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 determine the true statement regarding the angles in quadrilateral ABCD inscribed in circle E, we can use the following property of cyclic quadrilaterals:
The opposite angles of a cyclic quadrilateral sum to 180°. This means:
- \( m∠A + m∠C = 180° \)
- \( m∠B + m∠D = 180° \)
Given:
- \( m∠AB = 54° \) (let's call this \( m∠A \))
- \( m∠BC = 115° \) (let's call this \( m∠B \))
- \( m∠CD = 86° \) (let's call this \( m∠C \))
First, let's calculate \( m∠D \):
Using the properties of cyclic quadrilaterals:
1. To find \( m∠D \), we know that \( m∠B + m∠D = 180° \):
\[
115° + m∠D = 180°
\]
\[
m∠D = 180° - 115° = 65°
\]
2. Now, to find \( m∠A + m∠C \):
\[
m∠A + m∠C = 180°
\]
\[
54° + m∠C = 180°
\]
\[
m∠C = 180° - 54° = 126°
\]
Therefore, we have:
- \( m∠A = 54° \)
- \( m∠B = 115° \)
- \( m∠C = 126° \)
- \( m∠D = 65° \)
Now, we can evaluate the given statements:
1. \( m∠AD = 115° \): **False**
2. \( m∠B = 95.5° \): **False** (it's actually \( 115° \))
3. \( m∠ADC = 84.5° \): This refers to the angle subtended by arc \( AC \), which is not directly calculated here.
4. \( m∠D = 169° \): **False** (it is actually \( 65° \))
Given these evaluations, none of the provided statements appear to be true based on our calculations.
However, to focus on \( m∠ADC \), since \( m∠A + m∠D = 119° \) does not directly match a value of \( 84.5°\), it indicates that it is necessary to reconsider if it is referring to arc measurements.
You should verify the exact circumstance of the angles described in \( m∠ADC \) to settle if it may hold an argument.
If the calculation regarding any specific angle does not match the typical cyclic quadrilateral properties provided and which leads us to revisit the constraints used for \( A, B, C, D \) which helps specify more on circum-circle placements.
The opposite angles of a cyclic quadrilateral sum to 180°. This means:
- \( m∠A + m∠C = 180° \)
- \( m∠B + m∠D = 180° \)
Given:
- \( m∠AB = 54° \) (let's call this \( m∠A \))
- \( m∠BC = 115° \) (let's call this \( m∠B \))
- \( m∠CD = 86° \) (let's call this \( m∠C \))
First, let's calculate \( m∠D \):
Using the properties of cyclic quadrilaterals:
1. To find \( m∠D \), we know that \( m∠B + m∠D = 180° \):
\[
115° + m∠D = 180°
\]
\[
m∠D = 180° - 115° = 65°
\]
2. Now, to find \( m∠A + m∠C \):
\[
m∠A + m∠C = 180°
\]
\[
54° + m∠C = 180°
\]
\[
m∠C = 180° - 54° = 126°
\]
Therefore, we have:
- \( m∠A = 54° \)
- \( m∠B = 115° \)
- \( m∠C = 126° \)
- \( m∠D = 65° \)
Now, we can evaluate the given statements:
1. \( m∠AD = 115° \): **False**
2. \( m∠B = 95.5° \): **False** (it's actually \( 115° \))
3. \( m∠ADC = 84.5° \): This refers to the angle subtended by arc \( AC \), which is not directly calculated here.
4. \( m∠D = 169° \): **False** (it is actually \( 65° \))
Given these evaluations, none of the provided statements appear to be true based on our calculations.
However, to focus on \( m∠ADC \), since \( m∠A + m∠D = 119° \) does not directly match a value of \( 84.5°\), it indicates that it is necessary to reconsider if it is referring to arc measurements.
You should verify the exact circumstance of the angles described in \( m∠ADC \) to settle if it may hold an argument.
If the calculation regarding any specific angle does not match the typical cyclic quadrilateral properties provided and which leads us to revisit the constraints used for \( A, B, C, D \) which helps specify more on circum-circle placements.