identifying properties of angles intercepting the same arc
use the diagram to complete the statements.
the measure of angle ejb is one - half the measure of angle boe.
the measure of angle bde is dropdown the measure of angle boe.
the measure of angle oed is dropdown the measure of angle obd.
(diagram: a circle with center o, points j, b, e on the circle, point d outside the circle connected to b and e. there are line segments oj, ob, oe, jb, bd, ed.)

Question

Accepted Answer

### For the second statement (measure of angle BDE and angle BOE):
# Explanation:
## Step1: Recall circle angle properties
In a circle, the measure of an inscribed angle is half the central angle intercepting the same arc. But here, we analyze the relationship between angle BDE and angle BOE. First, note that $ BE $ is a chord, $ \angle BOE $ is a central angle. Now, $ BD $ and $ ED $ are tangents? Wait, no, $ E $ and $ B $ are on the circle, $ D $ is outside. Wait, actually, $ \angle BDE $: Let's think about the arc $ BE $. The central angle is $ \angle BOE $, and the inscribed angle over arc $ BE $ would be half of $ \angle BOE $, but wait, $ \angle BDE $: Wait, maybe $ BD $ and $ ED $ are tangents? Wait, $ OE $ and $ OB $ are radii, so $ OE \perp ED $ and $ OB \perp BD $ (if $ D $ is external, and $ BD, ED $ are tangents). Then quadrilateral $ OBD E $ has two right angles at $ OE D $ and $ OB D $. Wait, but the key property: the measure of an angle formed by two tangents (or a tangent and a secant) outside the circle is related to the intercepted arcs. But in this case, for the angle $ \angle BDE $: Wait, actually, $ \angle BDE $ and $ \angle BOE $: Let's recall that the measure of an angle formed by two tangents from an external point $ D $ to a circle with center $ O $, the angle between the tangents ($ \angle BDE $) and the central angle ($ \angle BOE $) intercepting the same arc $ BE $ have a relationship: $ \angle BDE = 180^\circ - \angle BOE $? No, wait, no. Wait, the measure of the angle between two tangents is equal to $ 180^\circ - $ the measure of the central angle intercepting the same arc. Wait, no, actually, the measure of the angle formed by two tangents from an external point is equal to half the difference of the measures of the intercepted arcs. But if the arc is minor arc $ BE $, and the major arc $ BE $, but in this case, if $ OE $ and $ OB $ are radii, and $ ED, BD $ are tangents, then $ \angle OED = \angle OBD = 90^\circ $. Then in quadrilateral $ OBD E $, the sum of angles is $ 360^\circ $, so $ \angle BDE + \angle BOE + 90^\circ + 90^\circ = 360^\circ $, so $ \angle BDE + \angle BOE = 180^\circ $, so $ \angle BDE = 180^\circ - \angle BOE $? No, that can't be. Wait, maybe I made a mistake. Wait, the first statement: $ \angle EJB $ is an inscribed angle over arc $ BE $, so its measure is half of $ \angle BOE $ (central angle over arc $ BE $). Now, for $ \angle BDE $: Wait, maybe $ BD $ and $ ED $ are tangents, so $ \angle BDE $: Wait, no, the second statement: the measure of $ \angle BDE $ is what compared to $ \angle BOE $. Wait, maybe the diagram shows that $ BD $ and $ ED $ are tangents, so $ \angle BDE $ and $ \angle BOE $: Wait, actually, the measure of an angle formed by two tangents is equal to $ 180^\circ - $ the central angle? No, wait, the correct property: the measure of an angle formed by two tangents drawn from an external point to a circle is equal to half the difference of the measures of the intercepted arcs. But if the intercepted arcs are the minor arc $ BE $ and the major arc $ BE $, but if the circle is such that $ OE $ and $ OB $ are radii, and $ EJ B $ is a triangle, maybe $ EJ $ is a diameter? Wait, $ EJ $ is a vertical line through $ O $, so $ EJ $ is a diameter. Then arc $ BE $ and arc $ JE B $? Wait, maybe I'm overcomplicating. Let's go back to the options. The options are "equal to", "one - half", "twice", "180 minus". Wait, the first statement: $ \angle EJB $ is an inscribed angle, so it's half of $ \angle BOE $ (central angle), that's correct. Now, for $ \angle BDE $: If $ BD $ and $ ED $ are tangents, then $ \angle OED = \angle OBD = 90^\circ $. Then in quadrilateral $ OBD E $, $ \angle BDE + \angle BOE + 90 + 90 = 360 $, so $ \angle BDE + \angle BOE = 180 $, so $ \angle BDE = 180 - \angle BOE $? No, that would be "180 minus", but wait, the options have "180 minus" as an option? Wait, the dropdown has "equal to", "one - half", "twice", "180 minus". Wait, maybe I made a mistake. Wait, no, the correct relationship for the angle between two tangents ($ \angle BDE $) and the central angle ($ \angle BOE $) intercepting the same arc: the measure of $ \angle BDE $ is equal to $ 180^\circ - $ the measure of $ \angle BOE $? No, wait, actually, the measure of an angle formed by two tangents is equal to half the difference of the measures of the intercepted arcs. The intercepted arcs are the major arc $ BE $ and the minor arc $ BE $. The measure of the angle between the tangents is $ \frac{1}{2}(\text{measure of major arc } BE - \text{measure of minor arc } BE) $. But the central angle $ \angle BOE $ is the measure of the minor arc $ BE $. The major arc $ BE $ is $ 360^\circ - \angle BOE $. So the angle between the tangents $ \angle BDE = \frac{1}{2}((360^\circ - \angle BOE) - \angle BOE) = \frac{1}{2}(360^\circ - 2\angle BOE) = 180^\circ - \angle BOE $. So $ \angle BDE = 180^\circ - \angle BOE $, which means the measure of $ \angle BDE $ is "180 minus" the measure of $ \angle BOE $? Wait, but the options in the dropdown for the second statement: the options are "equal to", "one - half", "twice", "180 minus". Wait, maybe I'm wrong. Wait, let's check the third statement: "The measure of angle OED is [ ] the measure of angle OBD". $ OED $ and $ OBD $: since $ OE $ and $ OB $ are radii, and $ ED $ and $ BD $ are tangents (if $ D $ is external), then $ OE \perp ED $ and $ OB \perp BD $, so $ \angle OED = \angle OBD = 90^\circ $, so they are equal. So the third statement should be "equal to". But back to the second statement: $ \angle BDE $ and $ \angle BOE $. Wait, maybe the diagram is such that $ BD $ and $ ED $ are not tangents, but $ D $ is on the circumference? No, $ D $ is outside. Wait, maybe the correct answer for the second statement is "one - half"? No, that contradicts. Wait, no, the first statement: $ \angle EJB $ is an inscribed angle, so it's half of $ \angle BOE $ (central angle), correct. Now, $ \angle BDE $: if $ D $ is such that $ BD $ and $ ED $ are secants, but no, the diagram shows $ E $ and $ B $ on the circle, $ D $ outside. Wait, maybe the key is that $ \angle BDE $ and $ \angle EJB $ are related? No, $ \angle EJB $ is half of $ \angle BOE $, and if $ \angle BDE $ is equal to $ \angle EJB $, then it would be half, but that doesn't fit. Wait, I think I made a mistake in the tangent part. Let's start over.

