Question

determining interior and exterior angles in circles
given:
in circle i,

• ( mwidehat{mx} = 24.5^circ )
• ( mangle mlx = 38.5^circ )

use the interactive diagram to determine ( mwidehat{wq} ).

	measure	reason
( mwidehat{mx} )	( 24.5^circ )	given
( mwidehat{wq} )

the ( mwidehat{wq} ) is \\( \square \\) degrees
reason:

Explanation:

Step1: Recall the property of angles in a circle

The measure of an inscribed angle is related to the arc it intercepts. Also, we can use the fact that the sum of angles in a triangle or the properties of central and inscribed angles. Here, we know that the measure of an angle formed by two chords (or a chord and a radius) can be related to the arcs. Wait, actually, let's look at triangle \( MLX \). Wait, no, maybe we can use the fact that the central angle or the inscribed angle. Wait, given \( m\angle MLX = 38.5^\circ \) and \( m\widehat{MX}=24.5^\circ \). Wait, maybe we can use the formula for the angle formed by a tangent and a chord, but here it's a circle with center? Wait, no, \( L \) is a point, maybe \( I \) is the center? Wait, the diagram shows \( I \) as the center? Wait, maybe we can use the property that the measure of an angle formed by two chords (or a chord and a radius) is equal to half the sum or half the difference of the intercepted arcs. Wait, let's think again.

Wait, the angle \( \angle MLX \) is an inscribed angle? No, maybe \( L \) is a point on the circle? Wait, no, the diagram shows \( M \), \( X \), \( W \), \( Q \) on the circle, \( L \) is a point inside? Wait, maybe the key is that in triangle \( MLX \), we can find some arc, but actually, the problem is about the arc \( \widehat{WQ} \). Wait, maybe we can use the fact that the sum of arcs around a circle is \( 360^\circ \), but no, let's use the angle \( \angle MLX = 38.5^\circ \) and arc \( \widehat{MX}=24.5^\circ \). Wait, the formula for the measure of an angle formed by a chord and a radius: the angle between a chord and a radius is equal to half the measure of the intercepted arc. Wait, no, the inscribed angle theorem: an inscribed angle is half the measure of its intercepted arc. Wait, if \( \angle MLX \) is an angle formed by two chords, then the measure of \( \angle MLX \) is half the sum or half the difference of the intercepted arcs. Wait, maybe the formula is \( m\angle = \frac{1}{2}(m\widehat{\text{arc1}} - m\widehat{\text{arc2}}) \) if it's an exterior angle, or \( \frac{1}{2}(m\widehat{\text{arc1}} + m\widehat{\text{arc2}}) \) if it's an interior angle. Wait, but here we have \( m\angle MLX = 38.5^\circ \) and \( m\widehat{MX}=24.5^\circ \). Let's assume that \( \angle MLX \) is an angle formed by a chord \( MX \) and a line \( LW \) (or something), but maybe the correct approach is:

The measure of an angle formed by two chords (or a chord and a radius) is equal to half the measure of the intercepted arc. Wait, no, let's use the fact that the sum of the arc \( \widehat{MX} \) and the arc related to \( \angle MLX \) and then find \( \widehat{WQ} \). Wait, maybe the angle \( \angle MLX \) is equal to half the measure of the arc \( \widehat{WQ} \) minus half the measure of arc \( \widehat{MX} \)? Wait, no, let's think of the formula for an angle formed by two secants, a secant and a tangent, or two chords. If the angle is inside the circle, it's half the sum of the intercepted arcs; if outside, half the difference. But here, \( L \) is inside? Wait, the diagram shows \( L \) inside the circle? Wait, \( I \) is the center. Wait, maybe \( L \) is a point on the diameter? Wait, no, let's look at the given: \( m\angle MLX = 38.5^\circ \), \( m\widehat{MX}=24.5^\circ \). Let's use the formula for the measure of an angle formed by a chord and a radius: the angle between a chord and a radius is equal to half the measure of the intercepted arc. Wait, no, the inscribed angle theorem: an inscribed angle is half the central angle. Wait, maybe \( \angle MLX \) is an angle such that \( m\angle MLX = \frac{1}{2}(m\widehat{WQ} - m\widehat{MX}) \)? Wait, that might be the case. Let's solve for \( m\widehat{WQ} \).

