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In the given figure, AB is a diameter of the circle such that ∠A = 35° and ∠Q = 25°, find ∠PBR. - Mathematics

Answer in Brief

In the given figure, AB is a diameter of the circle such that ∠A = 35° and ∠Q = 25°, find ∠PBR.

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Solution

Let us first consider the triangle ΔABQ.

It is known that in a triangle the sum of all the interior angles add up to 180°.

So here in our triangle ΔABQ we have,

`angleBAQ + angleAQB + angleABQ ` = 180°

                                  `angleABQ = 180° - angleBAQ - angle AQB`

                                                = 180° - 35° - 25° 

                                  `angle ABQ`  = 120° 

By a property of the circle we know that an angle formed in a semi-circle will be 90°..

In the given circle since ‘AB’ is the diameter of the circle the angle `angleAPB`  which is formed in a semi-circle will have to be 90°.

So, we have  `angleAPB` = 90°

Now considering the triangle  ΔAPB  we have,

`angleAPB + angleBAP + angleABP ` = 180°

                                 `angleABP  = 180° - angleAPB - angleBAP`

                                               = 180° - 90° - 35 °

                                  `angleABP`  = 55°

From the given figure it can be seen that,

`angleABP + anglePBQ = angle ABQ`

              `anglePBQ = angleABQ - angleABP`

                            = 120°- 55°

                `anglePBQ` = 65°

Now, we can also say that,

`anglePBQ + anglePBR` = 180°

                   `anglePBR = 180° - anglePBQ`

                                 =180° - 65 °

                   `anglePBR `  = 115°

Hence the measure of the angle  `anglePBR`  is 115°.

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APPEARS IN

RD Sharma Mathematics for Class 9
Chapter 15 Circles
Q 6 | Page 108
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