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∠D=169° m angle upper D equals 169 degrees m∠B=95.5°
Answers
GPT-4o mini
To analyze the given quadrilateral \(ABCD\) inscribed in circle \(E\) based on the provided angle measures, we need to employ the properties of cyclic quadrilaterals. The key property is that opposite angles in a cyclic quadrilateral sum up to \(180^\circ\).
Given:
- \(m\angle AB = 54^\circ\)
- \(m\angle BC = 115^\circ\)
- \(m\angle CD = 86^\circ\)
We can find the measures of the angles \(m\angle DA\) and \(m\angle AD\) using the cyclic property:
1. To find \(m\angle DA\):
\[
m\angle AB + m\angle CD = 180^\circ \Rightarrow 54^\circ + m\angle DA = 180^\circ
\]
\[
m\angle DA = 180^\circ - 54^\circ = 126^\circ
\]
2. To find \(m\angle AD\):
\[
m\angle BC + m\angle AD = 180^\circ \Rightarrow 115^\circ + m\angle AD = 180^\circ
\]
\[
m\angle AD = 180^\circ - 115^\circ = 65^\circ
\]
Next, let's check the available responses based on our calculations:
- \(m\angle ADC\) is not directly calculable but it should respect the cyclic property as well.
- \(m\angle D = 169^\circ\) and \(m\angle B = 95.5^\circ\) are not reflective of our existing calculations for angles \(DA\) and \(AD\).
Based on the calculations, the correct statement appears to be linked to:
- \(m\angle AD\) = 65°, and since \(m\angle DA\) = 126°, and checking the angles systematically shows alignment with the properties.
From the research on all results given, the statements don't reflect any specific outcome that might verify the original measure setup; hence, it's crucial to deduce based on the circle's properties and measure if erroneous inscriptions arise from possible checks, especially with measures rounds around circles.
**Conclusion: none of the options directly states** a valid conclusion based on simultaneous checks of measure values vis-a-vis cyclic properties beyond established refused values, thus verifying the angles to ensure no additional mismatched can be established without indicating the prior known cyclic properties.
Given:
- \(m\angle AB = 54^\circ\)
- \(m\angle BC = 115^\circ\)
- \(m\angle CD = 86^\circ\)
We can find the measures of the angles \(m\angle DA\) and \(m\angle AD\) using the cyclic property:
1. To find \(m\angle DA\):
\[
m\angle AB + m\angle CD = 180^\circ \Rightarrow 54^\circ + m\angle DA = 180^\circ
\]
\[
m\angle DA = 180^\circ - 54^\circ = 126^\circ
\]
2. To find \(m\angle AD\):
\[
m\angle BC + m\angle AD = 180^\circ \Rightarrow 115^\circ + m\angle AD = 180^\circ
\]
\[
m\angle AD = 180^\circ - 115^\circ = 65^\circ
\]
Next, let's check the available responses based on our calculations:
- \(m\angle ADC\) is not directly calculable but it should respect the cyclic property as well.
- \(m\angle D = 169^\circ\) and \(m\angle B = 95.5^\circ\) are not reflective of our existing calculations for angles \(DA\) and \(AD\).
Based on the calculations, the correct statement appears to be linked to:
- \(m\angle AD\) = 65°, and since \(m\angle DA\) = 126°, and checking the angles systematically shows alignment with the properties.
From the research on all results given, the statements don't reflect any specific outcome that might verify the original measure setup; hence, it's crucial to deduce based on the circle's properties and measure if erroneous inscriptions arise from possible checks, especially with measures rounds around circles.
**Conclusion: none of the options directly states** a valid conclusion based on simultaneous checks of measure values vis-a-vis cyclic properties beyond established refused values, thus verifying the angles to ensure no additional mismatched can be established without indicating the prior known cyclic properties.