So, \( m\angle MLX = \frac{1}{2}(m\widehat{WQ} - m\widehat{MX}) \)

We know \( m\angle MLX = 38.5^\circ \) and \( m\widehat{MX}=24.5^\circ \)

Multiply both sides by 2: \( 2 \times 38.5^\circ = m\widehat{WQ} - 24.5^\circ \)

Calculate left side: \( 77^\circ = m\widehat{WQ} - 24.5^\circ \)

Then, \( m\widehat{WQ} = 77^\circ + 24.5^\circ = 101.5^\circ \)? Wait, no, that doesn't seem right. Wait, maybe the formula is \( m\angle = \frac{1}{2}(m\widehat{\text{arc1}} + m\widehat{\text{arc2}}) \). Wait, no, let's check again.

Wait, the correct formula for an angle formed by two chords intersecting inside the circle is \( m\angle = \frac{1}{2}(m\widehat{\text{arc1}} + m\widehat{\text{arc2}}) \). But if the angle is formed by a chord and a tangent, it's half the intercepted arc. Wait, maybe \( L \) is outside? No, the diagram shows \( L \) inside? Wait, the problem says "Use the interactive diagram", but since we can't interact, we have to use the given.

Wait, another approach: the sum of the arc \( \widehat{MX} \) and the arc corresponding to the angle \( \angle MLX \) and then find \( \widehat{WQ} \). Wait, maybe the angle \( \angle MLX \) is equal to half the measure of arc \( \widehat{WQ} \) plus half the measure of arc \( \widehat{MX} \)? No, that would be if it's outside. Wait, I think I made a mistake. Let's start over.

Given \( m\angle MLX = 38.5^\circ \), \( m\widehat{MX}=24.5^\circ \). We need to find \( m\widehat{WQ} \).

The formula for the measure of an angle formed by two chords intersecting inside the circle is: \( m\angle = \frac{1}{2}(m\widehat{\text{arc1}} + m\widehat{\text{arc2}}) \). But if the angle is formed by a chord and a radius, then the angle is equal to half the measure of the intercepted arc. Wait, no, the inscribed angle theorem: an inscribed angle is half the measure of its intercepted arc. So if \( \angle MLX \) is an inscribed angle intercepting arc \( \widehat{MX} \), but \( m\widehat{MX}=24.5^\circ \), then the inscribed angle should be \( 12.25^\circ \), but it's given as \( 38.5^\circ \), so that's not it.

Wait, maybe \( L \) is the center? No, \( I \) is the center. Wait, maybe the triangle \( MLX \) has angles, and we can find the arc \( \widehat{WQ} \) using the fact that the sum of arcs. Wait, the total around the circle is \( 360^\circ \), but we need another approach.

Wait, the key is that the measure of the angle \( \angle MLX \) is equal to half the difference of the measures of the intercepted arcs. Wait, if \( L \) is outside the circle, then the angle is half the difference of the intercepted arcs. But if \( L \) is inside, it's half the sum. Wait, let's assume that \( \angle MLX \) is an angle formed by two secants (or a secant and a chord) outside the circle, so the formula is \( m\angle = \frac{1}{2}(m\widehat{\text{larger arc}} - m\widehat{\text{smaller arc}}) \). So here, \( m\angle MLX = \frac{1}{2}(m\widehat{WQ} - m\widehat{MX}) \). Then:

\( 38.5^\circ = \frac{1}{2}(m\widehat{WQ} - 24.5^\circ) \)

Multiply both sides by 2: \( 77^\circ = m\widehat{WQ} - 24.5^\circ \)

Then, \( m\widehat{WQ} = 77^\circ + 24.5^\circ = 101.5^\circ \)? Wait, that seems high, but maybe. Wait, no, maybe the formula is the other way. Wait, if the angle is inside, then \( m\angle = \frac{1}{2}(m\widehat{\text{arc1}} + m\widehat{\text{arc2}}) \). But then \( 38.5^\circ = \frac{1}{2}(24.5^\circ + m\widehat{WQ}) \), so \( 77^\circ = 24.5^\circ + m\widehat{WQ} \), so \( m\widehat{WQ} = 77^\circ - 24.5^\circ = 52.5^\circ \). No, that doesn't make sense.

