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°
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, we need to use the properties of inscribed angles and the arcs they intercept in a circle.
In a cyclic quadrilateral (one inscribed in a circle), the opposite angles are supplementary. We can use this property to find the missing angle ∠A, ∠C, and ∠D.
Given:
- \( m\angle A = 54° \)
- \( m\angle B = 115° \)
- \( m\angle C = 86° \)
Using the property of opposite angles in a quadrilateral, we can find \( m\angle D \):
\[
m\angle A + m\angle C + m\angle B + m\angle D = 360°
\]
We can express \( m\angle D \) as:
\[
m\angle D = 360° - (m\angle A + m\angle B + m\angle C)
\]
Substituting the values we have:
\[
m\angle D = 360° - (54° + 115° + 86°)
\]
Calculating the sum inside the parentheses:
\[
54° + 115° + 86° = 255°
\]
Now we can calculate \( m\angle D \):
\[
m\angle D = 360° - 255° = 105°
\]
Now let's check the options given:
1. **m∠D = 169°** - This is false since we found \( m\angle D = 105° \).
2. **m∠B = 95.5°** - We know \( m\angle B = 115° \) as given, so this is false.
3. **mAD with arc = 115°** - This statement is not given, and we do not have sufficient information to determine if it's true or false.
4. **mADC with arc = 84.5°** - There's no calculated angle corresponding to this to affirm its truth.
From the available statements, we can state clearly that \( m∠D\) is not equal to the angles listed nor do we have information about arc measures calculated.
Thus, none of the provided statements are correct based on the computations and inscribed angle theorem confirming the value of \( m\angle D = 105° \).
Final Answer: None of the statements are true based on calculated angles and their supplemental relationships in the context of the properties of inscribed angles in a circle.
In a cyclic quadrilateral (one inscribed in a circle), the opposite angles are supplementary. We can use this property to find the missing angle ∠A, ∠C, and ∠D.
Given:
- \( m\angle A = 54° \)
- \( m\angle B = 115° \)
- \( m\angle C = 86° \)
Using the property of opposite angles in a quadrilateral, we can find \( m\angle D \):
\[
m\angle A + m\angle C + m\angle B + m\angle D = 360°
\]
We can express \( m\angle D \) as:
\[
m\angle D = 360° - (m\angle A + m\angle B + m\angle C)
\]
Substituting the values we have:
\[
m\angle D = 360° - (54° + 115° + 86°)
\]
Calculating the sum inside the parentheses:
\[
54° + 115° + 86° = 255°
\]
Now we can calculate \( m\angle D \):
\[
m\angle D = 360° - 255° = 105°
\]
Now let's check the options given:
1. **m∠D = 169°** - This is false since we found \( m\angle D = 105° \).
2. **m∠B = 95.5°** - We know \( m\angle B = 115° \) as given, so this is false.
3. **mAD with arc = 115°** - This statement is not given, and we do not have sufficient information to determine if it's true or false.
4. **mADC with arc = 84.5°** - There's no calculated angle corresponding to this to affirm its truth.
From the available statements, we can state clearly that \( m∠D\) is not equal to the angles listed nor do we have information about arc measures calculated.
Thus, none of the provided statements are correct based on the computations and inscribed angle theorem confirming the value of \( m\angle D = 105° \).
Final Answer: None of the statements are true based on calculated angles and their supplemental relationships in the context of the properties of inscribed angles in a circle.