Applying the angles of intersecting chords and secants theorems, we would have:
1. m∠YZW = 129°
3. m(QS) = 256°
5. m∠EFG = 41°
7. m(CF) = 147°
9. x = 16
11. x = 8
12. x = 12
What is the Angles of Intersecting Chords Theorem?
The angles of intersecting chords theorem states that the measure of the vertical angle formed when two chords intersect equals half the sum of the measure of the intercepted arcs.
What is the Angles of Intersecting Secants Theorem?
The angles of intersecting secants theorem states that if two lines intersect outside a circle, the measure of the angle formed equals half of the difference of the measure of the intercepted arcs.
1. m∠YZW = 1/2(144 + 114) [angles of intersecting chords theorem]
m∠YZW = 129°
3. m(QS) = 2(m∠PQS) [central anglle theorem]
Substitute
m(QS) = 2(128)
m(QS) = 256°
5. m∠EFG = 1/2(121 - 39) [angles of intersecting secants theorem]
m∠EFG = 41°
7. m∠DEF = 1/2[m(CF) - m(DF)] [angles of intersecting secants theorem]
Substitute
53 = 1/2[m(CF) - 41]
2(53) = m(CF) - 41
106 = m(CF) - 41
106 + 41 = m(CF)
m(CF) = 147°
9. 5x - 7 = 1/2(119 + 27) [angles of intersecting chords theorem]
2(5x - 7) = 146
10x - 14 = 146
10x = 146 + 14
10x = 160
x = 160/10
x = 16
11. 2x + 7 = 1/2(78 - 32) [angles of intersecting secants theorem]
2(2x + 7) = 46
4x + 14 = 46
4x = 46 - 14
4x = 32
x = 8
12. m∠BCF = 1/2[m(ED) + m(BF)] [angles of intersecting chords theorem]
Substitute
11x - 9 = 1/2(9x - 3 + 15x - 39)
2(11x - 9) = 24x - 42
22x - 18 = 24x - 42
22x - 24x = 18 - 42
-2x = -24
x = 12
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