the measure of minor arc jl is 60°. what is the measure of angle jkl? 110° 120° 130° 140°
the measure of minor arc jl is 60°. what is the measure of angle jkl? 110° 120° 130° 140°
Answer
Explanation:
Step1: Recall inscribed - angle theorem
The measure of an inscribed angle is half the measure of its intercepted arc.
Step2: Identify the intercepted arc
The inscribed angle $\angle{JKL}$ intercepts the minor arc $\overset{\frown}{JL}$, and the measure of arc $\overset{\frown}{JL}=60^{\circ}$.
Step3: Calculate the measure of the inscribed angle
Let $m\angle{JKL}$ be the measure of angle $\angle{JKL}$. According to the inscribed - angle theorem, $m\angle{JKL}=\frac{1}{2}m\overset{\frown}{JL}$. Substituting $m\overset{\frown}{JL} = 60^{\circ}$, we get $m\angle{JKL}=\frac{1}{2}\times60^{\circ}=30^{\circ}$. But this is wrong. If we assume $\angle{JKL}$ is a central - angle formed by two radii and an exterior point, we use the formula $m\angle{JKL}=\frac{1}{2}(360^{\circ}-m\overset{\frown}{JL})$.
Step4: Calculate the correct angle
Substitute $m\overset{\frown}{JL} = 60^{\circ}$ into the formula $m\angle{JKL}=\frac{1}{2}(360 - 60)^{\circ}$. First, calculate $360-60 = 300^{\circ}$. Then, $\frac{1}{2}\times300^{\circ}=150^{\circ}$. There is no such option. If we assume $\angle{JKL}$ is an angle formed by a tangent and a chord with the intercepted arc $\overset{\frown}{JL}$, the measure of the angle formed by a tangent and a chord is half of the intercepted arc. But if we consider the non - standard case where $\angle{JKL}$ is an angle such that the arc $\overset{\frown}{JL}$ is the minor arc and we use the formula for the angle between two chords intersecting outside the circle: $m\angle{JKL}=\frac{1}{2}(m\overset{\frown}{JL_{major}}-m\overset{\frown}{JL})$. The major arc $\overset{\frown}{JL_{major}}=360 - 60=300^{\circ}$. Then $m\angle{JKL}=\frac{1}{2}(300 - 60)^{\circ}=\frac{1}{2}\times240^{\circ}=120^{\circ}$.
Answer:
$120^{\circ}$