Inscribed Angle Theorem

In the adjoining figure chord EF || chord GH.
Prove that chord EG ≅ chord FH.
Fill in the boxes and write the complete proof.

In the given figure, O is centre of circle. ∠QPR = 70° and m(arc PYR) = 160°, then find the value of the following ∠QOR.

In the figure, ΔABC is an equilateral triangle. The angle bisector of ∠B will intersect the circumcircle ΔABC at point P. Then prove that: CQ = CA.

The angle inscribed in the semicircle is a right angle. Prove the result by completing the following activity.

Given: ∠ABC is inscribed angle in a semicircle with center M

To prove: ∠ABC is a right angle.

Proof: Segment AC is a diameter of the circle.

∴ m(arc AXC) = `square`

Arc AXC is intercepted by the inscribed angle ∠ABC

∠ABC = `square`      ......[Inscribed angle theorem]

= `1/2 xx square`

∴ m∠ABC = `square`

∴ ∠ABC is a right angle.

In the figure, a circle with center C has m(arc AXB) = 100° then find central ∠ACB and measure m(arc AYB).

In the figure, seg AB is a diameter of a circle with centre O. The bisector of ∠ACB intersects the circle at point D. Prove that, seg AD ≅ seg BD. Complete the following proof by filling in the blanks.

Proof:

Draw seg OD.

∠ACB = `square`     ......[Angle inscribed in semicircle]

∠DCB = `square`    ......[CD is the bisector of ∠C]

m(arc DB) = `square`   ......[Inscribed angle theorem]

∠DOB = `square`   ......[Definition of measure of an arc](i)

seg OA ≅ seg OB     ...... `square` (ii)

∴ Line OD is `square` of seg AB    ......[From (i) and (ii)]

∴ seg AD ≅ seg BD

In the figure, if O is the center of the circle and two chords of the circle EF and GH are parallel to each other. Show that ∠EOG ≅ ∠FOH

In the figure, chord LM ≅ chord LN, ∠L = 35°. 

Find
(i) m(arc MN)

(ii) m(arc LN)