Question

if m∠ihj is 72°, what is m∠ijh? 198°, 108°, 45°, 18° the vertices of triangle hij lie on circle k, as shown.

Explanation:

Step1: Recall triangle - angle - sum property

The sum of the interior angles of a triangle is 180°. In \(\triangle IHJ\), we know that \(m\angle IHJ + m\angle HIJ+m\angle IJH=180^{\circ}\).

Step2: Assume \(\angle HIJ = 90^{\circ}\) (if \(IH\) is a diameter, the inscribed - angle theorem states that an angle inscribed in a semi - circle is a right angle). Let's assume this is the case (since no other information about the angles is given and a common situation in circle - triangle problems). We are given \(m\angle IHJ = 72^{\circ}\).

We substitute the known values into the angle - sum formula: \(72^{\circ}+90^{\circ}+m\angle IJH = 180^{\circ}\).

Step3: Solve for \(m\angle IJH\)

\[

$$\begin{align*} m\angle IJH&=180^{\circ}-(72^{\circ} + 90^{\circ})\\ &=180^{\circ}-162^{\circ}\\ &=18^{\circ} \end{align*}$$

\]

Answer:

\(18^{\circ}\)