The central angle theorem states that an inscribed angle is half the central angle intercepting the same arc. $ \angle EJB $ is an inscribed angle over arc $ BE $, $ \angle BOE $ is the central angle over arc $ BE $, so $ \angle EJB=\frac{1}{2}\angle BOE $, which matches the first statement.

Now, for $ \angle BDE $: Let's consider triangle $ BDE $ and the circle. If $ BD $ and $ ED $ are tangents, then $ OB \perp BD $ and $ OE \perp ED $ (radius is perpendicular to tangent at point of contact). So $ \angle OBD = \angle OED = 90^\circ $. In quadrilateral $ OBD E $, the sum of interior angles is $ 360^\circ $, so $ \angle OBD+\angle BDE+\angle OED+\angle BOE = 360^\circ $. Substituting the known right angles: $ 90^\circ+\angle BDE + 90^\circ+\angle BOE=360^\circ $. Simplifying: $ \angle BDE+\angle BOE = 180^\circ $, so $ \angle BDE = 180^\circ-\angle BOE $. Therefore, the measure of $ \angle BDE $ is "180 minus" the measure of $ \angle BOE $.

For the third statement: $ \angle OED $ and $ \angle OBD $: Since $ OB $ and $ OE $ are radii, and $ BD $ and $ ED $ are tangents (from point $ D $ to the circle), the radius is perpendicular to the tangent at the point of contact. So $ OB \perp BD $ (so $ \angle OBD = 90^\circ $) and $ OE \perp ED $ (so $ \angle OED = 90^\circ $). Therefore, $ \angle OED=\angle OBD $, so the measure of $ \angle OED $ is "equal to" the measure of $ \angle OBD $.

### Final Answers:
- For the second statement (measure of $ \angle BDE $ and $ \angle BOE $): The measure of angle $ BDE $ is $ \boldsymbol{180 \text{ minus}} $ the measure of angle $ BOE $.
- For the third statement (measure of $ \angle OED $ and $ \angle OBD $): The measure of angle $ OED $ is $ \boldsymbol{\text{equal to}} $ the measure of angle $ OBD $.

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QUESTION IMAGE

Question

Explanation:

For the second statement (measure of angle BDE and angle BOE):

Step1: Recall circle angle properties

Answer:

Step1: Recall circle angle properties

Final Answers: