Answer:
The measures of the angles of ΔABC are
m∠A = 51° , m∠B = 64.5 , m∠C = 64.5°
Step-by-step explanation:
* Lets revise some facts in the circle
- The measure of the circle is 360°
- Equal chords intercept equal arcs
- The measure of an inscribed angle equals half the measure
of the intercepted arc
* Lets solve the problem
- ΔABC is isosceles withe base BC
∴ AB = AC
- The vertex angle is A
∵ ∠A is an inscribed angle subtended by arc BC
∴ m∠A = 1/2 the measure of arc BC
∵ The measure of arc BC = 102° ⇒ given
∴ m∠A = 1/2(102) = 51°
- In ΔABC
∵ AB = AC ⇒ proved
∴ m∠B = m∠C ⇒ base angles of isosceles Δ
- The sum of the interior angles in any Δ = 180°
∵ m∠A = 51 ⇒ proved
∴ m∠B = m∠C = (180 - 51) ÷ 2 = 129 ÷ 2 = 64.5°
* The measures of the angles of ΔABC are
m∠A = 51° , m∠B = 64.5° , m∠C = 64.5°
