Answer:
- ∠ADC = 58°
- ∠ADB = 37°
- BD is a diameter
Explanation:
You want to know the measures of angles ADB and ADC given that inscribed angle BAC is 21° and exterior angle CBE is 58°.
Exterior angle
Angle CBE is exterior to triangle CBA with remote interior angles BAC (21°) and ACB. The exterior angle is equal to the sum of the remote interior angles, so ...
∠CAB +∠ACB = ∠CBE
21° +∠ACB = 58°
∠ACB = 37° . . . . . . . . subtract 21°
a) ∠ADC
The measure of an inscribed angle is half the measure of the arc it intercepts. This means the measures of all inscribed angles that intercept the same arc are the same.
∠ADB = ∠ACB = 37°
∠BDC = ∠CAB = 21°
By the Angle Addition theorem, ...
∠ADC = ∠ADB +∠BDC = 37° +21°
∠ADC = 58°
b) ∠ADB
As we just saw, ...
∠ADB = 37°
c) BD
For angle CAD = 69°, the total of the inscribed angles with vertex A is ...
∠CAD +∠CAB = ∠BAD
69° +21° = 90° = ∠BAD
This means angle BAD intercepts an arc BD of 180°, hence segment BD is a diameter.
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The attached figure is accurately drawn.