Geometry - Lines


Geometry is basically the study of different properties of point, line and plane.

Angle
Right angle: An angle of 90° is called a right angle.
Acute angle: An angle of less than 90° is called an acute angle.
Obtuse angle: An angle of greater than 90° is called an obtuse angle.
Reflex angle: An angle of greater than 180° is called reflex angle.
Complementary angles: If the sum of the two angles is 90°, then they are called complementary angles.
Supplementary angles:  If the sum of the two angles is 180°, then they are called supplementary angles.

Angle Relations: 

Vertically opposite angles: If two lines intersect each other, then the vertically opposite angles are equal.
$\angle $a = $\angle $c and $\angle $b = $\angle $d
a, c and b, d are vertically opposite angles.


Angles in parallel lines: If a transversal intersects two parallel lines:


a. each pair of consecutive interior angles are supplementary.
$\angle $4 + $\angle $5 = 180°
$\angle $3 + $\angle $6 = 180°

b. each pair of alternate interior angles are equal.
$\angle $5 =$\angle $3, $\angle $6 = $\angle $4
$\angle $5 = $\angle $3 =$\angle $1 =$\angle $7
$\angle $6 = $\angle $4 = $\angle $2 = $\angle $8

Collinear: Three or more than three points are said to be collinear, if there is a line which contains them
all.
Concurrent: Three or more than three lines are said to be concurrent, if there is a point which lies on them
all.

Proportionality Theorem:

In the above diagram, $\displaystyle\frac{{AB}}{{BC}} = \frac{{DE}}{{EF}}$ or $\displaystyle\frac{{AC}}{{BC}} = \frac{{DF}}{{EF}}$

Solved Examples:

1. In the adjoining figure, find $\angle $ C. If AB // CD

Sol:   $\angle $ E = 180 - 120 = 60
$\angle $ E + $\angle $ X + $\angle $ F = 180 $ \Rightarrow $ 60 + 50 + F = 180 $ \Rightarrow $ F = 70
Now $\angle $ G = 180 - F = 110.
As AB // CD, $\angle $G = $\angle $C
So $\angle $C = 110

2. Find the value of x in following figure.

Sol: As C // E, $\angle $B = $\angle $Y
Also $\angle $Y + $\angle $Z = 180 $ \Rightarrow $  $\angle $Z = 180 - 130 = 50
Now in $\Delta $ AGF, $\angle $A + $\angle $Z + $\angle $X = 180  $ \Rightarrow $ $\angle $X = 180 - 20 - 50 = 110

3. Find $\angle $OBD in the given figure. It is given that EC || AO.


Sol:   From the diagram, in $\Delta $OBD, 110 + 20 + $\angle $B = 180 $ \Rightarrow $ $\angle $B = 50
4. If the two lines n and m are parallel, then find the value of $\phi $ in the given figure.

Sol:  Draw a parellel to n. Now from the diagram $\phi $ = 50 + 30 = 80 (Alternate interior angles are equal)
Alternate method:  you can also solve this problem by calculating interior angles of the triangle.

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