Question

with reference to the figure, match the angles and arcs to their measures. ∠eob, 122°, ∠dfa, $widehat{ce}$, 114°, ∠oca, 124°, 58°, ∠cob

Explanation:

Step1: Find ∠OCA

Since AC is a tangent to the circle at C, OC is perpendicular to AC (tangent - radius property), so ∠OCA = 90°. Wait, no, maybe I misread. Wait, the angle at A is 114°, so the adjacent angle on the straight line: ∠OCA? Wait, no, let's find other angles. First, at point D, the angle is 124°, so the adjacent angle (linear pair) is 180 - 124 = 56°? Wait, maybe better to find ∠DFA.

Step2: Find ∠DFA

The sum of exterior angles of a triangle? Wait, the two tangent lines from F to the circle, so FE = FB (tangents from a point to a circle are equal), and OC is perpendicular to AC, OE perpendicular to FE. So quadrilateral OEFC and OBCF? Wait, maybe first find the angles at A and D. At A, the angle between tangent and secant? Wait, the angle at A: the straight line, so the angle adjacent to 114° is 180 - 114 = 66°? Wait, no, the figure has a triangle with vertices D, A, F. At D, angle is 124°, so the internal angle at D is 180 - 124 = 56°, at A, internal angle is 180 - 114 = 66°, so ∠DFA = 180 - 56 - 66 = 58°. So ∠DFA = 58°.

Step3: Find arc CE

The central angle for arc CE: since OE and OC are radii, ∠EOC. Wait, the tangents from F: ∠DFA = 58°, so the angle between tangents is 58°, so the central angle ∠EOB? Wait, no, the angle between two tangents is equal to 180° minus the central angle between the points of contact. So ∠DFA = 58°, so central angle ∠EOB = 180 - 58 = 122°? Wait, no, the formula is angle between tangents = 180° - central angle. So if ∠DFA = 58°, then central angle ∠EOB = 180 - 58 = 122°? Wait, maybe. Then arc CE: the measure of arc CE is equal to the central angle ∠EOC? Wait, maybe I made a mistake. Let's re - check.

Wait, at point A, the angle between tangent AC and line FA is 114°, so the angle between tangent and radius OC is 90°, so ∠OAC = 180 - 114 = 66°? No, OC is perpendicular to AC, so ∠OCA = 90°, so in triangle OAC, but maybe not. Wait, let's list the given angles and arcs:

We have ∠DFA = 58° (from step 2: 180 - (180 - 124) - (180 - 114) = 180 - 56 - 66 = 58°).

Then ∠EOB: the central angle. The angle between tangents ∠DFA = 58°, so the central angle ∠EOB = 180 - 58 = 122° (because the angle between two tangents is supplementary to the central angle between the points of contact).

Arc CE: the measure of arc CE is equal to the central angle ∠EOC. Wait, but we also have ∠COB. Wait, maybe ∠COB: since AC and BC are tangents? Wait, no, AC is tangent at C, AB is tangent at B? Wait, the circle is tangent to AC at C, to AE at E, and to AB at B? Wait, maybe OC is perpendicular to AC, OB is perpendicular to AB, so ∠OCA = 90°, ∠OBA = 90°. Then in quadrilateral O C A B, ∠CAB = 180 - 114 = 66°, so ∠COB = 180 - 66 = 114° (since sum of angles in quadrilateral is 360°, 90 + 90 + 66 + ∠COB = 360 → ∠COB = 114°).

Then arc CE: the central angle ∠EOC. We know that ∠EOB = 122°, ∠COB = 114°, so ∠EOC = 360 - 122 - 114 = 124°? Wait, no, that can't be. Wait, maybe the circle is a circle with center O, and CE, CB are arcs. Wait, maybe I messed up. Let's try again.

1. ∠DFA: as calculated, 58°, so ∠DFA ↔ 58°.

2. ∠EOB: 122°, so ∠EOB ↔ 122°.

3. ∠COB: 114°, so ∠COB ↔ 114°.

4. Arc CE: the central angle for arc CE is ∠EOC. Since ∠EOB = 122°, ∠COB = 114°, no, that's more than 180. Wait, maybe the angle at D is 124°, so the arc DE? No, maybe the measure of arc CE is 124°? Wait, the angle at D is 124°, which is an external angle, equal to the measure of the arc CE (by the tangent - secant angle theorem: the measure of an angle formed by a tangent and a secant is half the difference of the measures of the intercepted arcs. Wait, at D, the angle is 124°, which is formed by tangent DE and secant DC? No, D is on the tangent line, E is the point of tangency. Wait, the tangent - secant angle theorem: the measure of an angle formed outside the circle by a tangent and a secant is half the difference of the measures of the intercepted arcs. So at D, angle is 124°, so 124° = 1/2 (measure of arc CE - measure of the other arc). Wait, maybe the measure of arc CE is 124°, so ∠D (124°) is related to arc CE.

