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°
Answers
GPT-4o mini
To solve the problem, we need to analyze the angles and arcs related to quadrilateral \(ABCD\) inscribed in circle \(E\).
1. **Given Angles**:
- \(m\angle A = 54^\circ\)
- \(m\angle B = 115^\circ\)
- \(m\angle C = 86^\circ\)
2. **Using the Inscribed Angle Theorem**:
The inscribed angle theorem states that the measure of an inscribed angle is half the measure of the arc that it subtends.
3. **Finding the remaining angle \(D\)**:
Since \(ABCD\) is a cyclic quadrilateral, the opposite angles are supplementary. Thus, we can use the relationship:
\[
m\angle A + m\angle C = 180^\circ
\]
and
\[
m\angle B + m\angle D = 180^\circ.
\]
Therefore, we can find \(m\angle D\):
\[
m\angle D = 180^\circ - m\angle B = 180^\circ - 115^\circ = 65^\circ.
\]
4. **Finding the Arc Measures**:
We can find the measures of arcs based on the angles:
- Arc \(BC\) subtends angle \(A\):
\[
m\arc{BC} = 2 \times m\angle A = 2 \times 54^\circ = 108^\circ.
\]
- Arc \(CD\) subtends angle \(B\):
\[
m\arc{CD} = 2 \times m\angle B = 2 \times 115^\circ = 230^\circ.
\]
- Arc \(DA\) subtends angle \(C\):
\[
m\arc{DA} = 2 \times m\angle C = 2 \times 86^\circ = 172^\circ.
\]
- Arc \(AB\) subtends angle \(D\):
\[
m\arc{AB} = 2 \times m\angle D = 2 \times 65^\circ = 130^\circ.
\]
5. **Finding \(m \angle ADC\)**:
The angle \( \angle ADC \) can also be determined using the fact that it subtends the arc \(AC\):
\[
m\arc{AC} = m\arc{AB} + m\arc{BC} = 130^\circ + 108^\circ = 238^\circ.
\]
Thus,
\[
m\angle ADC = \frac{1}{2} m\arc{AC} = \frac{1}{2} \times 238^\circ = 119^\circ.
\]
Now, looking at the responses given:
- \(m\angle B = 95.5^\circ\) – False.
- \(m\arc{AD} = 115^\circ\) – False.
- \(m\angle ADC = 84.5^\circ\) – False.
- \(m\angle D = 169^\circ\) – False.
Ultimately, none of the responses are true based on these calculations. If you had to choose among these statements, you may need to reevaluate, but given the assignments and the calculations above, none of the given statements accurately reflect the results.
1. **Given Angles**:
- \(m\angle A = 54^\circ\)
- \(m\angle B = 115^\circ\)
- \(m\angle C = 86^\circ\)
2. **Using the Inscribed Angle Theorem**:
The inscribed angle theorem states that the measure of an inscribed angle is half the measure of the arc that it subtends.
3. **Finding the remaining angle \(D\)**:
Since \(ABCD\) is a cyclic quadrilateral, the opposite angles are supplementary. Thus, we can use the relationship:
\[
m\angle A + m\angle C = 180^\circ
\]
and
\[
m\angle B + m\angle D = 180^\circ.
\]
Therefore, we can find \(m\angle D\):
\[
m\angle D = 180^\circ - m\angle B = 180^\circ - 115^\circ = 65^\circ.
\]
4. **Finding the Arc Measures**:
We can find the measures of arcs based on the angles:
- Arc \(BC\) subtends angle \(A\):
\[
m\arc{BC} = 2 \times m\angle A = 2 \times 54^\circ = 108^\circ.
\]
- Arc \(CD\) subtends angle \(B\):
\[
m\arc{CD} = 2 \times m\angle B = 2 \times 115^\circ = 230^\circ.
\]
- Arc \(DA\) subtends angle \(C\):
\[
m\arc{DA} = 2 \times m\angle C = 2 \times 86^\circ = 172^\circ.
\]
- Arc \(AB\) subtends angle \(D\):
\[
m\arc{AB} = 2 \times m\angle D = 2 \times 65^\circ = 130^\circ.
\]
5. **Finding \(m \angle ADC\)**:
The angle \( \angle ADC \) can also be determined using the fact that it subtends the arc \(AC\):
\[
m\arc{AC} = m\arc{AB} + m\arc{BC} = 130^\circ + 108^\circ = 238^\circ.
\]
Thus,
\[
m\angle ADC = \frac{1}{2} m\arc{AC} = \frac{1}{2} \times 238^\circ = 119^\circ.
\]
Now, looking at the responses given:
- \(m\angle B = 95.5^\circ\) – False.
- \(m\arc{AD} = 115^\circ\) – False.
- \(m\angle ADC = 84.5^\circ\) – False.
- \(m\angle D = 169^\circ\) – False.
Ultimately, none of the responses are true based on these calculations. If you had to choose among these statements, you may need to reevaluate, but given the assignments and the calculations above, none of the given statements accurately reflect the results.