Question

6.5.4 quiz: intersecting chord theorem
question 1 of 10
what is the measure of \\(\overarc{jk}\\)?
a. \\(45^\circ\\)
b. \\(41^\circ\\)
c. \\(37^\circ\\)
d. \\(29^\circ\\)

Explanation:

Step1: Recall secant-chord angle formula

The measure of an angle formed by a tangent and a chord is half the measure of its intercepted arc. So, $\text{Angle } = \frac{1}{2} \times \text{Intercepted Arc}$

Step2: Define intercepted arc components

The intercepted arc for the $45^\circ$ angle is $\widehat{MH} + \widehat{JK}$. We know $\widehat{MH}$ is twice the inscribed angle over it: $\widehat{MH} = 2 \times 37^\circ = 74^\circ$

Step3: Solve for $\widehat{JK}$

Substitute into the formula:
$45^\circ = \frac{1}{2} \times (\widehat{JK} + 74^\circ)$
Multiply both sides by 2: $90^\circ = \widehat{JK} + 74^\circ$
Subtract $74^\circ$: $\widehat{JK} = 90^\circ - 74^\circ$

Answer:

D. $29^\circ$