Question

circle f is intersected by $overleftrightarrow{hi}$ and $overleftrightarrow{ij}$ and $mangle hij = 101^circ$
what is the measure of $overarc{hj}$?
enter your answer in the box.
$moverarc{hj} = square^circ$

Explanation:

Step1: Recall the property of a circle's total degrees

A full circle is \( 360^\circ \). We know one arc is \( 281^\circ \), and we need to find the measure of arc \( \widehat{HJ} \).

Step2: Calculate the measure of arc \( \widehat{HJ} \)

The sum of the measures of the two arcs (the given \( 281^\circ \) arc and arc \( \widehat{HJ} \)) should be \( 360^\circ \) (since they make up the whole circle). So we use the formula \( m\widehat{HJ}=360^\circ - 281^\circ \).
Calculating that, \( 360 - 281 = 79 \). Wait, but also, we can use the angle formed outside? Wait, no, looking at the diagram, the angle \( \angle HIJ = 101^\circ \), and the angle formed by a secant and a tangent (or two secants) outside the circle: the measure of the angle is half the difference of the measures of the intercepted arcs. Wait, maybe I made a mistake earlier. Let's correct.

Wait, \( \angle HIJ \) is formed by a secant \( \overleftrightarrow{HI} \) and a tangent \( \overleftrightarrow{IJ} \) (since \( IJ \) is a tangent at \( J \), as it touches the circle at one point). The formula for the measure of an angle formed by a secant and a tangent outside the circle is \( m\angle = \frac{1}{2}( \text{measure of the major arc} - \text{measure of the minor arc} ) \).

Let \( m\widehat{HJ} = x \) (the minor arc), and the major arc \( HJ \) would be \( 360 - x \). Then \( m\angle HIJ = \frac{1}{2}( (360 - x) - x ) \)? Wait, no, the angle outside is half the difference of the intercepted arcs, where the intercepted arcs are the major arc and the minor arc. Wait, actually, the formula is \( m\angle = \frac{1}{2}( \text{major arc} - \text{minor arc} ) \). Here, \( \angle HIJ = 101^\circ \), major arc is \( 281^\circ \) (given), so \( 101 = \frac{1}{2}(281 - x) \), where \( x \) is the minor arc \( \widehat{HJ} \). Wait, solving for \( x \): multiply both sides by 2: \( 202 = 281 - x \), then \( x = 281 - 202 = 79 \). Wait, but also, the total circle is \( 360 \), so \( 360 - 281 = 79 \), which matches. So the measure of arc \( \widehat{HJ} \) is \( 79^\circ \).

Wait, let's recheck. The angle formed outside the circle by a secant and a tangent: the measure of the angle is half the difference of the measures of the intercepted arcs. The intercepted arcs are the major arc (the larger one) and the minor arc (the smaller one) that are cut off by the secant and tangent. So here, the secant is \( HI \) (intersecting the circle at \( H \) and... wait, \( HI \) intersects the circle at \( H \), and \( IJ \) is tangent at \( J \). So the intercepted arcs are \( \widehat{HJ} \) (minor) and the rest of the circle (major arc \( HJ \), which is \( 281^\circ \) as given). So the formula is \( m\angle HIJ = \frac{1}{2}( \text{major arc} - \text{minor arc} ) \). So \( 101 = \frac{1}{2}(281 - m\widehat{HJ}) \). Multiply both sides by 2: \( 202 = 281 - m\widehat{HJ} \). Then \( m\widehat{HJ} = 281 - 202 = 79 \). Alternatively, since the total circle is \( 360 \), and the major arc is \( 281 \), the minor arc is \( 360 - 281 = 79 \). So both methods give the same result.

Answer:

\( 79 \)