Answer:
The measures of the angles of the triangle are 51° , 64.5° , 64.5°
OR
The measures of the angles of the triangle are 129° , 25.5° , 25.5°
Step-by-step explanation:
* Lets explain the meaning of the inscribed triangle in a circle
- If a triangle inscribed in a circle, then the vertices of the triangle lie
on the circumference of the circle and each vertex is an inscribed
angle in the circle subtended by the opposite arc
- Fact in the circle the measure of the inscribed angle is 1/2 the
measure of its subtended arc
* Now lets solve the problem
- Δ ABC is an isosceles with the base BC
∴ AB = AC
∴ m∠B = m∠C
- Δ ABC is inscribed in a circle
∴ ∠A is inscribed angle subtended by arc BC (minor or major)
# The measure of the minor arc is less than 180° and the measure of
the major arc is greater then 180° and the sum of the two arcs
equals the measure of the circle which is 360°
∴ ∠B subtended by arc AC
∴ ∠C subtended by arc AB
∵ The measure of the arc BC = 102°
- There is two cases in this question
(1) If the angle A subtended by the minor arc BC
(2) If the angle A subtended by the major arc BC
- Lets solve case (1)
∵ ∠A is an inscribed angle subtended by the minor arc BC
∴ m∠A = 1/2 the measure of the arc BC
∵ The measure of the arc BC is 102°
∴ m∠A = 1/2 × 102 = 51°
∵ The sum of the measures of the interior angles of a triangle is 180°
∴ m∠A + m∠B + m∠C = 180°
∴ 51 + m∠B + m∠C = 180° ⇒ subtract 51 from both sides
∴ m∠B + m∠C = 129°
∵ m∠B = m∠C ⇒ isosceles Δ
∴ m∠B = m∠C = 129/2 = 64.5°
* The measures of the angles of the triangle are 51° , 64.5° , 64.5°
- Lets solve case (2)
∵ ∠A is an inscribed angle subtended by the major arc BC
∴ m∠A = 1/2 the measure of the arc BC
∵ The measure of the minor arc BC is 102°
∵ The measure of the circle is 360°
∴ The measure of the major arc = 360 - 102 = 258°
∴ m∠A = 1/2 × 258 = 129°
∵ The sum of the measures of the interior angles of a triangle is 180°
∴ m∠A + m∠B + m∠C = 180°
∴ 129 + m∠B + m∠C = 180° ⇒ subtract 129 from both sides
∴ m∠B + m∠C = 51°
∵ m∠B = m∠C ⇒ isosceles Δ
∴ m∠B = m∠C = 51/2 = 25.5°
* The measures of the angles of the triangle are 129° , 25.5° , 25.5°
