Home
Class 9
MATHS
Prove that the angle in a segment shorte...

Prove that the angle in a segment shorter than a semicircle is greater than a right angle.

Text Solution

Verified by Experts

Given : `/_BAC` in a segment smaller than a semicircle in circle C(O,r).
To Prove : `/_ BAC gt 90^(@)`.
Construction : Join `BO` and `CO.`
Proof : We know that the angle subtended by an arc of a circle at the centre is doubled the angle subtended by it at any point on the remaining part of the circle.
Here , arc BDC subtends reflex `/_ BOC` at the centre and `/_ BAC` at a point A on the remaining part of the circle.
`:. ` reflex `/_ BOC = 2 /_ BAC`.
But, arc `BDC` being a major arc, we have reflex `/_ BOC gt 180^(@)`.
`:. 2/_ BAC gt 180^(@) implies /_ BAC gt 90^(@)`
Promotional Banner

Similar Questions

Explore conceptually related problems

Prove that the angle in a segment greater than a semi-circle is less than a right angle.

The angle in a semicircle measures

If two isosceles triangles have a common base, prove that the line segment joining their vertices bisects the common base at right angles.

Prove that the area of the semicircle drawn on the hypotenuse of a right angled triangle is equal to the sum of the areas of the semicircles drawn on the other two sides of the triangle

Prove that the sum of the angles in the four segments exterior to a cyclic quadrilateral is equal to 6 right angles.

Prove that in a triangle,other then an an eguilateral triangle,angle opposite the longest side is greater than (2)/(3) of a right angle

Fill in the blanks with acute, obtuse, right or straight : (a) An angle whose measure is less than that of a right angle is______. (b) An angle whose measure is greater than that of a right angle is ______. (c) An angle whose measure is the sum of the measures of two right angles is _____. (d) When the sum of the measures of two angles is that of a right angle, then each one of them is ______. (e) When the sum of the measures of two angles is that of a straight angle and if one of them is acute then the other should be _______.

The side BC of a ABC is produced on both sides.Show that the sum of the exterior angles so formed is greater than /_A by two right angles.