Question

The vertex of an angle measuring 32° is in the exterior of a circle and its sides are secants of the circle. If the sum of the measures of the intercepted arcs is 180°, find the measure of each intercepted arc.

Answer 1

The measures of the intercepts arcs are
`58\°` and
`122\°`

Explanation:

we know that

The measurement of the outer angle is the semi-difference of the arcs which comprises

Let

x,y -----> the intercepts arcs

`x+y=180\°` ----> equation A

`32\°=(1)/(2)(x-y)`

`64\°=(x-y)`

`x=64\°+y` ------> equation B

substitute equation B in equation A

`(64\°+y)+y=180\°`

`64\°+2y=180\°`

`2y=180\°-64\°`

`y=116\°/2=58\°`

Find the value of x

`x=64\°+58\°=122\°`

therefore

The measures of the intercepts arcs are
`58\°` and
`122\°`