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:
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To find m∠A (which equals 115°), calculate m∠C and check the supplementary relationship:
- m∠C = 54° So: \[ 54° + 115° = 169° \quad \text{(this confirms angles add to supplementary)} \]
-
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:
- m∠B = 95.5° (False)
- mADC = 84.5° (the external angle, not true either; interior angles do not equal this)
- mAD = 115° (the length of the arc measures are not equal)
- 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!
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