Also, ∠OCA: since OC is radius and AC is tangent, ∠OCA = 90°? But 90° is not in the options. Wait, the options have 124°, 114°, 58°, 122°. So maybe my earlier approach is wrong.

Let's list the elements:

Elements to match: ∠EOB, ∠DFA, arc CE, ∠OCA, ∠COB, and measures 122°, 58°, 114°, 124°.

From the figure, at A, the angle between the tangent and the line is 114°, so ∠OCA: if OC is perpendicular to AC, then ∠OCA = 90°, but that's not an option. Wait, maybe the angle at A is 114°, and ∠OCA is 90°, no. Wait, maybe the angle ∠OCA is 90°, but it's not in the options, so maybe I misidentify the angles.

Wait, the given angles and arcs:

- ∠EOB

- ∠DFA

- arc CE

- ∠OCA

- ∠COB

Measures: 122°, 58°, 114°, 124°

Let's assume:

- ∠DFA = 58° (as the sum of angles in triangle DAF: 180 - (180 - 124) - (180 - 114) = 58°)

- ∠EOB = 122° (since the angle between tangents is 58°, so central angle is 180 - 58 = 122°)

- ∠COB = 114° (since at A, the angle between tangent and the line is 114°, and OC and OB are radii, so ∠COB = 114°)

- arc CE = 124° (since at D, the angle is 124°, and by tangent - secant theorem, the angle is half the arc, but maybe arc CE is 124°)

- ∠OCA: maybe 90°, but not in options. Wait, no, the options have ∠OCA and 124°? Wait, maybe ∠OCA is 90°, but it's not there. Wait, maybe I made a mistake. Let's try to match:

1. ∠DFA ↔ 58° (because the triangle's angle sum gives 58°)

2. ∠EOB ↔ 122° (angle between tangents and central angle)

3. ∠COB ↔ 114° (from the angle at A)

4. arc CE ↔ 124° (from the angle at D)

5. ∠OCA: maybe 90°, but not in options. Wait, the options have ∠OCA and 124°? Wait, maybe ∠OCA is 90°, but it's not there. Wait, the given measures are 122°, 58°, 114°, 124°, and the elements are ∠EOB, ∠DFA, arc CE, ∠OCA, ∠COB. So:

- ∠DFA = 58°

- ∠EOB = 122°

- ∠COB = 114°

- arc CE = 124°

- ∠OCA: maybe 90°, but not in options. Wait, maybe the angle ∠OCA is 90°, but it's not listed. Wait, the problem says "match the angles and arcs to their measures", so the available measures are 122°, 58°, 114°, 124°, and the elements are ∠EOB, ∠DFA, arc CE, ∠OCA, ∠COB. So:

- ∠DFA ↔ 58°

- ∠EOB ↔ 122°

- ∠COB ↔ 114°

- arc CE ↔ 124°

- ∠OCA: maybe there's a mistake, but according to the options, ∠OCA ↔ 90° is not there, so maybe I messed up. Wait, maybe the angle at A is 114°, and ∠OCA is 90°, but no. Alternatively, maybe ∠OCA is 124°? No, that doesn't make sense. Wait, let's proceed with the matches we can do:

- ∠DFA = 58°

- ∠EOB = 122°

- ∠COB = 114°

- arc CE = 124°

- ∠OCA: maybe the problem has a typo, but based on the options, we can match as follows:

∠DFA ↔ 58°

∠EOB ↔ 122°

∠COB ↔ 114°

$\widehat{CE}$ ↔ 124°

∠OCA: maybe not, but the given elements and measures:

Elements: ∠EOB, ∠DFA, $\widehat{CE}$, ∠OCA, ∠COB

Measures: 122°, 58°, 114°, 124° (and 58°, 122° etc.)

So the matches are:

∠DFA ↔ 58°

∠EOB ↔ 122°

∠COB ↔ 114°

$\widehat{CE}$ ↔ 124°

∠OCA: maybe there's an error, but based on the given, these are the matches.

Answer:

∠DFA ↔ 58°
∠EOB ↔ 122°
∠COB ↔ 114°
$\widehat{CE}$ ↔ 124°