use the diagram to complete the statements.
the measure of minor arc ef is $\square$$^\circ$.
the measure of angle dab is $\square$ (options: 35.5, 54.5, 71, 109)
(diagram: a circle with center c, points a, b, d on the left half, e, f on the right half; angle at c: $\angle ecf = 109^\circ$; arcs with red marks on left and right sides)

Question

Accepted Answer

### For the measure of minor arc EF:
# Explanation:
## Step1: Identify the central angle
The central angle for arc EF is ∠ECF (or ∠ACF? Wait, the center is C, and ∠ECF? Wait, no, the angle given is 109°? Wait, no, the diagram has two arcs with red marks, meaning they are equal. Wait, the total around a circle is 360°, but maybe the two arcs with red marks are equal, and the central angle for arc EF: Wait, actually, the central angle for arc EF is equal to the angle at C, but wait, no, the diagram shows that the two arcs (the ones with red ticks) are congruent, so their central angles are equal. Wait, the angle at C is 109°? Wait, no, maybe the other arc: Wait, the total of a circle is 360°, but if we have two equal arcs (the ones with red marks) and the central angle for arc EF: Wait, actually, the measure of a minor arc is equal to the measure of its central angle. Wait, but looking at the diagram, the central angle for arc EF: Wait, no, the angle at C is 109°? Wait, no, maybe I made a mistake. Wait, the two arcs with red ticks are congruent, so their central angles are equal. Let's see, the angle at C is 109°, but the other arc (the one opposite) would be... Wait, no, the total around a point is 360°, but maybe the two arcs with red marks are each (360 - 109*2)/2? No, wait, maybe the central angle for arc EF is equal to the angle at C, but no, wait, the problem is about minor arc EF. Wait, actually, the measure of minor arc EF is equal to the measure of the central angle ∠ECF? Wait, no, the diagram shows that the two arcs (the ones with red ticks) are congruent, so their central angles are equal. Let's calculate: the total circle is 360°, the central angle given is 109°, so the other two central angles (for the equal arcs) would be (360 - 109*2)/2? Wait, no, 360 - 2*109 = 360 - 218 = 142, then 142/2 = 71? Wait, no, that doesn't make sense. Wait, maybe the central angle for arc EF is 71°? Wait, no, the first part: the measure of minor arc EF is 71°? Wait, no, let's re-examine. Wait, the central angle for arc EF: if the angle at C is 109°, then the minor arc EF would be 109°? No, that can't be. Wait, maybe the two arcs with red marks are equal, so their central angles are equal. Let's say the two equal arcs have central angles of x, and the other two arcs (including EF) have central angles. Wait, the angle given is 109°, so maybe the central angle for arc EF is 71°, because 180 - 109 = 71? No, that's not right. Wait, actually, the measure of a minor arc is equal to the measure of its central angle. Wait, maybe the central angle for arc EF is 71°, so the minor arc EF is 71°? Wait, no, the first answer: the measure of minor arc EF is 71°? Wait, no, let's check again. Wait, the diagram has two arcs with red ticks, meaning they are congruent, so their central angles are equal. The central angle for arc EF: if the angle at C is 109°, then the other central angle (for the arc opposite) would be 109°, and the remaining two central angles (for the equal arcs) would be (360 - 109*2)/2 = (360 - 218)/2 = 142/2 = 71. So each of the equal arcs has a central angle of 71°, so the minor arc EF: wait, no, arc EF's central angle is 71°? Wait, maybe I messed up. Wait, the problem says "the measure of minor arc EF is [ ]°". Let's think again. The central angle for arc EF is equal to the angle at C, but if the two arcs with red ticks are equal, then their central angles are equal. So the total around the center is 360°, so 2x + 2*109 = 360? No, 2x + 109*2 = 360? 2x = 360 - 218 = 142, so x = 71. So the minor arc EF has a central angle of 71°, so its measure is 71°. Wait, but the first part: the measure of minor arc EF is 71°? Wait, no, the first box is for minor arc EF, and the second is for angle DAB. Wait, maybe the first part: the measure of minor arc EF is 71°, and the second part: angle DAB.

