Geometry is the mathematical study of shapes, sizes, and spatial relationships, anchored by theorems and formulas. It helps us understand everything from the angles of a triangle to the design of buildings. This article explores geometry's most powerful principles—from triangle theorems to circle theorems and polygon formulas.
Geometry Theorems
Geometry theorems are fundamental statements that help us understand properties of shapes, angles, lines, and figures.
Here are some key geometry theorems:
Triangle Theorems
Though there are many Geometry Theorems on Triangles, let us see some important geometry theorems.
Theorem 1:
In any triangle, the sum of the three interior angles is always 180°.
Theorem 2:
If a side of a triangle is extended, the exterior angle formed is equal to the sum of the two interior opposite angles.
Theorem 3:
In an isosceles triangle, the base angles are congruent.
Advanced Triangle Theorems
Theorem 4:
If a line is drawn parallel to one side of a triangle and intersects the midpoints of the other two sides, then the two sides are divided in the same ratio.
Example:
In triangle ABC, if DE is a line parallel to BC and intersects AB at D and AC at E, then according to the theorem:
AD/DB= AE/EC.
Theorem 5:
If the corresponding angles of two triangles are equal, then the corresponding sides are proportional, and the two triangles are similar.
Example:
For triangles ∆ABC and ∆PQR, if ∠A = ∠P, ∠B = ∠Q, and ∠C = ∠R, then according to the theorem:
AB/PQ = BC/QR = AC/PR,
and ∆ABC ~ ∆PQR.
Circle Theorems
Circle theorems help establish the relationships between various elements of a circle, such as tangents, angles, chords, radii, and sectors. In other words, these theorems describe the different angle properties of a circle.
Now, let’s explore the various geometric theorems associated with circles.
Theorem 1:
Angles in the same segment and on the same chord are always equal.
Theorem 2:
The angle at the center of a circle is twice the angle at the circumference.
Theorem 3:
The angle between the tangent and the side of the triangle is equal to the interior opposite angle.
Theorem 4:
Tangents from a common point (A) to a circle are always equal in length. AB=BC
Theorem 5:
The angle between the tangent and the radius is always 90°
Theorem 6:
In a cyclic quadrilateral, all vertices lie on the circumference of the circle. Opposites angles add up to 180°.
Length of Tangent:
When two circles touch each other externally, the length of their tangent can be calculated based on their radii.
Length of Direct common tangent:
A direct common tangent is a straight line that touches both circles on the same side. In the diagram given below, AB is a Direct common tangent and the circles with centres x and y lie on the same side of it.
Length of Transverse common tangent:
A transverse common tangent is a straight line that touches the circles in such a way that the circles lie on opposite sides of it. In the diagram given below, AB is a transverse common tangent and the circles with centres x and y lie on the opposite side of it.
Parallelogram Theorems
A parallelogram is a quadrilateral with both pairs of opposite sides being parallel.
Let’s explore some of the key theorems related to parallelograms:
Theorem 1:
A quadrilateral is a parallelogram if its opposite sides are congruent.
Theorem 2:
A quadrilateral must be a parallelogram if its opposite angles are congruent.
Theorem 3:
If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.
Theorem 4:
A quadrilateral is a parallelogram if one pair of opposite sides is both parallel and equal in length.
Miscellaneous Formula
The important formulas associated with a regular polygon are given below:
- Sum of all interior angles = (n-2) x 180
- each interior angle of an n side polygon = (n-2) x 180 /n
- Sum of all the exterior angles of polygon = 360
- each exterior angle of polygon = 360 /n
- The number of diagonals of an “n-sided” polygon = [n(n-3)]/2