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∠D=169°
m angle upper D equals 169 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
mADC=84.5°
inscribed in circle E
. The mAB=54°
, mBC=115°
, and mCD=86°
. Select the true statement.(1 point)
Responses
m∠D=169°
m angle upper D equals 169 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
mADC=84.5°
Answers
GPT-4o mini
To solve for the angles in the quadrilateral ABCD inscribed in circle E, we can apply the properties of cyclic quadrilaterals.
1. The sum of the opposite angles in a cyclic quadrilateral is always 180 degrees. Thus:
- \( m\angle A + m\angle C = 180^\circ \)
- \( m\angle B + m\angle D = 180^\circ \)
2. The measures of \( \angle A \), \( \angle B \), and \( \angle C \) are given:
- \( m\angle AB = 54^\circ \)
- \( m\angle BC = 115^\circ \)
- \( m\angle CD = 86^\circ \)
3. Let's find the measure of angle \( D \):
- First, find \( m\angle A \) using the fact that the angles around point E must sum up to 360 degrees:
\[
m\angle A + m\angle B + m\angle C + m\angle D = 360^\circ
\]
- We already have:
\[
m\angle B = 115^\circ
\]
- Rearranging gives:
\[
m\angle A + 115^\circ + 86^\circ + m\angle D = 360^\circ
\]
- This simplifies to:
\[
m\angle A + m\angle D = 360^\circ - (115^\circ + 86^\circ) = 360^\circ - 201^\circ = 159^\circ
\]
4. Now we check each response:
- **m∠D = 169°**: Let's check if this is true by calculating \( m\angle D \) from \( m\angle B \).
- Since \( m\angle B + m\angle D = 180^\circ \):
\[
115^\circ + m\angle D = 180^\circ \implies m\angle D = 180^\circ - 115^\circ = 65^\circ
\]
- This statement is false.
- **mAD = 115°**: This is not directly derived from our information and seems incorrect without additional context.
- **m∠B = 95.5°**: Given \( m\angle B = 115^\circ \), this is false.
- **mADC = 84.5°**: To compute \( m\angle A + m\angle C \), we need:
- We calculated that:
\[
m\angle A + m\angle D = 159^\circ
\]
- And we know \( m\angle C = 86^\circ \):
- From the cyclic property, check:
\[
m\angle C + m\angle D = 180^\circ \implies 86^\circ + m\angle D = 180^\circ \implies m\angle D = 94^\circ
\]
This would lead to incorrect results too.
Based on all reductions, none of the statements offered is true. The calculated pairs didn't yield results that support any proposed answers entirely. If prompted to select one statement I would note that all obtained properties need additional review as no correlations line up to satisfy given responses.
1. The sum of the opposite angles in a cyclic quadrilateral is always 180 degrees. Thus:
- \( m\angle A + m\angle C = 180^\circ \)
- \( m\angle B + m\angle D = 180^\circ \)
2. The measures of \( \angle A \), \( \angle B \), and \( \angle C \) are given:
- \( m\angle AB = 54^\circ \)
- \( m\angle BC = 115^\circ \)
- \( m\angle CD = 86^\circ \)
3. Let's find the measure of angle \( D \):
- First, find \( m\angle A \) using the fact that the angles around point E must sum up to 360 degrees:
\[
m\angle A + m\angle B + m\angle C + m\angle D = 360^\circ
\]
- We already have:
\[
m\angle B = 115^\circ
\]
- Rearranging gives:
\[
m\angle A + 115^\circ + 86^\circ + m\angle D = 360^\circ
\]
- This simplifies to:
\[
m\angle A + m\angle D = 360^\circ - (115^\circ + 86^\circ) = 360^\circ - 201^\circ = 159^\circ
\]
4. Now we check each response:
- **m∠D = 169°**: Let's check if this is true by calculating \( m\angle D \) from \( m\angle B \).
- Since \( m\angle B + m\angle D = 180^\circ \):
\[
115^\circ + m\angle D = 180^\circ \implies m\angle D = 180^\circ - 115^\circ = 65^\circ
\]
- This statement is false.
- **mAD = 115°**: This is not directly derived from our information and seems incorrect without additional context.
- **m∠B = 95.5°**: Given \( m\angle B = 115^\circ \), this is false.
- **mADC = 84.5°**: To compute \( m\angle A + m\angle C \), we need:
- We calculated that:
\[
m\angle A + m\angle D = 159^\circ
\]
- And we know \( m\angle C = 86^\circ \):
- From the cyclic property, check:
\[
m\angle C + m\angle D = 180^\circ \implies 86^\circ + m\angle D = 180^\circ \implies m\angle D = 94^\circ
\]
This would lead to incorrect results too.
Based on all reductions, none of the statements offered is true. The calculated pairs didn't yield results that support any proposed answers entirely. If prompted to select one statement I would note that all obtained properties need additional review as no correlations line up to satisfy given responses.