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∠B=95.5°
m angle upper B equals 95.5 degrees
mAD=115°
m Modifying above upper A upper D with arc equals 115 degrees
mADC=84.5°
m Modifying above upper A upper D upper C with arc equals 84.5 degrees
m∠D=169°
Responses
m∠B=95.5°
m angle upper B equals 95.5 degrees
mAD=115°
m Modifying above upper A upper D with arc equals 115 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 solve the problem, we need to analyze the given angles of the quadrilateral ABCD inscribed in circle E.
1. **Understanding the angles**:
- In a cyclic quadrilateral, the opposite angles are supplementary, meaning they add up to 180°.
- The given angles are as follows:
- \(m \angle A + m \angle C = 180°\) (opposite angles)
- \(m \angle B + m \angle D = 180°\) (opposite angles)
2. **Given Angles**:
- \(m \angle A = 54°\)
- \(m \angle B = 115°\)
- \(m \angle C = x\) (unknown angle)
- \(m \angle D = y\) (unknown angle)
3. **Calculating Opposite Angles**:
Using the supplementary property of opposite angles, we can find angles C and D.
a. Finding angle C:
\[
m \angle A + m \angle C = 180°
\]
\[
54° + x = 180° \implies x = 180° - 54° = 126°
\]
Therefore, \(m \angle C = 126°\).
b. Finding angle D:
\[
m \angle B + m \angle D = 180°
\]
\[
115° + y = 180° \implies y = 180° - 115° = 65°
\]
Therefore, \(m \angle D = 65°\).
Now we have:
- \(m \angle A = 54°\)
- \(m \angle B = 115°\)
- \(m \angle C = 126°\)
- \(m \angle D = 65°\)
4. **Evaluating options**:
- **Option 1**: \(m \angle B = 95.5°\) **(False)**, as \(m \angle B = 115°\).
- **Option 2**: \(m AD = 115°\) **(False)**; without further information, we cannot equate the angle AD directly to the measurements given.
- **Option 3**: \(m \angle ADC = 84.5°\) **(False)**, as we calculated angles A and D.
- **Option 4**: \(m \angle D = 169°\) **(False)**, since \(m \angle D = 65°\).
None of the provided options appear to match the calculations. To ensure we follow logical course:
- Verifying angles sum: \(54° + 115° + 126° + 65° = 360°\).
In conclusion, the true relationships in cyclic quadrilaterals were confirmed, yet the descriptions provided for responses don’t hold accurate in the context of \(ABC\) being \(D\). Please double-check if assumptions were made against stated angles vs. diagram orientation. Each calculated angle stands independently as dictated under circle properties.
1. **Understanding the angles**:
- In a cyclic quadrilateral, the opposite angles are supplementary, meaning they add up to 180°.
- The given angles are as follows:
- \(m \angle A + m \angle C = 180°\) (opposite angles)
- \(m \angle B + m \angle D = 180°\) (opposite angles)
2. **Given Angles**:
- \(m \angle A = 54°\)
- \(m \angle B = 115°\)
- \(m \angle C = x\) (unknown angle)
- \(m \angle D = y\) (unknown angle)
3. **Calculating Opposite Angles**:
Using the supplementary property of opposite angles, we can find angles C and D.
a. Finding angle C:
\[
m \angle A + m \angle C = 180°
\]
\[
54° + x = 180° \implies x = 180° - 54° = 126°
\]
Therefore, \(m \angle C = 126°\).
b. Finding angle D:
\[
m \angle B + m \angle D = 180°
\]
\[
115° + y = 180° \implies y = 180° - 115° = 65°
\]
Therefore, \(m \angle D = 65°\).
Now we have:
- \(m \angle A = 54°\)
- \(m \angle B = 115°\)
- \(m \angle C = 126°\)
- \(m \angle D = 65°\)
4. **Evaluating options**:
- **Option 1**: \(m \angle B = 95.5°\) **(False)**, as \(m \angle B = 115°\).
- **Option 2**: \(m AD = 115°\) **(False)**; without further information, we cannot equate the angle AD directly to the measurements given.
- **Option 3**: \(m \angle ADC = 84.5°\) **(False)**, as we calculated angles A and D.
- **Option 4**: \(m \angle D = 169°\) **(False)**, since \(m \angle D = 65°\).
None of the provided options appear to match the calculations. To ensure we follow logical course:
- Verifying angles sum: \(54° + 115° + 126° + 65° = 360°\).
In conclusion, the true relationships in cyclic quadrilaterals were confirmed, yet the descriptions provided for responses don’t hold accurate in the context of \(ABC\) being \(D\). Please double-check if assumptions were made against stated angles vs. diagram orientation. Each calculated angle stands independently as dictated under circle properties.