1. Given: circle k(O), m
FE =56°,
FD=ED
Find: m∠EFO, m∠EFD

2. Given: circle k(O),
RU =50°,
UT =30°
Find: m∠RUS and m∠UST

1.

2.

Step-by-step explanation:

Q1. Given circle k(O).

The measure of the arc FE is 56°, this means the measure of the central angle FOE is 56° too.

Consider triangle FOE. This triangle is isosceles triangle with the base FE because FO = EO as radii of the circle.

Angles adjacent to the base of the isosceles triangle are congruent, so

The sum of the measures of all interior angles in the triangle FEO is always 180°, then

Angle FDE is inscribed angle subtended on the arc FE, hence its measure is the half of the central angle FOE:

Since FD = ED, thriangle FDE is isosceles triangle with congruent angles adjacent to the base FE. Then

Q2. If the measure of the arc RU is 50°, then the measure of the central angle ROU is 50° too.

If the measure of the arc UT is 30°, then the measure of the central angle TOU is 30° too.

Triangle ROU is isosceles triangle because RO = UO as radii of the circle. Angles adjacent to the base of the isosceles triangle are congruent, so

The sum of the measures of all interior angles in the triangle ROU is always 180°, then

Angle UST is inscribed angle subtended on the arc UT and has the measure that is half the measure of the central angle TOU:

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