To analyze quadrilateral ABCD inscribed in circle E, we can use some properties of angles and arcs in circles.
- Since ABCD is a cyclic quadrilateral, we know that the opposite angles must sum up to 180°.
Given:
- m∠A + m∠C = 180°
- m∠B + m∠D = 180°
From the information given:
- mAB (measure of arc AB) = 54°
- mBC (measure of arc BC) = 115°
- mCD (measure of arc CD) = 86°
To find the measures of angles B and D:
-
m∠B is equal to half the measure of the arc opposite to it, which is arc AD (which consists of arcs AB and CD).
- mAD = mAB + mCD = 54° + 86° = 140°
- Therefore, m∠B = (1/2) * mAD = (1/2) * 140° = 70°
-
m∠D is equal to half the measure of the arc opposite to it, which is arc BC.
- m∠D = (1/2) * mBC = (1/2) * 115° = 57.5°
Now we can apply the property of cyclic quadrilaterals:
- m∠B + m∠D = 180°
- So 70° + 57.5° = 127.5°; we can find m∠A and m∠C.
Now checking the choices:
- m∠B = 95.5° is false.
- m∠ADC = 84.5° needs to compare with the complementary angle. We found only measures for B and D.
- m∠AD = 115° needs the same verification as above, measure definition in this context.
- m∠D = 169° (false, as calculated).
From this analysis, we see:
- Most directly assumable common responses indicate mismatching for angle B and D in this arrangement.
In conclusion, to ascertain definitive selections, confirming interdependencies of respective angles directly aligns with circumstantial values of arcs and their oppositions will solidify these results effectively. The provided angles do not seem to hold true on any of the provided options without iterative adjustments or further clarifications on the arcs' measures or definitions.
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