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 ( mangle k ). (b) find ( hi ).

Explanation:

Step1: Use the property of isosceles triangles for part (a)

In an isosceles triangle, if two sides are equal (\(JK = JL = 11\)), then the base - angles are equal. Let \(\angle K=\angle L\). The sum of angles in a triangle is \(180^{\circ}\). We know \(\angle J = 38^{\circ}\). So, \(m\angle K+m\angle L + m\angle J=180^{\circ}\). Since \(m\angle K = m\angle L\), we can write \(2m\angle K+38^{\circ}=180^{\circ}\).

$$2m\angle K=180^{\circ}- 38^{\circ}$$

$$2m\angle K = 142^{\circ}$$

$$m\angle K=\frac{142^{\circ}}{2}$$

Step2: Use the property of congruent angles and sides for part (b)

In \(\triangle IOH\), if \(\angle I=\angle O\), then the sides opposite to these angles are equal. The side opposite \(\angle I\) is \(OH\) and the side opposite \(\angle O\) is \(IH\). Given \(OH = 8\), so \(HI=OH\).

Answer:

(a) \(m\angle K = 71^{\circ}\)
(b) \(HI = 8\)