Question
Draw the following diagram quadrilateral ABCD inscribed in circle E. The mAB = 54 degrees mBC=115 degrees and mCD=86 degrees .select the true statement. mAD=115 degrees. mADC=84.5 degrees. m<D=169 degrees. m<B=95.5 degrees
Answers
GPT-4o mini
To analyze quadrilateral ABCD inscribed in circle E with the given angles, let's denote the angles at each vertex:
1. \( m \angle ABC = 115^\circ \)
2. \( m \angle BCD = 86^\circ \)
3. \( m \angle CDA \) will be calculated.
4. \( m \angle DAB \) will be calculated.
In a cyclic quadrilateral (one inscribed in a circle), the opposite angles are supplementary, meaning they sum to \( 180^\circ \).
Using this property:
1. For angles \( ABC \) and \( CDA \):
\[
m \angle ABC + m \angle CDA = 180^\circ
\]
\[
115^\circ + m \angle CDA = 180^\circ
\]
\[
m \angle CDA = 180^\circ - 115^\circ = 65^\circ
\]
2. For angles \( BCD \) and \( DAB \):
\[
m \angle BCD + m \angle DAB = 180^\circ
\]
\[
86^\circ + m \angle DAB = 180^\circ
\]
\[
m \angle DAB = 180^\circ - 86^\circ = 94^\circ
\]
Now, the measures of the angles in quadrilateral ABCD are:
- \( m \angle ABC = 115^\circ \)
- \( m \angle BCD = 86^\circ \)
- \( m \angle CDA = 65^\circ \)
- \( m \angle DAB = 94^\circ \)
Now let's evaluate the statements provided:
1. \( mAD=115^\circ \) — This does not relate to angle measures directly.
2. \( mADC=84.5^\circ \) — \( m \angle ADC \) has not been calculated and is not equal to this value.
3. \( m<D=169^\circ \) — \( m \angle D = m \angle DAB + m \angle CDA \) isn't \( 169^\circ \).
4. \( m<B=95.5^\circ \) — This is not accurate since we found \( m \angle ABC = 115^\circ \).
Based on the calculations, none of the statements provided appear correct. If any statement has slight discrepancies, please check if they are referring to different angle measurements not revealed directly here, such as arc measures or other configurations.
Thus, it appears all suggested statements are incorrect or unrelated to the angles computed in the quadrilateral ABCD.
1. \( m \angle ABC = 115^\circ \)
2. \( m \angle BCD = 86^\circ \)
3. \( m \angle CDA \) will be calculated.
4. \( m \angle DAB \) will be calculated.
In a cyclic quadrilateral (one inscribed in a circle), the opposite angles are supplementary, meaning they sum to \( 180^\circ \).
Using this property:
1. For angles \( ABC \) and \( CDA \):
\[
m \angle ABC + m \angle CDA = 180^\circ
\]
\[
115^\circ + m \angle CDA = 180^\circ
\]
\[
m \angle CDA = 180^\circ - 115^\circ = 65^\circ
\]
2. For angles \( BCD \) and \( DAB \):
\[
m \angle BCD + m \angle DAB = 180^\circ
\]
\[
86^\circ + m \angle DAB = 180^\circ
\]
\[
m \angle DAB = 180^\circ - 86^\circ = 94^\circ
\]
Now, the measures of the angles in quadrilateral ABCD are:
- \( m \angle ABC = 115^\circ \)
- \( m \angle BCD = 86^\circ \)
- \( m \angle CDA = 65^\circ \)
- \( m \angle DAB = 94^\circ \)
Now let's evaluate the statements provided:
1. \( mAD=115^\circ \) — This does not relate to angle measures directly.
2. \( mADC=84.5^\circ \) — \( m \angle ADC \) has not been calculated and is not equal to this value.
3. \( m<D=169^\circ \) — \( m \angle D = m \angle DAB + m \angle CDA \) isn't \( 169^\circ \).
4. \( m<B=95.5^\circ \) — This is not accurate since we found \( m \angle ABC = 115^\circ \).
Based on the calculations, none of the statements provided appear correct. If any statement has slight discrepancies, please check if they are referring to different angle measurements not revealed directly here, such as arc measures or other configurations.
Thus, it appears all suggested statements are incorrect or unrelated to the angles computed in the quadrilateral ABCD.