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 mADC=84.5° m Modifying above upper A upper D upper C with arc equals 84.5 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 m∠D=169°
Answers
GPT-4o mini
To solve the problem regarding the quadrilateral ABCD inscribed in circle E with given angles, we first need to recall some properties of cyclic quadrilaterals (those inscribed in a circle).
### Given:
- \( m\angle AB = 54^\circ \)
- \( m\angle BC = 115^\circ \)
- \( m\angle CD = 86^\circ \)
### Finding \( m\angle DA \):
In a cyclic quadrilateral, the opposite angles sum up to \( 180^\circ \). That is:
- \( m\angle AB + m\angle CD = 180^\circ \)
- \( m\angle BC + m\angle DA = 180^\circ \)
1. **First, calculate \( m\angle DA \):**
We can find \( m\angle DA \) using the relationship between angles:
\[
m\angle DA = 180^\circ - m\angle BC
\]
Substituting the value for \( m\angle BC \):
\[
m\angle DA = 180^\circ - 115^\circ = 65^\circ
\]
2. **Next, calculate \( m\angle ADC \):**
To find \( m\angle ADC \) we can use the relationship with \( m\angle AB \):
\[
m\angle ADC = 180^\circ - m\angle AB = 180^\circ - 54^\circ = 126^\circ
\]
Now we have the angles of the quadrilateral:
- \( m\angle AB = 54^\circ \)
- \( m\angle BC = 115^\circ \)
- \( m\angle CD = 86^\circ \)
- \( m\angle DA = 65^\circ \)
- \( m\angle ADC = 126^\circ \)
### Check the Options Given:
1. \( m\angle ADC \approx 84.5^\circ \) **(This is incorrect as \( 126^\circ \) is not \( 84.5^\circ \))**
2. \( m\angle AD \text{ with arc } \approx 115^\circ \) **(Involves angle subtended by the arc, not applicable here)**
3. \( m\angle B \approx 95.5^\circ \) **(This is also incorrect)**
4. \( m\angle D = 169^\circ \) **(This is incorrect as well)**
### Conclusion:
None of the options provided are correct based on how we've calculated the angles in quadrilateral ABCD. The angles calculated show how the cyclic nature of the quadrilateral works, with opposite angles summing to \( 180^\circ \).
The accurate measure of \( m\angle DA \) is \( 65^\circ \), and \( m\angle ADC = 126^\circ \). Therefore, additional insight or re-evaluation of the given choices might be necessary.
### Given:
- \( m\angle AB = 54^\circ \)
- \( m\angle BC = 115^\circ \)
- \( m\angle CD = 86^\circ \)
### Finding \( m\angle DA \):
In a cyclic quadrilateral, the opposite angles sum up to \( 180^\circ \). That is:
- \( m\angle AB + m\angle CD = 180^\circ \)
- \( m\angle BC + m\angle DA = 180^\circ \)
1. **First, calculate \( m\angle DA \):**
We can find \( m\angle DA \) using the relationship between angles:
\[
m\angle DA = 180^\circ - m\angle BC
\]
Substituting the value for \( m\angle BC \):
\[
m\angle DA = 180^\circ - 115^\circ = 65^\circ
\]
2. **Next, calculate \( m\angle ADC \):**
To find \( m\angle ADC \) we can use the relationship with \( m\angle AB \):
\[
m\angle ADC = 180^\circ - m\angle AB = 180^\circ - 54^\circ = 126^\circ
\]
Now we have the angles of the quadrilateral:
- \( m\angle AB = 54^\circ \)
- \( m\angle BC = 115^\circ \)
- \( m\angle CD = 86^\circ \)
- \( m\angle DA = 65^\circ \)
- \( m\angle ADC = 126^\circ \)
### Check the Options Given:
1. \( m\angle ADC \approx 84.5^\circ \) **(This is incorrect as \( 126^\circ \) is not \( 84.5^\circ \))**
2. \( m\angle AD \text{ with arc } \approx 115^\circ \) **(Involves angle subtended by the arc, not applicable here)**
3. \( m\angle B \approx 95.5^\circ \) **(This is also incorrect)**
4. \( m\angle D = 169^\circ \) **(This is incorrect as well)**
### Conclusion:
None of the options provided are correct based on how we've calculated the angles in quadrilateral ABCD. The angles calculated show how the cyclic nature of the quadrilateral works, with opposite angles summing to \( 180^\circ \).
The accurate measure of \( m\angle DA \) is \( 65^\circ \), and \( m\angle ADC = 126^\circ \). Therefore, additional insight or re-evaluation of the given choices might be necessary.