The game of billiards is played with balls placed on a rectangula… - Vidyadip

Question

The game of billiards is played with balls placed on a rectangular table. One ball is struck with the
end of a stick, called a cue. The ball bounces into other balls and relects off the sides of the table. In a real game, the ball may spin, but for mathematical purposes, it is considered that the ball travels in a straight line with the same relection and incidence angles.

On a billiard table $ABCD,$ the ball placed at $O$ is struck with the cue.
$1.$ What is the value of $\angle a + \angle d?$
$2.$ Why is the line $OM$ parallel to $PN?$

✓

Answer

$1. 90$
$90^\circ $
$2.$ Mathematically valid proof
● Let angles on line $AMB$ be $a, x$ and $b$ and angles on line $BNC$ be $c, y$ and $d.$
$x = 180 – (a + b) …..1$
$y = 180 – (c + d) …...2$
Adding $1$ and $2,$
$x + y = 360 – (a + b + c + d)$
$= 360 – (2a + 2c)$
$= 360 – 2 \times 90 = 180$
Thus, lines $OM$ and $NP$ are parallel.

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