Two parallel horizontal lines and are cut by two distinct transversals and

MATHEMATICS · Multi-part · CBSE · Mathematics · Lines and Angles · Lines and Angles

Problem

Two parallel horizontal lines

p

and

q

are cut by two distinct transversals

m

and

n

. The points of intersection are as follows:

m

meets

p

P

and

q

R

;

n

meets

p

Q

and

q

S

. At

P

the acute angle between

m

and

p

is denoted by

\angle \alpha

. At

R

the angle between

m

and

q

that is situated in the corresponding position to

\angle \alpha

is denoted by

\angle \beta

. At

Q

the interior angle between

n

and

p

on the right of

n

is denoted by

\angle \gamma

, and at

S

the exterior angle between

n

and

q

on the right of

n

is denoted by

\angle \delta

. (i) State the relationship between

\angle \alpha

and

\angle \beta

and give a reason. (ii) State the relationship between

\angle \gamma

and

\angle \delta

and give a reason. (iii) Decide, with justification, whether

\angle \alpha + \angle \delta

is always equal to

180^{\circ}

, always equal to

90^{\circ}

, or has no fixed value.