Question

quadrilateral ABCD inscribed in circle E. the ⌒mAB = 54, ⌒mBC = 115, and ⌒mCD = 86
select the true statement

⌒mADC = 84.5

m/_D = 169

m/_B = 95.5

⌒mAD = 115

Answer 1

To solve this problem, we can use the properties of a cyclic quadrilateral (a quadrilateral inscribed in a circle). Specifically, the opposite angles of a cyclic quadrilateral are supplementary, meaning that the sum of the measures of opposite angles equals 180 degrees.

Also, the measure of an arc in a circle corresponds to the inscribed angle that subtends the arc, which is half the measure of the arc.

Given:
- m(arc AB) = 54
- m(arc BC) = 115
- m(arc CD) = 86

First, we can find the measure of arc DA:

The total measure of a circle is 360 degrees. Thus,

m(arc DA) = 360 - (m(arc AB) + m(arc BC) + m(arc CD))
= 360 - (54 + 115 + 86)
= 360 - 255
= 105 degrees

Now, we can find the inscribed angles corresponding to each arc:

1. m∠B (opposite arc CD) = (m(arc CD)) / 2 = 86 / 2 = 43 degrees
2. m∠C (opposite arc DA) = (m(arc DA)) / 2 = 105 / 2 = 52.5 degrees
3. m∠D (opposite arc AB) = (m(arc AB)) / 2 = 54 / 2 = 27 degrees
4. m∠A (opposite arc BC) = (m(arc BC)) / 2 = 115 / 2 = 57.5 degrees

Now we check each statement:

1. **⌒m ADC = 84.5** - Incorrect; we found m∠A is 57.5 degrees and m∠C is 52.5 degrees, not needing the total.

2. **m/_D = 169** - Incorrect; we found m∠D = 27 degrees.

3. **m/_B = 95.5** - Incorrect; we found m∠B = 43 degrees.

4. **⌒m AD = 115** - This refers to the measure of arc AD, which is the sum of arcs AB and BC = 54 + 115 = 169 degrees. Hence, if referring to the arc it could be right contextually, but 115 only for comparison.

Thus, none of the listed statements match exactly correct findings under these definitions.

Please confirm if the task or angle/arc references should be altered or recalibrated!