In any circle, the angle between a chord and a tangent through one of the end points of the chord is equal to the angle in the alternate segment.
Problem 1 :
Find x.
Solution:
∠BCD = 45
BC, AB, AC are chords of the circle. DE is the tangent.
The angle between chord BC and tangent DE is 45 degree,
∠BCD = ∠BAC (alternate segments)
∠BAC = 45
Problem 2 :
Find x.
Solution :
A and C are two endpoints of the chord AC.
Angle between the chord BC and the tangent DE = x
∠BCE = x
BC, AB, AC are chords of the circle. DE is the tangent.
The angle between chord BC and tangent DE is 45 degree,
∠BCE = ∠BAC (alternate segments)
∠BAC = 50
∠BCE = 50
Problem 3 :
Solution :
∠BAC = 64, ∠BCA = 50 (vertically opposite angles)
In triangle ABC,
∠BAC + ∠BCA + ∠CBA = 180
64 + 50 + ∠CBA = 180
114 + ∠CBA = 180
∠CBA = 180 - 114
∠CBA = 66
Problem 4 :
Solution :
∠ABC = x
AB is a chord, angle between one endpoint of the chord (A) and tangent is 145.
Alternate segment theorem ∠ABC = 145.
Problem 5 :
Solution :
∠CBA = 62 (Using alternate segment theorem)
∠BCD = 62
CB = AC = 62 (Sides of isosceles triangle)
In triangle ABC,
∠CAB + ∠ABC + ∠ACB = 180
62 + 62 + x = 180
124 + x = 180
x = 180 - 124
x = 56
Problem 6 :
Solution :
In triangle ABC,
∠CBA + ∠ACB + ∠CAB = 180
Since it is isosceles triangle and CB = CA
∠CBA + 120 + ∠CBA = 180
2 ∠CBA = 180-120
∠CBA = 120/2
∠CBA = 60
∠CBA = x (Alternate segment)
x = 60
Problem 7 :
Solution :
In triangle ABC, sum of interior angles of a triangle.
∠ABC = 180 - (40 + 25)
∠ABC = 180 - 65
∠ABC = 115
x = 115 (Alternate segment)
Problem 8 :
Solution :
BC = BA
In triangle ABC, sum of interior angles of a triangle.
∠ABC + ∠ACB + ∠BAC = 180
86 + ∠ACB + ∠ACB = 180
86 + 2∠ACB = 180
2∠ACB = 180 - 86
2∠ACB = 94
∠ACB = 94/2
∠ACB = 47
Problem 9 :
Solution :
CB = CA
In triangle ABC, sum of interior angles of a triangle.
∠ABC + ∠ACB + ∠BAC = 180
∠ABC + ∠ACB + 72 = 180
72 + 2∠ACB = 180
2∠ACB = 180 - 72
2∠ACB = 108
∠ACB = 108/2
∠ACB = 54
May 21, 24 08:51 PM
May 21, 24 08:51 AM
May 20, 24 10:45 PM