Question

for each part below, use the figure to fill in the blank. if necessary, you may learn what the markings on a figure indicate. (a) find hi. (b) find m∠k.

Explanation:

Step1: Analyze triangle \( \triangle HIO \)

In \( \triangle HIO \), since \( \angle I=\angle O \), the triangle is isosceles. In an isosceles triangle, the sides opposite equal angles are equal. So \( HI = HO \). Given \( HO = 7 \), then \( HI=7 \).

Step2: Analyze triangle \( \triangle LKJ \)

In \( \triangle LKJ \), \( LK = KJ=6 \). So \( \triangle LKJ \) is isosceles. Let \( m\angle K=x \). Using the angle - sum property of a triangle (\( m\angle L + m\angle K+m\angle J=180^{\circ} \)). Since \( \angle L=\angle J = 40^{\circ} \) (angles opposite equal sides \( LK = KJ \)), we have \( x+40^{\circ}+40^{\circ}=180^{\circ} \). Then \( x=180^{\circ}-(40^{\circ}+40^{\circ})=100^{\circ} \).

Answer:

(a) \( HI = 7 \)
(b) \( m\angle K=100^{\circ} \)