Question

Find each of the following angles and arcs formed in circle R

Answer 1

ANSWERS

a. m∠RTV = 90°

b. m∠TRU = 129°

c. m∠TVU = 51°

d. m∠TSU = 64.5°

e. mST = 101°

f. mSTU = 230°

Step-by-step explanation

a. Angle RTV is formed by the segments RT - which is the radius of the circle, and TV, which is tangent to the circle at point T. The radius always forms a right angle with the tangent lines.

Hence, the measure of angle RTV is 90°.

b. Angle TRU is a central angle because its vertex is at the center of the circle, and it intersects arc TU, whose measure is 129°. The measure of a central angle and the measure of the intercepted arc are equal. Hence, the measure of angle TRU is 129°.

c. RTVU is a quadrilateral. We know that three of the interior angles' measures are 90°, 90°, and 129°. The measures of the interior angles of any quadrilateral add up to 360°, so the measure of angle TVU is,

`m\angle TVU=360\degree-129\degree-90\degree-90\degree=51\degree`

Hence, the measure of angle TVU is 51°.

d. Angle TSU is an inscribed angle that intercepts the same arc as angle TRU, arc TU. Its measure of half the measure of the intercepted arc,

`m\angle TSU=(1)/(2)mTU=(1)/(2)\cdot129\degree=64.5\degree`

Hence, the measure of angle TSU is 64.5°.

e. The sum of all consecutive arcs in a circle is 360°,

`mST+mTU+mUS=360\degree`

We know that mTU = 129° and mUS = 130°. Solving for mST,

`mST=360\degree-mTU-mUS=360\degree-129\degree-130\degree=101\degree`

Hence, the measure of arc ST is 101°.

f. Similarly, arc STU is the composition of the consecutive arcs ST and TU, so its measure is the sum of those measures,

`mSTU=mST+mTU=101\degree+129\degree=230\degree`

Hence, the measure of arc STU is 230°.