Answer :
Let's go through each part of the question step-by-step:
### Part a)
Question: State, with a reason, which other angle is equal to angle \( ABD \).
Solution:
Since \( A, B, C \), and \( D \) are points on the circumference of a circle, and angle \( ABD \) and angle \( ABC \) are subtended by the same arc \( AD \), we can conclude that:
[tex]\[ \angle ABD = \angle ABC \][/tex]
Reason: \(\triangle ABC\) and \(\triangle ABD\) are subtended by the same arc \( AD \), so we have:
[tex]\[ \angle ABC = \angle ABD \][/tex]
### Part b)
Question: State, with a reason, which other angle is equal to angle \( AEB \).
Solution:
Given that \( AE \) and \( BE \) are straight lines and they intersect at point \( E \), angle \( AEB \) and angle \( CED \) are vertical angles. Hence, we conclude:
[tex]\[ \angle AEB = \angle CED \][/tex]
Reason: \( AEB \) and \( CED \) are vertical angles, so they are equal:
[tex]\[ \angle AEB = \angle CED \][/tex]
### Part c)
Question: If angle \( ABC = 88^\circ \), state, with a reason, the size of angle \( ADC \).
Solution:
Given that \(\angle ABC = 88^\circ\), we need to find the measure of angle \( ADC \). Since \(A, B, C, \) and \(D\) form a cyclic quadrilateral (a four-sided figure where each vertex lies on the circumference of a circle), the opposite angles in a cyclic quadrilateral are supplementary. Therefore:
[tex]\[ \angle ABC + \angle ADC = 180^\circ \][/tex]
Given \(\angle ABC = 88^\circ\), we can substitute this value in:
[tex]\[ 88^\circ + \angle ADC = 180^\circ \][/tex]
Solving for \(\angle ADC\), we get:
[tex]\[ \angle ADC = 180^\circ - 88^\circ = 92^\circ \][/tex]
Reason: In a cyclic quadrilateral, the opposite angles are supplementary, hence:
[tex]\[ \angle ADC = 180^\circ - \angle ABC = 92^\circ \][/tex]
Final Answer:
- Part a) \(\angle ABD = \angle ABC\), because \(\triangle ABC\) and \(\triangle ABD\) are subtended by the same arc \( AD \).
- Part b) \(\angle AEB = \angle CED\), because \( AEB \) and \( CED \) are vertical angles.
- Part c) [tex]\(\angle ADC = 92^\circ\)[/tex], because in a cyclic quadrilateral, the opposite angles are supplementary.
### Part a)
Question: State, with a reason, which other angle is equal to angle \( ABD \).
Solution:
Since \( A, B, C \), and \( D \) are points on the circumference of a circle, and angle \( ABD \) and angle \( ABC \) are subtended by the same arc \( AD \), we can conclude that:
[tex]\[ \angle ABD = \angle ABC \][/tex]
Reason: \(\triangle ABC\) and \(\triangle ABD\) are subtended by the same arc \( AD \), so we have:
[tex]\[ \angle ABC = \angle ABD \][/tex]
### Part b)
Question: State, with a reason, which other angle is equal to angle \( AEB \).
Solution:
Given that \( AE \) and \( BE \) are straight lines and they intersect at point \( E \), angle \( AEB \) and angle \( CED \) are vertical angles. Hence, we conclude:
[tex]\[ \angle AEB = \angle CED \][/tex]
Reason: \( AEB \) and \( CED \) are vertical angles, so they are equal:
[tex]\[ \angle AEB = \angle CED \][/tex]
### Part c)
Question: If angle \( ABC = 88^\circ \), state, with a reason, the size of angle \( ADC \).
Solution:
Given that \(\angle ABC = 88^\circ\), we need to find the measure of angle \( ADC \). Since \(A, B, C, \) and \(D\) form a cyclic quadrilateral (a four-sided figure where each vertex lies on the circumference of a circle), the opposite angles in a cyclic quadrilateral are supplementary. Therefore:
[tex]\[ \angle ABC + \angle ADC = 180^\circ \][/tex]
Given \(\angle ABC = 88^\circ\), we can substitute this value in:
[tex]\[ 88^\circ + \angle ADC = 180^\circ \][/tex]
Solving for \(\angle ADC\), we get:
[tex]\[ \angle ADC = 180^\circ - 88^\circ = 92^\circ \][/tex]
Reason: In a cyclic quadrilateral, the opposite angles are supplementary, hence:
[tex]\[ \angle ADC = 180^\circ - \angle ABC = 92^\circ \][/tex]
Final Answer:
- Part a) \(\angle ABD = \angle ABC\), because \(\triangle ABC\) and \(\triangle ABD\) are subtended by the same arc \( AD \).
- Part b) \(\angle AEB = \angle CED\), because \( AEB \) and \( CED \) are vertical angles.
- Part c) [tex]\(\angle ADC = 92^\circ\)[/tex], because in a cyclic quadrilateral, the opposite angles are supplementary.
