Answer:
The measure of the smallest angle is 77°
Explanation:
* Lats study some facts about the circle
* In any circle; If the vertices of a quadrilateral lie on its
circumference then the quadrilateral is called a cyclic quadrilateral
- In any cyclic quadrilateral each two opposite angles are supplementary
that means their sum = 180°
* In any circle; if the vertex of an angle lies on the circumference
is called an inscribed angle
- The inscribed angle subtended by the opposite arc
- The measure of the inscribed angle = 1/2 the measure of
the subtended arc
* Now we can solve our question
∵ R , A , T , E all on the the circumference of the circle
∴ RATE is a cyclic quadrilateral
∵ Angle RAT is an inscribed angle subtended by arc RET
∵ The measure of arc RE = 121°
∵ The measure of arc ET = 51°
∴ The measure of arc RET = 121 + 51 = 172°
∵ m∠RAT = (1/2) measure of arc RET
∴ m∠RAT = (1/2) × 172 = 86°
∵ Angle ATE is an inscribed angle subtended by arc ARE
∵ The measure of arc AR = 85°
∵ The measure of arc RE = 121°
∴ The measure of arc ARE = 85 + 121 = 206°
∵ m∠ATE = (1/2) measure of arc ARE
∴ m∠ATE = (1/2) × 206 = 103°
* Now lets find the remainder angles by using cyclic quadrilateral
∵ RATE is a cyclic quadrilateral
∴ m∠RAT + m∠RET = 180°
∵ m∠RAT = 86°
∴ m∠RET = 180 - 86 = 94°
* Similar;
∵ RATE is a cyclic quadrilateral
∴ m∠ATE + m∠ARE = 180°
∵ m∠ATE = 103°
∴ m∠ARE = 180 - 103 = 77°
* The measure of the smallest angle is 77°