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CAT 2025 Lesson : Circles - Angle Properties

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6. Angle Properties

Properties Figure
Property 1:All angles subtended by an arc on the same side of its segment are equal.
∠\angleAWB =∠= \angleAXB and ∠\angleAZB =∠= \angleAYB

Sum of angles subtended in the major and minor arcs =180o= 180^\mathrm{o}
∠\angleAWB +∠+ \angleAYB =180o= 180^\mathrm{o}
Property 2:Angles subtended by the end points of the diameter at any point on the circle is 90o90^\mathrm{o}

As XZ is the diameter, ∠\angleXAZ =∠= \angleXBZ =∠= \angleXCZ =90o= 90^\mathrm{o}
Property 3:The angle subtended by two points of the circle at the centre is twice the angle subtended by those two points at any other point in the major segment, i.e., Central Angle =2×= 2 \times Inscribed Angle

Where ∠\angleAXB =θ= \theta, the central angle ∠\angleAOB =2θ= 2 \theta

Example 6

In the adjoining figure, chord ED is parallel to the diameter AC of the circle. If ∠\angleCBE =65o= 65^\mathrm{o}, then what is the value of ∠\angleDEC?
[CAT 2004]

(1) 35o35^\mathrm{o}                     (2) 55o55^\mathrm{o}                     (3) 45o45^\mathrm{o}                     (4) 25o25^\mathrm{o}                    

Solution

Central angle =2×= 2 \times Inscribed Angle
∴\therefore ∠\angleCOE =2×= 2 \times ∠\angleCBE =130o= 130^\mathrm{o}

As OC == OE == Radius,
△\triangleEOC is isosceles and ∠\angleOEC =∠= \angleOCE =25o= 25^\mathrm{o}

As AC | | ED,
∠\angleOCE =∠= \angleOED =25o= 25^\mathrm{o} (Alternate Interior Angles)

Answer: (4) 25o25^\mathrm{o}


Example 7

In the figure below, A, B and C lie on the circle with centre O such that O lies on AD, and AC == CD. If ∠\angleACD =140o= 140^\mathrm{o}, then ∠\angleABC = ?

Solution

As △\triangleACD is isosceles, ∠\angleCDA =∠= \angleCAD = 20o20^\mathrm{o}

∠\angleACE =90o= 90^\mathrm{o} (Angle subtended by diameter AE)

In △\triangleACE, ∠\angleAEC =180−90−20=70o= 180 - 90 - 20 = 70^\mathrm{o}

Angles subtended in major & minor arc are supplementary.
∴\therefore ∠\angleABC +∠+ \angleAEC == 180o180^\mathrm{o}
⇒ ∠\angleABC =180o−70o=110o= 180^\mathrm{o} - 70^\mathrm{o} = 110^\mathrm{o}

Answer: 110o110^\mathrm{o}


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