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In the figure, $AB \parallel CD$ and $\angle 1$ and $\angle 2$ are in the ratio $3 : 2$. Determine all angles from 1 to 8.
"
Given:
$AB \parallel CD$ and $\angle 1$ and $\angle 2$ are in the ratio $3 : 2$.
To do:
We have to find all angles from 1 to 8.
Solution:
We know that,
Vertically opposite angles are equal.
Corresponding angles are equal.
Therefore,
$AB \parallel CD$ and $l$ is transversal.
$\angle 1 : \angle 2 = 3 : 2$
Let $\angle 1 = 3x$
This implies,
$\angle 2 = 2x$
$\angle 1 + \angle 2 = 180^o$ (Linear pair)
$3x + 2x = 180^o$
$5x = 180^o$
$x = \frac{180^o}{5}$
$x = 36^o$
Therefore,
$\angle 1 = 3x = 3(36^o) = 108^o$
$\angle 2 = 2x = 2(36^o) = 72^o$
$\angle 1 = \angle 3$ (Vertically opposite angles)
$\angle 2 = \angle 4$ (Vertically opposite angles)
$\angle 3 = 108^o$
$\angle 4 = 72^o$
$\angle 1 = \angle 5$ (Corresponding angles)
$\angle 2 = \angle 6$ (Corresponding angles)
$\angle 5 = 108^o$
$\angle 6 = 72^o$
$\angle 4 = \angle 8$ (Corresponding angles)
$\angle 3 = \angle 7$ (Corresponding angles)
$\angle 8 = 72^o$
$\angle 7 = 108^o$
Hence, $\angle 1 = 108^o, \angle 2 = 72^o, \angle 3 = 108^o, \angle 4 = 72^o, \angle 5 = 108^o, \angle 6 = 72^o, \angle 7 = 108^o, \angle 8 = 72^o$.