Can four angles of a cyclic quadrilateral be in A. P.? why?
Answers
Topic
Arithmetic Progression
To Check
Whether four angles of a cyclic quadrilateral can be in Arithmetic Progression or not?
Solving
Let's assume that angles are in Arithmetic Progression.
Then angles will be,
a - 3d, a - d, a + d, a + 3d
where a and d are constant.
This is an AP with
First term = a - 3d and
Common Difference = 2d
Sum of all angles in a quadrilateral = 360°.
(a - 3d) + (a - d) + (a + d) + (a + 3d) = 360°
4a = 360°
a = 90°
Sum of opposite angles in cyclic quadrilateral is 180°.
Here, opposite angles are ( a - 3d ) and ( a + d ); and ( a - d ) and ( a + 3d ).
( a - 3d ) + ( a + d ) = 180°
2a - 2d = 180°
Put a = 90°
180° - 2d = 180°
d = 0°
Similarly,
( a - d ) + ( a + 3d ) = 180°.
2a - 2d = 180°
Put a = 90°.
180° - 2d = 180°
d = 0°.
So, angles obtained are
a - 3d = 90° - 0° = 90°
a - d = 90° - 0° = 90°
a + d = 90° + 0° = 90°
a + 3d = 90° + 0° = 90°
So, angles are
90°, 90°, 90° and 90°.
The angles are in a constant AP.
Answer
Yes, four angles of a Cyclic Quadrilateral can be in AP because a parallelogram that is Rectangle can be formed inside a Circle.
Learn More :-
Arithmetic Progression ( AP )
A sequence with a common difference between consecutive terms is known as Arithmetic Progression.
Constant Arithmetic Progression
It is an special case of AP when common difference between consecutive terms is 0 (zero).