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Theorem 5 - The Degree Measure of an Arc of a Circle, is Twice the Angle Subtended by it at any point of the Alternate Segment of the Circle with Respect to the Arc

Theorem 5 : The degree measure of an arc of a circle is twice the angle subtended by it at any point of the alternate segment of the circle with respect to the arc.
Given : An arc PQ of a circle C(O, r) with a point R in arc other than P or Q.
To Prove :
POQ =2 PRQ
Construction : Join RO and draw the ray ROM.
Proof : There will be three cases as
(i) is a minor arc
(ii) is a semi-circle
(iii) is a major arc


In each of these three cases, the exterior angle of a triangle is equal to the sum of two interior opposite angles.
Therefore,
POM = PRO + RPO(i) 
 
MOQ = ORQ + RQO(ii)
 In
ΔOPR and ΔOQR
Now, OP = OR and OR = OQ  (radii of the same circle)

PRO = RPO and ORQ = RQO (angles opposite to the equal sides are equal)
Hence,
POM = 2PRO(iii)
And
MOQ = 2 ORQ (iv)

Case (i) adding equations (iii) and (iv) we get

POM + MOQ = 2PRO + 2ORQ
∴ ∠POQ = 2(PRO + ORQ)
= 2
PRQ
∴ ∠ POQ = 2PRQ

Case (ii)
POM + QOM = 180° = POQ
..
POQ = 2 ORP + 2ORQ
    = 2
PRQ

Case (iii)
POM + QOM
(180° -
POR) + (180° - QOR)
= [(360° - (
POR + QOR)]
= 360° -
POQ
=
POQ \∠POQ = 2 ORP + 2 ORQ = 2 PRQ
 

Example :

In the following figures find the angles x, y, z.

Solution :

Refer to figure (i)
AB passes through centre O

AB is a diameter and ACB = z + y = 90° (Angle in a semi-circle)
In
ΔABC,
A + B + C = 180°
x° + 55° + 90° = 180°
x = 35°
\ ∠BDC = BCA = 35°        (Angles in the same segment of the circles)
Now in
ΔBCD, B + DCB + BDC = 180°
30o + 55o + yo + 35o = 180o

y = 60o
Now,
ABD = ACD = 30° = z
x = 35°, y = 60° and z = 30°
Referring to figure (ii), we have 2x° + 3x° = 180° [opposite angles of a cyclic quad are supplementary).
...  5x = 180°

x = 36°
Again 3y° + 6y° = 180°

9y = 180°
y = 20°
Referring to figure (iii) we have

A + C = 180°
x + 2x = 180°
3x = 180°
x = 60°
Now
BOD = 2BAD [Angle at the centre is twice the angle in the alternate segment]
z = 2x = 2×60° = 120°

 


 




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