Question

An isosceles triangle ABC with the base BC is inscribed into a circle. Find the measure of angles of it, if measure of arc BC=102°.

Answer 1

The central angle BOC is twice the measure of arc BC, so it is 204°. Since the circle is 360°, the sum of the other two angles is 156°. Being an isosceles triangle, angles ABC and ACB are equal, thus each is 78°.

Step-by-step explanation:

To find the measure of the angles of an isosceles triangle ABC inscribed in a circle, with the measure of arc BC being 102°, we utilize the properties of a circle and the fact that an isosceles triangle has two equal angles. In a circle, the angle subtended by an arc at the center is twice the angle subtended on the circumference. Thus, if the measure of arc BC is 102°, the central angle BOC would be 204° (since it's twice the angle at the circumference).

This central angle separates the circle into two segments, one of which is the base of the isosceles triangle. The other two angles of the triangle, angles A and ABC or ACB (since it's an isosceles triangle, these angles are equal), will each occupy the remaining portion of the circle that amounts to 360° - 204° = 156°. Since the angles at B and C are equal in an isosceles triangle, we divide the remaining 156° by 2 to find each angle. Therefore, angle A = 204°, and each of the other two angles is 78°.

Answer 2

25.5°, 25.5°, 129° and 51°, 64.5°, 64.5°

Step-by-step explanation:

if you read the problem carefully you will notice that there are actually 2 ways of solving this problem. the first is by looking at the arc as if it is across from angle A, while the second is so the arc is across from side BC and bisected by point A (because the triangle is isosceles).

first way:

102/2=51°=m∠A

180-51=129°=m∠B+m∠C

129/2=m∠B=m∠C=64.5°

answers: 64.5°, 64.5°, 51°

second way:

102/2=51

51/2=m∠B=m∠C=25.5°

180-51=129°=m∠A

answers: 25.5°, 25.5°, 129°

Hope this helps!:)