Question

1. find the value of x in the figure below.

(figure of triangle ghi with h=126°, i=27°, hi=18 in, gi=34 in, gh=x)
options: 17 inches, 34 inches, 18 inches, 9 inches

Explanation:

Step1: Find angle at G

Sum of angles in a triangle is \(180^\circ\). So, \(\angle G = 180^\circ - 126^\circ - 27^\circ = 27^\circ\).

Step2: Identify triangle type

Since \(\angle G=\angle I = 27^\circ\), triangle \(GHI\) is isosceles with \(GH = HI\)? Wait, no, wait: Wait, \(\angle I = 27^\circ\), \(\angle G = 27^\circ\), so sides opposite equal angles are equal. Side opposite \(\angle I\) is \(GH\) (length \(x\)), side opposite \(\angle G\) is \(HI\) (length 18 in)? Wait, no, wait: Wait, side \(HI\) is 18 in, angle at \(I\) is \(27^\circ\), angle at \(G\) is \(27^\circ\), so side opposite angle \(I\) (which is \(GH\), length \(x\)) and side opposite angle \(G\) (which is \(HI\), length 18 in) should be equal? Wait, no, wait: Wait, angle at \(G\) is \(27^\circ\), angle at \(I\) is \(27^\circ\), so triangle is isosceles with \(GH = HI\)? Wait, no, \(HI\) is 18 in, so \(GH = x = 18\) in? Wait, no, wait, let's check again. Wait, side \(GI\) is 34 in, angle at \(H\) is \(126^\circ\), angles at \(G\) and \(I\) are both \(27^\circ\), so it's an isosceles triangle with \(GH = HI\)? Wait, \(HI\) is 18 in, so \(GH = x = 18\) in? Wait, but let's confirm. Wait, in triangle \(GHI\), angles at \(G\) and \(I\) are equal (\(27^\circ\) each), so the sides opposite those angles are equal. Side opposite angle \(I\) (which is angle at \(I\), \(27^\circ\)) is \(GH\) (length \(x\)), and side opposite angle \(G\) (angle at \(G\), \(27^\circ\)) is \(HI\) (length 18 in). Therefore, \(x = 18\) inches.

Answer:

18 inches (corresponding to the option "18 inches")