Answered

Join the growing community of curious minds on IDNLearn.com and get the answers you need. Explore thousands of verified answers from experts and find the solutions you need, no matter the topic.

[tex]$A, B, C,$[/tex] and [tex]$D$[/tex] are four points on the circumference of a circle. [tex]$AEC$[/tex] and [tex]$BED$[/tex] are straight lines.

a) State, with a reason, which other angle is equal to angle [tex]$ABD$[/tex].
[tex]$\square$[/tex]

b) State, with a reason, which other angle is equal to angle [tex]$AEB$[/tex].
[tex]$\square$[/tex]

c) If angle [tex]$ABC = 88^\circ$[/tex], state, with a reason, the size of angle [tex]$ADC$[/tex].
[tex]$92^\circ$[/tex] [tex]$\square$[/tex]

Sagot :

GinnyAnsweridn
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.
We appreciate your contributions to this forum. Don't forget to check back for the latest answers. Keep asking, answering, and sharing useful information. IDNLearn.com is your go-to source for accurate answers. Thanks for stopping by, and come back for more helpful information.