Question

what is the measure of ∠jgk shown in the diagram below? a. 112° b. 138° c. 56° d. 69°

Explanation:

Step1: Recall the inscribed - angle theorem

The measure of an inscribed angle is half the measure of its intercepted arc. Let the measure of arc $JK$ be $x$.

Step2: Use the property of the angle - formed by two secants

The measure of the angle formed by two secants ($\angle G$) outside the circle is given by $\angle G=\frac{1}{2}(x - 15)$. We know that $\angle G$ is formed by two secants and its measure is related to the arcs intercepted by the secants.

Step3: Set up the equation

Let's assume the central - angle corresponding to arc $JK$ is $x$ and the central - angle corresponding to the arc with measure $15^{\circ}$ is $15^{\circ}$. The measure of the angle $\angle G$ (formed by two secants) is given by the formula $\angle G=\frac{1}{2}(x - 15)$. But we can also use another approach. The sum of the arcs of a circle is $360^{\circ}$. Let's assume the arc $JK$ has measure $x$.
We know that the measure of the inscribed - angle related to the arc $JK$ and the given information. If we consider the fact that the measure of the angle formed by two secants $\angle G = 15^{\circ}$, and using the formula $\angle G=\frac{1}{2}(m\overset{\frown}{JK}-m\overset{\frown}{HI})$. Let $m\overset{\frown}{JK}=x$ and $m\overset{\frown}{HI} = 15^{\circ}$.
We know that the measure of an inscribed - angle $\angle JGK$ and the arcs it intercepts. The measure of the angle formed by two secants gives us the equation $15=\frac{1}{2}(x - 15)$.
First, multiply both sides of the equation by $2$: $30=x - 15$.
Then, add $15$ to both sides: $x=45$. But this is wrong.
The correct formula for the measure of an inscribed angle $\angle JGK$: If we consider the fact that the measure of an inscribed angle is half of the measure of its intercepted arc.
The measure of the arc intercepted by $\angle JGK$ is $112^{\circ}$ (by using the property of angles and arcs in a circle).
The measure of $\angle JGK=\frac{1}{2}\times112^{\circ}=56^{\circ}$.

Answer:

C. $56^{\circ}$