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 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∠B=95.5° m angle upper B equals 95.5 degrees m∠D=169°
pick one
pick one
Answers
GPT-4o mini
To analyze the given quadrilateral \(ABCD\) inscribed in circle \(E\), we need to use some properties of cyclic quadrilaterals.
1. The sum of the measures of opposite angles in a cyclic quadrilateral is \(180^\circ\).
From the information provided:
- \(m \angle AB = 54^\circ\)
- \(m \angle BC = 115^\circ\)
- \(m \angle CD = 86^\circ\)
Now, we can find \(m \angle AD\):
According to the properties of cyclic quadrilaterals:
- \(m \angle AB + m \angle CD = 180^\circ\)
- \(m \angle BC + m \angle AD = 180^\circ\)
First, let's find \(m \angle AD\).
Using \(m \angle AB\) and \(m \angle CD\):
\[
m \angle AB + m \angle CD = 54^\circ + 86^\circ = 140^\circ \quad \text{(not equal to 180)}
\]
Now, calculate \(m \angle AD\):
Using \(m \angle BC\):
\[
m \angle BC + m \angle AD = 115^\circ + m \angle AD = 180^\circ
\]
\[
m \angle AD = 180^\circ - 115^\circ = 65^\circ
\]
Now we can verify the statements:
1. \(m \angle ADC = 84.5^\circ\) (not possible, as we calculated \(m \angle AD = 65^\circ\))
2. \(m \angle A\) with arc equals \(84.5^\circ\) (not possible, as we have no such calculation)
3. \(m \angle AD = 115^\circ\) (not possible, as \(m \angle AD = 65^\circ\))
4. \(m \angle B = 95.5^\circ\) (not directly calculable based just on what we computed)
5. \(m \angle D = 169^\circ\) (not plausible based on angle sums)
Since our rule states sums of opposite angles must equal 180°, the only direct related relationship that we calculated correctly points to \(m \angle AD\) being \(65^\circ\).
Let's verify again with angle B and D:
If we assume \(m \angle D\) would be \(m \angle BC = 115\), that would imply opposite angles must fit itself for angle 65. Hence, the 169 assumption would contradict this setup.
Thus, based on cyclic properties and angle allocations here, **there may not be a correct option** among the ones given without empirical supporting external context.
Feel free to check angles B and D using specifics, but ensure every computation pairs correctly (e.g., remaining calculations too etc). The reframed focus is critical to validate cyclic properties firmly. Let me know if you need further elucidation!
1. The sum of the measures of opposite angles in a cyclic quadrilateral is \(180^\circ\).
From the information provided:
- \(m \angle AB = 54^\circ\)
- \(m \angle BC = 115^\circ\)
- \(m \angle CD = 86^\circ\)
Now, we can find \(m \angle AD\):
According to the properties of cyclic quadrilaterals:
- \(m \angle AB + m \angle CD = 180^\circ\)
- \(m \angle BC + m \angle AD = 180^\circ\)
First, let's find \(m \angle AD\).
Using \(m \angle AB\) and \(m \angle CD\):
\[
m \angle AB + m \angle CD = 54^\circ + 86^\circ = 140^\circ \quad \text{(not equal to 180)}
\]
Now, calculate \(m \angle AD\):
Using \(m \angle BC\):
\[
m \angle BC + m \angle AD = 115^\circ + m \angle AD = 180^\circ
\]
\[
m \angle AD = 180^\circ - 115^\circ = 65^\circ
\]
Now we can verify the statements:
1. \(m \angle ADC = 84.5^\circ\) (not possible, as we calculated \(m \angle AD = 65^\circ\))
2. \(m \angle A\) with arc equals \(84.5^\circ\) (not possible, as we have no such calculation)
3. \(m \angle AD = 115^\circ\) (not possible, as \(m \angle AD = 65^\circ\))
4. \(m \angle B = 95.5^\circ\) (not directly calculable based just on what we computed)
5. \(m \angle D = 169^\circ\) (not plausible based on angle sums)
Since our rule states sums of opposite angles must equal 180°, the only direct related relationship that we calculated correctly points to \(m \angle AD\) being \(65^\circ\).
Let's verify again with angle B and D:
If we assume \(m \angle D\) would be \(m \angle BC = 115\), that would imply opposite angles must fit itself for angle 65. Hence, the 169 assumption would contradict this setup.
Thus, based on cyclic properties and angle allocations here, **there may not be a correct option** among the ones given without empirical supporting external context.
Feel free to check angles B and D using specifics, but ensure every computation pairs correctly (e.g., remaining calculations too etc). The reframed focus is critical to validate cyclic properties firmly. Let me know if you need further elucidation!