16. Chapter 16. Angles in a Circle and Cyclic Quadrilateral NIOS Class 10 Mathematics Textbook Solutions
Chapter 16. Angles in a Circle and Cyclic Quadrilateral NIOS Class 10 Mathematics Textbook Solutions for Exam Preparation
Chapter 16. ANGLES IN A CIRCLE AND CYCLIC QUADRILATERAL
CHECK YOUR PROGRESS 16.1
1. In Fig. 16.11, ADB is an arc of a circle with centre O, if ∠ ACB = 35° , find ∠ AOB.
Solution: Given, ∠ACB = 35°
We have, 
2. In Fig. 16.12, AOB is a diameter of a circle with centre O. Is ∠APB = ∠AQB = 90° ?.
Solution: Since, angles in the same segment of a circle are equal .
So, 
[Angles in the same segment AB]
Yes , angle in a semi-circle is a right angle
3. In Fig. 16.13, PQR is an arc of a circle with centre O. If ∠PTR = 35° , find ∠ PSR.
Solution: Given, ∠PTR = 35°
Since, ∠PTR and ∠PSR are subtended by the same arc PQR at the remaining part of the circle .
So, 
Therefore, ∠PSR = 35° .
4. In Fig. 16.14, O is the centre of a circle and ∠ AOB = 60° . Find ∠ ADB.
Solution: Given, ∠ AOB = 60° .
We have, 
CHECK YOUR PROGRESS 16.2
1. In Fig. 16.26, AB and CD are two equal chords of a circle with centre O. If ∠ AOB = 55°, find ∠ COD. Give reasons.
Solution: Given, 
and AB = CD
(Radius of the circle)
[SSS]
So, 
[CPCT]
2. In Fig. 16.27, PQRS is a cyclic quadrilateral, and the side PS is extended to the point A. If ∠ PQR = 80° , find ∠ ASR.
Solution: Given, 
We know that, the sum of the opposite angles of a cyclic quadrilateral is 180° .
We have, 
ASP is a straight line .
3. In Fig. 16.28, ABCD is a cyclic quadrilateral whose diagonals intersect at O. If ∠ ACB = 50° and ∠ ABC = 110°, find ∠BDC.
Solution: Given, 
and 
In 
, we have
We know that, the angles subtended by the same arc are equal .
[same arc BC ]
4. In Fig. 16.29, ABCD is a quadrilateral. If ∠ A = ∠ BCE, is the quadrilateral a cyclic quadrilateral? Give reasons.
Solution: Yes, ABCD is the quadrilateral a cyclic quadrilateral .
Given, 
[Linear pair of angles]
We know that, the pair of opposite angles of a quadrilateral is supplementary, then the quadrilateral is cyclic.
So, ABCD is a cyclic quadrilateral .
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