Wait, maybe I got the formula wrong. Let's check the inscribed angle theorem again. The measure of an inscribed angle is half the measure of its intercepted arc. So if an angle is formed by two chords, one of which is a radius, then the angle between the chord and the radius is equal to half the measure of the intercepted arc. Wait, no, the radius is perpendicular to the tangent, but here it's a chord.

Wait, maybe the problem is that \( \angle MLX \) is an angle in triangle \( MLX \), and we can find the arc \( \widehat{WQ} \) using the fact that the central angle is twice the inscribed angle. Wait, I'm confused. Let's try another way.

Given \( m\angle MLX = 38.5^\circ \) and \( m\widehat{MX}=24.5^\circ \). Let's assume that the arc \( \widehat{WQ} \) is related to the angle \( \angle MLX \) by the formula \( m\widehat{WQ} = 2 \times m\angle MLX + m\widehat{MX} \). Wait, that would be \( 2 \times 38.5 + 24.5 = 77 + 24.5 = 101.5 \). So maybe that's the answer.

Step2: Calculate \( m\widehat{WQ} \)

Using the formula for the angle formed by two secants (or a secant and a chord) outside the circle: \( m\angle = \frac{1}{2}(m\widehat{\text{larger arc}} - m\widehat{\text{smaller arc}}) \)

We have \( m\angle MLX = 38.5^\circ \) and \( m\widehat{MX}=24.5^\circ \)

So, \( 38.5^\circ = \frac{1}{2}(m\widehat{WQ} - 24.5^\circ) \)

Multiply both sides by 2: \( 2 \times 38.5^\circ = m\widehat{WQ} - 24.5^\circ \)

\( 77^\circ = m\widehat{WQ} - 24.5^\circ \)

Add \( 24.5^\circ \) to both sides: \( m\widehat{WQ} = 77^\circ + 24.5^\circ = 101.5^\circ \)

Wait, but let's check again. If the angle is inside the circle, the formula is \( m\angle = \frac{1}{2}(m\widehat{\text{arc1}} + m\widehat{\text{arc2}}) \). But then \( 38.5 = 0.5(24.5 + m\widehat{WQ}) \), so \( 77 = 24.5 + m\widehat{WQ} \), so \( m\widehat{WQ} = 77 - 24.5 = 52.5 \). But that contradicts. So which is it?

Wait, the diagram shows \( L \) outside the circle? No, the diagram shows \( L \) inside? Wait, the diagram has \( M \), \( X \), \( W \), \( Q \) on the circle, \( L \) is a point inside? No, the line from \( M \) to \( X \) passes through \( L \), and \( L \) is inside the circle? Wait, \( I \) is the center. So \( L \) is inside the circle. Then the angle \( \angle MLX \) is an angle inside the circle, formed by two chords \( ML \) and \( XL \). Then the formula is \( m\angle = \frac{1}{2}(m\widehat{MX} + m\widehat{WQ}) \). Wait, no, that would be if the angle is formed by two chords intersecting inside the circle, then the angle is half the sum of the intercepted arcs. So:

\( m\angle MLX = \frac{1}{2}(m\widehat{MX} + m\widehat{WQ}) \)

Then, \( 38.5 = \frac{1}{2}(24.5 + m\widehat{WQ}) \)

Multiply both sides by 2: \( 77 = 24.5 + m\widehat{WQ} \)

Subtract 24.5: \( m\widehat{WQ} = 77 - 24.5 = 52.5 \)

Ah, that makes more sense. So I had the formula reversed. If the angle is inside the circle (formed by two intersecting chords), then the measure of the angle is half the sum of the measures of the intercepted arcs. So that's the correct formula. So:

\( m\angle = \frac{1}{2}(m\widehat{\text{arc1}} + m\widehat{\text{arc2}}) \)

So solving for \( m\widehat{WQ} \):

\( 2 \times m\angle MLX = m\widehat{MX} + m\widehat{WQ} \)

\( m\widehat{WQ} = 2 \times m\angle MLX - m\widehat{MX} \)

\( m\widehat{WQ} = 2 \times 38.5 - 24.5 \)

\( m\widehat{WQ} = 77 - 24.5 = 52.5^\circ \)

Yes, that makes sense. So the correct formula is for an angle inside the circle, formed by two intersecting chords, the angle is half the sum of the intercepted arcs. So that's the key.

Answer:

\( 52.5 \)