## Step2: Calculate angle DAB
Angle DAB is an inscribed angle? Wait, no, AB is a chord, AD is a chord, and CD is a radius? Wait, C is the center, so CA and CD are radii, so triangle CAD is isoceles? Wait, no, AB and AD: wait, angle DAB is an angle formed by a chord and a radius? Wait, no, CA and CD are radii, so CA = CD. The central angle for arc BD (the arc with red tick) is 71°, so the inscribed angle over that arc would be half? Wait, no, angle DAB: let's see, AB is a chord, AD is a chord, and the central angle for arc BD is 71°, so the inscribed angle at A (angle DAB) would be half of that? Wait, no, inscribed angle is half the central angle. Wait, the central angle for arc BD is 71°, so the inscribed angle over arc BD would be 71°/2 = 35.5°? Wait, but the options are 35.5, 54.5, 71, 109. So angle DAB is 35.5°? Wait, no, wait, maybe the central angle for arc BD is 71°, so the inscribed angle at A (angle DAB) is half of that, so 71/2 = 35.5°. So angle DAB is 35.5°, and minor arc EF is 71°.

Wait, let's correct:

For minor arc EF:

## Step1: Determine the central angle
The total around a circle is 360°. The two arcs with red ticks are congruent, so their central angles are equal. Let the measure of each of these central angles be x. The central angle for arc EF (and its opposite) is 109°? Wait, no, the diagram shows a central angle of 109° (∠ECF or ∠DCF? Wait, the center is C, and the angle at C is 109°, so the central angle for arc EF is 109°? No, that can't be, because minor arc is less than 180°. Wait, 109° is less than 180°, so it is a minor arc. Wait, maybe I was wrong earlier. The measure of minor arc EF is equal to the measure of its central angle, which is 109°? But that contradicts the earlier thought. Wait, the problem's first box: "The measure of minor arc EF is [ ]°". Let's check the options. Wait, the options for the second part are 35.5, 54.5, 71, 109. So maybe the first part is 71°, and the second is 35.5°.

Wait, let's start over.

1. Measure of minor arc EF:

The central angle for arc EF is equal to the angle at the center, C. Looking at the diagram, the two arcs with red marks are congruent, so their central angles are equal. Let’s denote the measure of each of these central angles as $ x $. The central angle given (for the other arc) is $ 109^\circ $. Since the total around a circle is $ 360^\circ $, we have:

$$ 2x + 2 \times 109^\circ = 360^\circ $$

Wait, no, maybe there are two arcs with central angle $ 109^\circ $ and two equal arcs. Wait, no, the diagram has two arcs with red ticks (meaning they are equal) and the central angle for arc EF is $ 109^\circ $? No, that can't be. Wait, maybe the central angle for arc EF is $ 71^\circ $, because $ 180^\circ - 109^\circ = 71^\circ $? No, that's not a rule. Wait, actually, the measure of a minor arc is equal to the measure of its central angle. If the central angle for arc EF is $ 71^\circ $, then the minor arc EF is $ 71^\circ $.

2. Measure of angle DAB:

Angle DAB is an inscribed angle? Wait, no, AB is a chord, AD is a chord, and CA and CD are radii. The central angle for arc BD is $ 71^\circ $ (from the first part), so the inscribed angle over arc BD would be half of that, so $ \frac{71^\circ}{2} = 35.5^\circ $. So angle DAB is $ 35.5^\circ $.

So:

- Minor arc EF: $ 71^\circ $
- Angle DAB: $ 35.5^\circ $

# Answer:
The measure of minor arc EF is $ \boldsymbol{71}^\circ $.
The measure of angle DAB is $ \boldsymbol{35.5}^\circ $.

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QUESTION IMAGE

Question

Explanation:

For the measure of minor arc EF:

Step1: Identify the central angle

Answer: