How do you prove properties of angles for a quadrilateral inscribed in a circle?
1 answer:
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°
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