Question

angles central and inscribed angles try it the measure of arc ab is °. the measure of angle aob is 25, 50, 98, 212 °. the measure of angle bda is 25, 50, 98, 212 °. (circle with center o, points a, b, d on circumference. arc ad is 212°, arc bd is 98°)

Explanation:

Step1: Recall the total degrees in a circle

A circle has a total of \( 360^\circ \). We know one arc is \( 212^\circ \) and another is \( 98^\circ \). Let the measure of arc \( AB \) be \( x \). So, \( x + 212^\circ+ 98^\circ= 360^\circ \).

Step2: Solve for \( x \)

First, add \( 212^\circ \) and \( 98^\circ \): \( 212 + 98 = 310 \). Then, \( x=360 - 310 = 50 \). So the measure of arc \( AB \) is \( 50^\circ \).

For the central angle \( \angle AOB \), the measure of a central angle is equal to the measure of its intercepted arc. Since arc \( AB \) is \( 50^\circ \), \( \angle AOB = 50^\circ \)? Wait, no, wait. Wait, maybe I misread. Wait, the inscribed angle or central angle. Wait, actually, the arc \( AB \): total circle is \( 360 \), so \( 360 - 212 - 98 = 50 \), so arc \( AB \) is \( 50^\circ \). Then, the central angle \( \angle AOB \) (if \( O \) is the center) would be equal to arc \( AB \), so \( 50^\circ \)? Wait, but the options for angle \( AOB \) are 25, 50, 98, 212. Wait, maybe the inscribed angle. Wait, no, let's re - check.

Wait, the arc \( BD \) is \( 98^\circ \), arc \( AD \) is \( 212^\circ \)? Wait, no, the diagram: points \( A \), \( B \), \( D \) on the circle, center \( O \). So the arcs: arc \( ABD \) is \( 212^\circ \), arc \( BD \) is \( 98^\circ \), so arc \( AB \) is \( 212 - 98 = 114 \)? No, that can't be. Wait, I think I made a mistake. Wait, the total circle is \( 360 \), so the sum of all arcs: arc \( AB \), arc \( BD \), and the other arc (arc \( DA \))? Wait, the diagram shows arc \( DA \) as \( 212^\circ \), arc \( BD \) as \( 98^\circ \), so arc \( AB \) is \( 360-(212 + 98)=360 - 310 = 50^\circ \). So arc \( AB \) is \( 50^\circ \). Then, the central angle \( \angle AOB \) (with \( O \) as center) is equal to arc \( AB \), so \( 50^\circ \). Then, for the inscribed angle \( \angle BDA \), an inscribed angle is half the measure of its intercepted arc. Wait, arc \( AB \) is \( 50^\circ \), no, wait, maybe arc \( AB \) is \( 50 \), arc \( BD \) is \( 98 \), so arc \( AD \) is \( 212 \). Wait, maybe the angle \( \angle BDA \): if it's an inscribed angle intercepting arc \( AB \), no, maybe intercepting arc \( AB \) or arc \( AB \) and others. Wait, no, let's start over.

1. Measure of arc \( AB \):
A full circle is \( 360^\circ \). The two given arcs are \( 212^\circ \) and \( 98^\circ \). So arc \( AB=360-(212 + 98)=360 - 310 = 50^\circ \).

2. Measure of \( \angle AOB \):
The central angle \( \angle AOB \) (where \( O \) is the center) has the same measure as its intercepted arc \( AB \). So \( \angle AOB = 50^\circ \)? Wait, but the options for \( \angle AOB \) are 25, 50, 98, 212. So 50 is an option.

3. Measure of \( \angle BDA \):
An inscribed angle is half the measure of its intercepted arc. If \( \angle BDA \) intercepts arc \( AB \), no, wait, maybe it intercepts arc \( AB \) or arc \( AB \) and others. Wait, no, if arc \( AB \) is \( 50^\circ \), arc \( BD \) is \( 98^\circ \), then arc \( AD \) is \( 212^\circ \). Wait, maybe \( \angle BDA \) intercepts arc \( AB \), but that would be half of \( 50 \), which is 25, but the options for \( \angle BDA \) are 25, 50, 98, 212. Wait, maybe \( \angle BDA \) intercepts arc \( AB \) and arc \( BD \)? No, wait, maybe I messed up the arcs. Wait, the arc \( BD \) is \( 98^\circ \), so the inscribed angle over arc \( BD \) would be half, but no. Wait, maybe the central angle for arc \( BD \) is \( 98^\circ \), so the inscribed angle would be \( 49^\circ \), but that's not an option. Wait, maybe the angle \( \angle BDA \) is equal to the central angle? No, inscribed angle is half. Wait, maybe the problem has some typos, but based on the arc \( AB \) calculation: \( 360 - 212 - 98 = 50 \), so arc \( AB \) is \( 50^\circ \), \( \angle AOB \) (central angle) is \( 50^\circ \), and \( \angle BDA \): if arc \( AB \) is \( 50^\circ \), and \( \angle BDA \) is an inscribed angle over arc \( AB \), then it's \( 25^\circ \), but the options for \( \angle BDA \) include 25. Wait, maybe:

- Arc \( AB \): \( 50^\circ \)
- \( \angle AOB \) (central angle) : \( 50^\circ \) (since central angle = arc)
- \( \angle BDA \) (inscribed angle) : \( 25^\circ \) (since inscribed angle = \( \frac{1}{2} \) arc \( AB \), \( \frac{50}{2}=25 \))

Yes, that makes sense. So:

- Measure of arc \( AB \): \( 50^\circ \)
- Measure of \( \angle AOB \): \( 50^\circ \)
- Measure of \( \angle BDA \): \( 25^\circ \)

Wait, but the options for \( \angle BDA \) are 25, 50, 98, 212. So 25 is an option. Let's confirm:

Total circle: \( 360^\circ \). Arc \( AB + \) arc \( BD + \) arc \( DA = 360 \). Arc \( DA = 212^\circ \), arc \( BD = 98^\circ \), so arc \( AB = 360 - 212 - 98 = 50^\circ \).

Central angle \( \angle AOB \): intercepts arc \( AB \), so \( \angle AOB = 50^\circ \).

Inscribed angle \( \angle BDA \): intercepts arc \( AB \), so \( \angle BDA=\frac{1}{2}\times arc\ AB=\frac{1}{2}\times50 = 25^\circ \).

And arc \( BD \) is \( 98^\circ \), so if there's a central angle for arc \( BD \), it would be \( 98^\circ \), but in the options, angle \( BDA \) is 25, 50, 98, 212. So \( \angle BDA = 25^\circ \), \( \angle AOB = 50^\circ \), arc \( AB = 50^\circ \).

Answer:

• The measure of arc \( AB \) is \( \boldsymbol{50}^\circ \).
• The measure of angle \( AOB \) is \( \boldsymbol{50}^\circ \).
• The measure of angle \( BDA \) is \( \boldsymbol{25}^\circ \).