**8.2 Angle between Tangent and Chord**

**1.**In the diagram above,

*ABC*is a tangent to the circle at point

*B.*

**2.**Chord

*PB*divides the circle into two segments, that is, the

**minor segment**and the

*PRB***major segment**

**PQB**.

**3.**With respect to ∠

*PBA*, ∠

*PQB*is known as the angle subtended by

**chord**in the

*BP***alternate segment**.

**With respect to ∠**

4.

4.

*QBC*, ∠

*BPQ*is known as the angle subtended by

**chord**in the

*BQ***alternate segment**.

**5.**The angle formed by the tangent and the chord which passes through the point of contact of the tangent is the same as the angle in the alternate segment which is subtended by the chord.

**6.**The relationships between the angles are:

Angle ∠

*ABP*= Angle ∠*BQP*Angle ∠

*CBQ*= Angle ∠*BPQ***Example 1:**

In the diagram, ABC is a tangent to the circle

*BDE*at

*B.*

The length of arc

*BD*is equal to the length of arc*DE.*Find the value of

*p.*

**Solution:**Angle

*BED =*82^{o}← (angle in alternate segment)Angle

*DBE =*82^{o}← (Arc*BD*= Arc*DE, BDE*is an isosceles triangle)Therefore

*p*= 180^{o}– 82^{o}– 82^{o }=**16**^{o}**Example 2:**

In the diagram,

*PQR*is a tangent to the circle*QSTU*at*Q.*

Find the value of

*y.*

**Solution:**Angle

*QUT**=*180

^{o}– 98

^{o}← (opposite angle in cyclic quadrilateral

*QSTU*)

= 82

^{o}Angle

*QTU =*75^{o}← (angle in alternate segment)Therefore

*y*= 180^{o}– (82^{o}+ 75^{o}) ← (Sum of interior angles in ∆*QTU*) =

**23**^{o}**Example 3:**

**(a)**

*x*

**(b)**

*y*

**Solution:****(a)**

∠

*UTS*+ ∠*UQS*= 180^{o}←(opposite angle in cyclic quadrilateral*QSTU*)105

^{o}+ ∠*UQS*= 180^{o} ∠

*UQS*= 75^{o}*x*+ 75

^{o}+ 20

^{o}= 180

^{o}←(the sum of angles on a straight line

*PQR*= 180

^{o})

*x*+ 95

^{o}= 180

^{o}

**x****= 85**

^{o}**(b)**

∠

*PQU*= ∠*QSU*← (angle in alternate segment)85

^{o}= 35^{o}+*y*

*y*= 50^{o}**Example 4:**

In the diagram,

*ABC*is a tangent to the circle

*BDE*with centre

*O*, at

*B*.

Find the value of

*x*.