# Plane Geometry

#### Similar Triangles

Triangle $ABC$ is said to be similar to triangle $PQR$, written $\triangle ABC \sim \triangle PQR$, if and only if one of the following equivalent conditions is satisfied:

1. $AB:BC:CA=PQ:QR:RP$
2. $\angle ABC =\angle PQR,\; \angle ACB=\angle PRQ,\; \angle BAC =\angle QPR$

Two similar triangles $ABC$ and $PQR$

#### Area of plane shapes

a. Area of triangle $=\frac12\times b\times h$

Here, $h =$ vertical height and $b =$ base.

b. Area of rectangle $=w\times h$

Here, $w =$ width and $h =$ height.

c. Area of trapezium or trapezoid $=\frac12 \left( a+b\right) \times h$

Here, $a =$ base 1, $b =$ base 2 and $h =$ height

d. Area of circle $=\pi r^2$

Circumference of circle $=2\pi r$

Here $r =$ radius of circle.

e. Area of ellipse $=\pi ab$

Here $a =$ semi minor axis and $b =$ semi major axis.

f. Area of sector $=\frac12 \times r^2 \times \theta$

Here $r =$ radius and $\theta =$ angle in radians.

#### Types of angles

Besides common types of angles such as acute angle, right angle, obtuse angle, straight angle and reflex angle, some other types of angles are as follows:

##### a. Congruent angles

Two angles which have same angle are known as congruent angles.

Here $\angle ABC$ and $\angle PQR$ are congruent angles.

##### b. Angles around a point

The sum of angles around a point is always $360^\circ$.

Here $\angle AOB+\angle BOC+\angle COD+\angle DOA = 360^\circ$.

Angles which have a common side and common vertex (corner point) and which do not overlap are known as adjacent angles.

Here, $\angle ABC$ and $\angle CBD$ share common vertex $B$ and common side $BC$ and they are not overlapping. So, $\angle ABC$ and $\angle CBD$ are adjacent angles.

##### d. Supplementary angles

Two angles are supplementary when their sum is equal to $180^\circ$.

Here, supplementary angles $\angle ABC + \angle CBD = 180^\circ$.

##### e. Complementary angles

Two angles are complementary when their sum is equal to $90^\circ$.

Here, complementary angles $\angle ABC + \angle CBD = 90^\circ$.

##### f. Angles made by transversal crossing parallel lines

A transversal is a line that crosses at least two other lines. When a transversal crosses parallel lines, many angles are formed. For example, when transversal $XY$ crosses parallel lines $AB$ and $CD$ at point $P$ and $Q$.

Vertically opposite angles

$\angle APX = \angle QPB = a \quad \angle CQP = \angle DQY = a$

$\angle XPB = \angle APQ = b \quad \angle PQD = \angle CQY = b$.

Alternate interior angles

$\angle BPQ = \angle CQP = a$.

$\angle APQ = \angle PQD = b$.

Alternate exterior angles

$\angle APX = \angle DQY = a$.

$\angle XPB = \angle CQY = b$.

Corresponding angles

$\angle APX = \angle CQP = a \qquad \angle QPB = \angle DQY = a$.

$\angle XPB = \angle PQD = b \qquad \angle APQ = \angle CQY = b$.