ANGLES FORMED BY SECANTS AND TANGENTS IN CIRCLE

For each circle, tangent segments are shown. Use the measurements given find the value of x.

Problem 1 :

Solution:

x = 1/2 (AE - AF)

= 1/2 (141° - 61°)

= 1/2 (80°)

x = 40°

Problem 2 :

Solution:

As per given by property of circle,

∠x = (180 - 70)/2

∠x = 110/2

∠x = 55°

Problem 3 :

Solution:

∠GHI + ∠GBI = 180°

58° + x = 180°

x = 180° - 58°

x = 122°

Problem 4 :

Solution:

Vertical angles are equal.

So, x = 38°

Problem 5 :

Solution:

AB = 160°

So, ∠ADB = 80°

∠ADB = ∠ACD + ∠DAC

So, ∠DAC = 45°

80° = 35° + ∠DAC

 ∠DAC = 80° - 35°

 ∠DAC = 45°

 ∠ABD = ∠DAC

So, x = 45°

Problem 6 :

Solution:

∠AEB = 1/2 ∠AB

= 1/2 × 50

= 25°

∠ABE = 1/2 ∠AE

= 1/2 × (4x + 5)

= (2x + 5/2)°

OA ⊥ DA (DA is a tangent line of circle O)

Hence ∠OAB = 1/2 (180° - 50°)

= 1/2 (130)°

= 65°

∠BAD = 90° - 65° = 25°

Hence, x + 25° = (5/2 + 2x)°

2x - x = 25 - (5/2)

x = 45/2

x = 22.5°