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
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
mADC=84.5°
m Modifying above upper A upper D upper C with arc equals 84.5 degrees
m∠D=169°
inscribed in circle E
. The mAB=54°
, mBC=115°
, and mCD=86°
. Select the true statement.(1 point)
Responses
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
mADC=84.5°
m Modifying above upper A upper D upper C with arc equals 84.5 degrees
m∠D=169°
Answers
GPT-4o mini
To analyze the quadrilateral \(ABCD\) inscribed in circle \(E\), we need to use the properties of cyclic quadrilaterals. One of the key properties is that opposite angles are supplementary (i.e., they sum to \(180^\circ\)).
We know the following angle measures:
- \(m \angle AB = 54^\circ\)
- \(m \angle BC = 115^\circ\)
- \(m \angle CD = 86^\circ\)
Since the sum of the opposite angles must equal \(180^\circ\), we can find \(m \angle AD\) and \(m \angle DA\):
1. To find \(m \angle AD\):
\[
m \angle AD + m \angle BC = 180^\circ
\]
\[
m \angle AD + 115^\circ = 180^\circ
\]
\[
m \angle AD = 180^\circ - 115^\circ = 65^\circ
\]
2. To find \(m \angle DA\):
\[
m \angle AB + m \angle CD = 180^\circ
\]
\[
54^\circ + m \angle DA = 180^\circ
\]
\[
m \angle DA = 180^\circ - 54^\circ = 126^\circ
\]
Now let's examine the options provided:
1. **\(m \angle AD = 115^\circ\)**: This is **false**, as calculated, \(m \angle AD = 65^\circ\).
2. **\(m \angle B = 95.5^\circ\)**: This is also **false**; \(m \angle B\) was given as \(54^\circ\).
3. **\(m \angle ADC = 84.5^\circ\)**: Since \(m \angle DA + m \angle CD\) should sum to \(180^\circ\), it can be calculated as:
\[
m \angle ADC + m \angle CD = 180^\circ \implies m \angle ADC + 86^\circ = 180^\circ
\]
\[
m \angle ADC = 94^\circ
\]
So this statement is **false** as \(m \angle ADC\) is \(94^\circ\), not \(84.5^\circ\).
4. **\(m \angle D = 169^\circ\)**: This is **false** too, since we derived \(m \angle DA = 126^\circ\).
None of the provided statements are true based on our calculations for the quadrilateral \(ABCD\) inscribed in circle \(E\). If there are any discrepancies or if additional information is provided, that could help refine the answer further.
We know the following angle measures:
- \(m \angle AB = 54^\circ\)
- \(m \angle BC = 115^\circ\)
- \(m \angle CD = 86^\circ\)
Since the sum of the opposite angles must equal \(180^\circ\), we can find \(m \angle AD\) and \(m \angle DA\):
1. To find \(m \angle AD\):
\[
m \angle AD + m \angle BC = 180^\circ
\]
\[
m \angle AD + 115^\circ = 180^\circ
\]
\[
m \angle AD = 180^\circ - 115^\circ = 65^\circ
\]
2. To find \(m \angle DA\):
\[
m \angle AB + m \angle CD = 180^\circ
\]
\[
54^\circ + m \angle DA = 180^\circ
\]
\[
m \angle DA = 180^\circ - 54^\circ = 126^\circ
\]
Now let's examine the options provided:
1. **\(m \angle AD = 115^\circ\)**: This is **false**, as calculated, \(m \angle AD = 65^\circ\).
2. **\(m \angle B = 95.5^\circ\)**: This is also **false**; \(m \angle B\) was given as \(54^\circ\).
3. **\(m \angle ADC = 84.5^\circ\)**: Since \(m \angle DA + m \angle CD\) should sum to \(180^\circ\), it can be calculated as:
\[
m \angle ADC + m \angle CD = 180^\circ \implies m \angle ADC + 86^\circ = 180^\circ
\]
\[
m \angle ADC = 94^\circ
\]
So this statement is **false** as \(m \angle ADC\) is \(94^\circ\), not \(84.5^\circ\).
4. **\(m \angle D = 169^\circ\)**: This is **false** too, since we derived \(m \angle DA = 126^\circ\).
None of the provided statements are true based on our calculations for the quadrilateral \(ABCD\) inscribed in circle \(E\). If there are any discrepancies or if additional information is provided, that could help refine the answer further.