Question

Find the angle measure indicated. Assume that lines which appear to be tangent are tangent.

Answer 1

13)  135°   14)  143°

Solution:

13) The lengths of tangents drawn from an external point are equal.

So, PA = PB

Tangent at any point of a circle is perpendicular to the radius of the circle through the point of contact.

OB ⊥ PB and OA ⊥ PA

So, ∠PBO = 90°, ∠PAO = 90°

Sum of the angles in a quadrilateral is 360°.

In quadrilateral PAOB,

⇒ ∠APB + ∠PAO + ∠PBO + ∠AOB = 360°

⇒ 45° + 90° + 90° + ∠AOB = 360°

⇒ ∠AOB = 360° – 225°

⇒ ∠AOB = 135°

Hence the missing angle is 135°

14)  Angle in a semi-circle is 90° .

∠ACB = 90°

Sum of the angles in triangle is 180° .

In ∆ABC, ∠ACB + ∠ABC + ∠CAB = 180°

⇒ 90° + 53° + ∠CAB = 180°

⇒ ∠CAB = 180° – 143°

⇒ ∠CAB = 37°

AD and DE are radius.

So, ∠DEA = 37°

Sum of the adjacent angles in a straight line is 180°.

⇒ ∠DEA + ∠DEB = 180°

⇒ 37° + ∠DEB = 180°

⇒ ∠DEB = 143°

Hence the missing angle is 143°