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
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°
Responses
m∠D=169°
m angle upper D equals 169 degrees
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°
Answers
GPT-4o mini
To solve the problem, we can use the properties of cyclic quadrilaterals (quadrilaterals inscribed in a circle), where opposite angles are supplementary, meaning their measures add up to \(180^\circ\).
Given the angles of the quadrilateral ABCD:
- \( m\angle A + m\angle C = 180^\circ \)
- \( m\angle B + m\angle D = 180^\circ \)
We are provided with:
- \( m\angle A = 54^\circ \)
- \( m\angle B = 115^\circ \)
- \( m\angle C = 86^\circ \)
1. **Finding \( m\angle D \)**:
Since \( m\angle B + m\angle D = 180^\circ \):
\[
115^\circ + m\angle D = 180^\circ
\]
\[
m\angle D = 180^\circ - 115^\circ = 65^\circ
\]
Thus, the statement \( m\angle D = 169^\circ \) is **false**.
2. **Finding \( m\angle A \)** using \( m\angle A + m\angle C = 180^\circ \):
\[
m\angle A + 86^\circ = 180^\circ
\]
\[
m\angle A = 180^\circ - 86^\circ = 94^\circ
\]
This also seems incorrect since \( m\angle A \) is given as \( 54^\circ \).
3. **Checking the statements provided**:
- \( m\angle D = 169^\circ \) is **false**.
- \( m\angle B = 95.5^\circ \) is **false** (since we have \( m\angle B = 115^\circ \)).
- \( mAD\text{(arc)} = 115^\circ\) which is true as arc measurements are consistent with the angles.
- To find \( mADC \):
- Compute the angles:
\[
m\angle A + m\angle D = 54^\circ + 65^\circ = 119^\circ
\]
- Hence,
\[
m\angle ADC = 180^\circ - 119^\circ = 61^\circ
\]
In conclusion, given the information provided and the calculations, the true statement is:
- \( mAD\text{(arc)} = 115^\circ \) since this corresponds with the measure adjacent to B being equal (the arcs supported by these angles).
Feel free to ask if you have any more queries!
Given the angles of the quadrilateral ABCD:
- \( m\angle A + m\angle C = 180^\circ \)
- \( m\angle B + m\angle D = 180^\circ \)
We are provided with:
- \( m\angle A = 54^\circ \)
- \( m\angle B = 115^\circ \)
- \( m\angle C = 86^\circ \)
1. **Finding \( m\angle D \)**:
Since \( m\angle B + m\angle D = 180^\circ \):
\[
115^\circ + m\angle D = 180^\circ
\]
\[
m\angle D = 180^\circ - 115^\circ = 65^\circ
\]
Thus, the statement \( m\angle D = 169^\circ \) is **false**.
2. **Finding \( m\angle A \)** using \( m\angle A + m\angle C = 180^\circ \):
\[
m\angle A + 86^\circ = 180^\circ
\]
\[
m\angle A = 180^\circ - 86^\circ = 94^\circ
\]
This also seems incorrect since \( m\angle A \) is given as \( 54^\circ \).
3. **Checking the statements provided**:
- \( m\angle D = 169^\circ \) is **false**.
- \( m\angle B = 95.5^\circ \) is **false** (since we have \( m\angle B = 115^\circ \)).
- \( mAD\text{(arc)} = 115^\circ\) which is true as arc measurements are consistent with the angles.
- To find \( mADC \):
- Compute the angles:
\[
m\angle A + m\angle D = 54^\circ + 65^\circ = 119^\circ
\]
- Hence,
\[
m\angle ADC = 180^\circ - 119^\circ = 61^\circ
\]
In conclusion, given the information provided and the calculations, the true statement is:
- \( mAD\text{(arc)} = 115^\circ \) since this corresponds with the measure adjacent to B being equal (the arcs supported by these angles).
Feel free to ask if you have any more queries!