Question

question
in circle j with m∠hlk = 21°, find the m∠hjk.
(there is a diagram of circle j with points h, k, l on the circumference and center j, with lines connecting the points as shown in the image)

Explanation:

Step1: Recall the inscribed angle theorem

The measure of an inscribed angle is half the measure of its intercepted arc. Also, the central angle is equal to the measure of its intercepted arc. Here, $\angle HLK$ is an inscribed angle, and $\angle HJK$ is a central angle intercepting the same arc $HK$.

Step2: Relate the inscribed angle and central angle

Let the measure of arc $HK$ be $m\overset{\frown}{HK}$. We know that the measure of an inscribed angle $\angle HLK$ is half the measure of its intercepted arc $HK$, so $m\angle HLK=\frac{1}{2}m\overset{\frown}{HK}$. The central angle $\angle HJK$ intercepts the same arc $HK$, so $m\angle HJK = m\overset{\frown}{HK}$.

Step3: Calculate the measure of $\angle HJK$

Given $m\angle HLK = 21^{\circ}$, from $m\angle HLK=\frac{1}{2}m\overset{\frown}{HK}$, we can find $m\overset{\frown}{HK}=2\times m\angle HLK$. Substituting the value of $m\angle HLK$, we get $m\overset{\frown}{HK}=2\times21^{\circ} = 42^{\circ}$. Since $m\angle HJK=m\overset{\frown}{HK}$, then $m\angle HJK = 42^{\circ}$.

Answer:

$42^{\circ}$