PROBLEMS ON INTERSECTING CHORDS INSIDE A CIRCLE

If two chords intersect inside a circle, then the measure of each angle formed is one half the sum of the measures of the arcs intercepted by the angle and its vertical angle.

m∠1 = 1/2(mCD + mAB),

m∠1 = 1/2(mAD + mBC)

Find the value of x.

Problem 1 :

Solution :

x˚ = 1/2(mAD + mBC)

x˚ = 1/2(162˚ + 134˚)

x = 1/2(296)

x = 148

Problem 2 :

Solution :

x˚ = 1/2(mAB + mCD)

x˚ = 1/2(25˚ + 75˚)

x = 1/2(100)

x = 50

Problem 3 :

Solution :

x˚ = 1/2(mAB + mCD)

x˚ = 1/2(130˚ + 96˚)

x = 1/2(226)

x = 113

Problem 4 :

Solution :

55˚ = 1/2(mAB + mCD)

55˚ = 1/2(x + 89˚)

110˚ = x + 89

x = 110 - 89

x = 21

Problem 5:

Solution :

59˚ = 1/2(mAB + mCD)

59˚ = 1/2(x + 70˚)

118˚ = x + 70

x = 118 - 70

x = 48

Problem 6 :

Solution :

129˚ = 1/2(mAD + mBC)

129˚ = 1/2(x + 72˚)

258˚ = x + 72

x = 258 - 72

x = 186

Find the value of x.

Problem 7 :

Solution :

x˚ = 1/2(mPS + mRQ)

x˚ = 1/2(106˚ + 174˚)

x = 1/2(280)

x = 140

Problem 8 :

Solution :

80˚ = 1/2(mAB + mCD)

80˚ = 1/2(x + 60˚)

160˚ = x + 60

x = 160 - 60

x = 100

Problem 9 :

Solution :

x˚ = 1/2(mAB + mCD)

x˚ = 1/2(190˚ + 70˚)

x = 1/2(260)

x = 130

Problem 10 :

Solution :

x˚ = 1/2(mAB + mCD)

x˚ = 1/2(66˚ + 70˚)

x = 1/2(136)

x = 68

Problem 11 :

Solution :

72˚ = 1/2(mAB + mCD)

72˚ = 1/2(x + 99˚)

144˚ = x + 99

x = 144 - 99

x = 45

Find the measure of ∠1.

Problem 12 :

Solution :

∠1 = 1/2(mAD + mBC)

∠1 = 1/2(55˚ + 65˚)

∠1 = 1/2(120)

∠1 = 60

Problem 13 :

Solution :

∠1 = 1/2(mAB + mBC)

∠1 = 1/2(92˚ + 88˚)

∠1 = 1/2(180)

∠1 = 90

Problem 14 :

Solution :

∠1 = 1/2(mAD + mBC)

∠1 = 1/2(168˚ + 110˚)

∠1 = 1/2(278)

∠1 = 139

Find the value of x.

Problem 15 :

Solution :

x˚ = 1/2(mAD + mBC)

x˚ = 1/2(162˚ + 134˚)

x = 1/2(296)

x = 148

Problem 16 :

Solution :

x˚ = 1/2(mAB + mCD)

x˚ = 1/2(25˚ + 75˚)

x = 1/2(100)

x = 50