Question

How do you prove properties of angles for a quadrilateral inscribed in a circle?

Answer 1

Answer:

The sum of the measures of the opposite angles in the quadrilateral inscribed in a circle is 180°

Step-by-step explanation:

* Look to the attached figure to understand how you solve the problem

- The quadrilateral is inscribed in a circle if its vertices lie on the

 circumference of the circle

- The inscribed quadrilateral in a circle is called cyclic quadrilateral

- The sum of the measures of the opposite angles in the cyclic

  quadrilateral is 180°

* Lets prove that

- The measure of any inscribed angle in a circle is half the measure

  of the subtended arc to this angle

- From the figure

∵ ∠ DAB is an inscribed angle subtended by the arc DCB

∴ m∠ DAB = 1/2 measure of arc DCB

∵ ∠ DCB is an inscribed angle subtended by the arc DAB

∴ m∠ DCB = 1/2 measure of arc DAB

∵ measure of arc DAB + measure of arc DCB = measure of the circle

∵ The measure of the circle is 360°

∴ measure of arc DAB + measure of arc DCB = 360°

∵ m∠ DAB = 1/2 measure of arc DCB ⇒ (1)

∵ m∠ DCB = 1/2 measure of arc DAB ⇒ (2)

- Add (1) and (2)

∴ m∠ DAB + m∠ DCB = 1/2 m of arc DAB + 1/2 m of arc DCB

∵ measure of arc DAB + measure of arc DCB = 360°

∴ m∠ DAB + m∠ DCB = 1/2 (360°)

∴ m∠ DAB + m∠ DCB = 180°

- By the same way we can prove that m∠ABC + m∠ADC = 180°

∴ The sum of the measures of the opposite angles in the cyclic

  quadrilateral is 180°

* The sum of the measures of the opposite angles in the quadrilateral

  inscribed in a circle is 180°