5) The sum of angle in the triangle DEF is 180 degrees
mFE = <D
Recall that <D+<E+<F = 180⁰
<D+63+90 = 180
<D = 180-153
<D = 27 degrees
Hence the measure of arc FE is 27degrees
6) For this circle geometry, we will use the theorem
The sum of Opposite side of a cyclic quadrilateral is 180 degrees.
A + C = 180
m<A + 101 = 180
m<A = 180-101
m<A = 79degrees
Similarly
B + D = 180
m<B + 68 = 180
m<B = 180-68
m<B = 112degrees
7) The sum of angle in a circle is 360, hence;
arcGJ+68+31+115 = 36p
arcGJ = 360 - 214
arcGJ = 146⁰
Since the angle at the centre is twice angle at the circumference, then;
<GHJ = 1/2 arcGJ
<GHJ = 1/2(146)
<GHJ = 73⁰
<GHJ = <GIJ = 73⁰ (angle in the same segment of the circle are equal)
8) Recall that the sum of Opposite side of a cyclic quadrilateral is 180 degrees.
P + R = 180
57 + <R = 180
m<R = 180-57
m<R = 123degrees
Similarly, m<Q+m<S = 180⁰
Since the triangle in a semi circle is a right angled triangle, hence m<Q = 90 degrees (triangle PQR is a right angled triangle)
m<S = 180 - 90
m<S = 90 degrees
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