Rate is a quadrilateral inscribed in the above circle with arc measures shown above. what is the measure of the smallest angle in rate?

The measure of the smallest angle is 77°

Step-by-step 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°