Question

the vertices of triangle hij lie on circle k, as shown. if ( mangle ihj ) is ( 42^{circ} ), what is ( mangle ijh )?

Explanation:

Step1: Recognize the right - angle

Since \(HJ\) is a diameter of circle \(K\), \(\angle HIJ = 90^{\circ}\) (angle inscribed in a semicircle is a right angle).

Step2: Use the triangle angle - sum property

In \(\triangle HIJ\), the sum of the interior angles of a triangle is \(180^{\circ}\). Let \(m\angle IJH=x\). We know that \(m\angle HIJ = 90^{\circ}\) and \(m\angle IHJ=42^{\circ}\).
By the angle - sum formula \(m\angle HIJ+m\angle IHJ + m\angle IJH=180^{\circ}\), substituting the values: \(90^{\circ}+42^{\circ}+x = 180^{\circ}\).

Step3: Solve for \(x\)

$$
LATEXBLOCK0
$$

Answer:

\(48^{\circ}\)