Chapter 6: Lines and Angles

From the figure, we know that,

AB and CD intersect each other at point O.

Let the two pairs of vertically opposite angles be,

1^st pair – ∠AOC and ∠BOD

2^nd pair – ∠AOD and ∠BOC

To prove:

Vertically opposite angles are equal,

i.e., ∠AOC = ∠BOD, and ∠AOD = ∠BOC

From the figure,

The ray AO stands on the line CD.

We know that,

If a ray lies on a line, then the sum of the adjacent angles is equal to 180°.

⇒ ∠AOC + ∠AOD = 180° (By linear pair axiom) … (i)

Similarly, the ray DO lies on line AOB.

⇒ ∠AOD + ∠BOD = 180° (By linear pair axiom) … (ii)

From equations (i) and (ii),

We have,

∠AOC + ∠AOD = ∠AOD + ∠BOD

⇒ ∠AOC = ∠BOD – – – – (iii)

Similarly, the ray BO lies on the line COD.

⇒ ∠DOB + ∠COB = 180° (By linear pair axiom) – – – – (iv)

Also, the ray CO lies on line AOB.

⇒ ∠COB + ∠AOC = 180° (By linear pair axiom) – – – – (v)

From equations (iv) and (v),

We have,

∠DOB + ∠COB = ∠COB + ∠AOC

⇒ ∠DOB = ∠AOC – – – – (vi)

Thus, from equation (iii) and equation (vi),

We have,

∠AOC = ∠BOD, and ∠DOB = ∠AOC

Therefore, we get vertically opposite angles are equal.

Hence Proved.

Given: △ ABC, produce BC to D, and the bisectors of ∠ABC and ∠ACD meet at point T.

To prove:

∠BTC = ½ ∠BAC

Proof:

In △ABC,∠ACD is an exterior angle.

We know that,

The exterior angle of a triangle is equal to the sum of two opposite angles,

Then,

∠ACD = ∠ABC + ∠CAB

Dividing L.H.S and R.H.S by 2,

⇒ ½ ∠ACD = ½ ∠CAB + ½ ∠ABC

⇒ ∠TCD = ½ ∠CAB + ½ ∠ABC …(1)

We know that,

The exterior angle of a triangle is equal to the sum of two opposite angles,

Then in △ BTC,

∠TCD = ∠BTC +∠CBT

⇒ ∠TCD = ∠BTC + ½ ∠ABC …(2)

From equations (1) and (2),

We get,

½ ∠CAB + ½ ∠ABC = ∠BTC + ½ ∠ABC

⇒ ½ ∠CAB = ∠BTC or ½ ∠BAC = ∠BTC

Hence, proved.

Let,

AB ║ CD

EF be the transversal passing through the two parallel lines at P and Q, respectively.

PR and QS are the bisectors of ∠EPB and ∠PQD.

We know that the corresponding angles of parallel lines are equal,

So, ∠EPB = ∠PQD

½ ∠EPB = ½ ∠PQD

∠EPR = ∠PQS

But, we also know that they are corresponding angles of PR and QS

Since the corresponding angles are equal,

We have,

PR ║ QS

Hence Proved.