Question

Draw the following diagram: quadrilateral ABCD inscribed in circle E . The mAB=54° , mBC=115° , and mCD=86° . Select the true statement.(1 point) Responses m∠B=95.5° m angle upper B equals 95.5 degrees mADC=84.5° m Modifying above upper A upper D upper C with arc equals 84.5 degrees mAD=115° m Modifying above upper A upper D with arc equals 115 degrees m∠D=169°

Answer 1

To solve this problem, we need to recall some properties of cyclic quadrilaterals (quadrilaterals inscribed in circles). The main property we will use is that the opposite angles of a cyclic quadrilateral are supplementary, meaning that the sum of the angles will equal 180°.

Let’s denote the angles of quadrilateral ABCD as follows:
- m∠A = a
- m∠B = b
- m∠C = c
- m∠D = d

From the data we have:
- mAB = 54° (this is the arc opposite angle C, so the angle C can be calculated as c= 54°)
- mBC = 115° (this is the arc opposite angle A, so the angle A can be calculated as a= 115°)
- mCD = 86° (this is the arc opposite angle B, so the angle B can be calculated as b= 86°)
- mDA = angle D (yet to find)

### Step 1: Calculate the angles.

Using the properties of a cyclic quadrilateral:
- Angle C + Angle A = 180°
- Angle B + Angle D = 180°

Substituting the values we have:
1. To find m∠A (which equals 115°), calculate m∠C and check the supplementary relationship:
- m∠C = 54°
So:
\[
54° + 115° = 169° \quad \text{(this confirms angles add to supplementary)}
\]

2. To find m∠B (which equals 86°), calculate m∠D:
- m∠B = 86°
Then:
\[
m∠D = 180° - m∠B = 180° - 86° = 94°
\]

### Step 2: Verify the statements.

Now we can check the provided options:
1. m∠B = 95.5° (False)
2. mADC = 84.5° (the external angle, not true either; interior angles do not equal this)
3. mAD = 115° (the length of the arc measures are not equal)
4. m∠D = 169° (False; m∠D = 94°)

So looking through the listed angles, all options provided are false or not possible based on cyclical properties of angle measures summing to 180°

Based on the work, we can firmly conclude that none of the statements provided post-calculation are correct within the scoped quadrilateral ABCD. If you have further details concerning the statements or corrections on angles, feel free to ask!