Question

cyclic quadrilateral is that the exterior angle is equal to the sum of remote interior angle

Answer 1

A quadrilateral is called Cyclic quadrilateral if its all vertices lie on the circle.

It has some special properties which other quadrilaterals, in general, need not have. Here we have proved some theorems on cyclic quadrilateral.

1) The opposite angles of a Cyclic - quadrilateral are supplementary.

Given : A cyclic quadrilateral ABCD.

Prove that : ∠A + ∠C = 180° ang ∠B + ∠D = 180°

Construction : Join AC and BD.

Statements

Reasons

1) ∠ACB = ∠ADB 1) Angles in the same segment.

2) ∠BAC = ∠BDC 2) Angles in the same segment

3)∠ACB + ∠BAC = ∠ADB + ∠BDC 3) Addition property

4) ∠ACB + ∠BAC = ∠ADC 4) Add ∠ABC on both sides.

5) ∠ABC + ∠ACB + ∠BAC = ∠ABC + ∠ADC 5) From Above.

6) 180o = ∠ABC + ∠ADC 6) Sum of the angle of a triangle is 180o

7) ∠B + ∠D = 180o 7) Opposite angles of cyclic quadrilateral.

8) ∠A + ∠B + ∠C + ∠D 8) Measure of a quadrilateral.

9) ∠A + ∠C = 360o - (∠B + ∠D) 9) From Above.

10) ∠A + ∠C = 360o - 180o = 180o 10) Angle sum property

11) ∠A + ∠C = 180o and ∠B + ∠D = 180o 11) From above .So opposite angles are supplementary.

Answer 2

Answer:

Step-by-step explanation:

Definition of cyclic quadrilateral:

A quadrilateral whose four vertices all fall on a circle is said to be cyclic. Additionally known as an inscribed quadrilateral. The circumcircle or circumscribed circle is the circle that has every vertex of any polygon along its perimeter.

Properties of cyclic quadrilateral:

The opposing angles in a cyclic quadrilateral are supplementary, or they add up to 180 degrees.  A cyclic quadrilateral's exterior angle is equal to its interior opposite angle. For instance, the internal angle ABC is equivalent to the external angle ADF.

Here, we demonstrate a few theorems on cyclic quadrilaterals.

A Cyclic-opposing quadrilateral's angles are supplementary.

Given: A cyclic quadrilateral ABCD is provided.

Show that A + C Equals 180 degrees and B + D = 180 degrees.

Join AC and BD to complete the project.

Statements

Reasons

1) Angles inside the same segment:  $ACB = ADB 1.$

2)  Parallel angles inside a segment: $BDC + BAC.$

3) Third-party property :$ACB+ BAC = ADB + BDC$.

4)  Fill both sides with ABC:  $ACB + BAC = ADC$.

5) From Above:  $ABC + ACBC + BAC = ABC + ADC.$

6)The sum of a triangle's angles is 180 degrees: $180 degrees = ABC + ADC$.

7) The opposite angles of a cyclic quadrilateral are: $B + D = 180 degrees.$

8) Measure of a quadrilateral: $A + B + C + D.$

9) From Above: $(A + C = 360o - (B + D)$.

10) Angle sum property: $A + C = 360° - 180° = 180°$.

11) From above. Therefore, opposing angles are supplementary. A + C = 180° and B + D = 180°.

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