = 50^\circ\), we calculate \(\angle C\) as:

$$ \angle C=180^\circ - 80^\circ - 50^\circ = 50^\circ $$

Step2: Identify the type of triangle

Since \(\angle B=\angle C = 50^\circ\), the triangle is isosceles with sides opposite equal angles being equal. The side opposite \(\angle B\) is \(x\) and the side opposite \(\angle C\) is \(8\) in? Wait, no, wait. Wait, side \(AB = 8\) in (opposite \(\angle C\)), side \(AC=x\) (opposite \(\angle B\)). Since \(\angle B=\angle C\), their opposite sides are equal. Wait, \(\angle B = 50^\circ\) (opposite side \(AC = x\)), \(\angle C=50^\circ\) (opposite side \(AB = 8\) in). Wait, no, wait: in triangle \(ABC\), side opposite \(\angle A\) is \(BC = 10\) in, side opposite \(\angle B\) is \(AC=x\), side opposite \(\angle C\) is \(AB = 8\) in. Since \(\angle B=\angle C = 50^\circ\), then \(AC = AB\). Wait, \(\angle B\) is at vertex \(B\), so side opposite \(\angle B\) is \(AC\) (length \(x\)), side opposite \(\angle C\) is \(AB\) (length \(8\) in). Since \(\angle B=\angle C\), then \(AC = AB\), so \(x = 8\) in? Wait, but wait, let's check again. Wait, \(\angle B = 50^\circ\), \(\angle C = 50^\circ\), so triangle \(ABC\) has \(\angle B=\angle C\), so it's isosceles with \(AB = AC\)? Wait, no: \(AB\) is from \(A\) to \(B\), length \(8\) in. \(AC\) is from \(A\) to \(C\), length \(x\). \(BC\) is from \(B\) to \(C\), length \(10\) in. Angles: \(\angle A = 80^\circ\), \(\angle B = 50^\circ\), \(\angle C=50^\circ\). So side opposite \(\angle B\) (which is \(AC\), length \(x\)) and side opposite \(\angle C\) (which is \(AB\), length \(8\) in) should be equal because \(\angle B=\angle C\). Therefore, \(x = 8\) in. Wait, but let's confirm with the Law of Sines to be sure. Law of Sines: \(\frac{AB}{\sin C}=\frac{AC}{\sin B}=\frac{BC}{\sin A}\). So \(\frac{8}{\sin 50^\circ}=\frac{x}{\sin 50^\circ}\), so \(x = 8\) in.

Answer:

8 inches (the option: 8 inches)