Geometry

Plane & Solid Geometry

Plane Geometry is about flat shapes (1D and 2D) like lines, circles and triangles. Solid Geometry is about three dimensional (3D) objects like cubes, prisms, cylinders and spheres.

Points and Lines

A point is basic unit of geometry. It has no dimension (0D, or no measurement). A point only has position. When two points are joined, you get a line-segment. Line segment has one dimension (1D) i.e. length. A line with one end point is called a ray.

If three or more points lie on the same line, they are called collinear points otherwise they are called non-collinear points.

Two lines will be either intersecting or non-intersecting (parallel). The distance between two parallel lines is the length of the common perpendiculars at different points on these parallel lines.

Angles

An angle is formed when two lines originate from the same end point. There are five different types of angles based on their measurement - acute angle, right angle, obtuse angle, straight angle and reflex angle. Two angles whose sum is 90° are called complementary angles, and two angles whose sum is 180° are called supplementary angles.

Two adjacent angles Have a common vertex and a common arm but no common interior.

When two lines intersect each other, they are called vertically opposite angles. They are equal in measure.

Parallel Lines and a Transversal

When two lines l and m meet, they intersect; the meeting point is called the point of intersection. When lines drawn on a sheet of paper do not meet, however far produced, they are called parallel lines.

Transversal is a line which intersects two or more lines at distinct points.

If a transversal intersects two parallel lines, then

  • (i) each pair of corresponding angles is equal
  • (ii) each pair of alternate interior angles is equal
  • (iii) each pair of interior angles on the same side of the transversal is supplementary

  • ∠1 = ∠5, ∠3 = ∠7, ∠2 = ∠6, ∠4 = ∠8
  • ∠3 = ∠6, ∠4 = ∠5
  • ∠3 + ∠5 = 180°, ∠4 + ∠6 = 180°

Polygons

A polygon is a closed plane figure bounded by straight lines.

  • Convex Polygon: A polygon in which none of its interior angles is more than 180°.
  • Concave Polygon: A polygon in which at least one angle is more than 180°.
  • Regular Polygon: A regular polygon has all its sides and angles equal.

A polygon is called a Triangle, Quadrilateral, Pentagon, Hexagon, Heptagon, Octagon, Nonagon and Decagon according as it contains 3, 4, 5, 6, 7, 8, 9, 10 sides respectively.

Regular Polygon

Exterior angle of regular polygon with n sides = 360°/n

Interior angle = 180° - Exterior angle

Convex Polygon

In a Convex Polygon of n sides,

Sum of all interior angles = (2n - 4) × 90°

Sum of all exterior angles = 360°

Number of diagonals of a polygon on n sides = n(n-3)/2

Triangles

Triangle is a closed figure formed by three intersecting lines. A triangle has three sides, three angles and three vertices. The sum of all the three angles of a triangle is 180°.

Median of a Triangle

The line segment joining a vertex of a triangle to the mid point of its opposite side is called a median of the triangle. A triangle has 3 medians. A median of a triangle divides it into two triangles of equal areas. All the three medians of a triangle meet at a point called the centroid of the triangle.

Altitude of a Triangle

The perpendicular line segment from a vertex of a triangle to its opposite side is called an altitude of the triangle. A triangle has 3 altitudes. All the three altitudes of a triangle meet at a point called the orthocentre of the triangle.

Angle Bisector

It is the bisector of an angle contained in the vertex of a triangle. All the three angle bisectors of a triangle meet at a point called the incentre of the triangle. The incentre is the centre of a circle which can be perfectly inscribed in the triangle.

Perpendicular Bisector

It is the line passing through the mid point of the side of a triangle and perpendicular to it. All the three perpendicular bisectors of a triangle meet at a point called the circumcentre of the triangle. The circumcentre is the centre of a circle which can be perfectly circumscribed about the triangle.

Exterior Angle

An exterior angle of a triangle is formed when a side of a triangle is produced. At each vertex, you have two ways of forming an exterior angle. The measure of any exterior angle of a triangle is equal to the sum of the measures of its interior opposite angles.

Congruence of Triangles

Two figures are congruent, if they are of the same shape and of the same size. Congruent objects are exact copies of one another. For example, two circles of the same radii are congruent and two squares of the same sides are congruent. Two angles are congruent if their measures are equal.

If two triangles ABC and PQR are congruent under the correspondence A ↔ P, B ↔ Q and C ↔ R, then symbolically, it is expressed as Δ ABC ≅ Δ PQR. There are five congruence rules in triangles - SAS, ASA, AAS, SSS and RHS. There is no such thing as AAA Congruence of two triangles.

SSS Congruence of two triangles

Two triangles are congruent if the three sides of the one are equal to the three corresponding sides of the other.

SAS Congruence of two triangles

Two triangles are congruent if two sides and the angle included between them in one of the triangles are equal to the corresponding sides and the angle included between them of the other triangle.

ASA Congruence of two triangles

Two triangles are congruent if two angles and the side included between them in one of the triangles are equal to the corresponding angles and the side included between them of the other triangle.

RHS Congruence of two right-angled triangles

Two right-angled triangles are congruent if the hypotenuse and a leg of one of the triangles are equal to the hypotenuse and the corresponding leg of the other triangle.

Similar Triangles

Two figures having the same shape but not necessarily the same size are called similar figures. All the congruent figures are similar but the converse is not true. Two polygons of the same number of sides are similar, if (i) their corresponding angles are equal and (ii) their corresponding sides are in the same ratio (proportion).

In addition to the five congruence rules, two triangles are similar with AA or AAA criteria. The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

Side Angle Relationship

Angles opposite to equal sides of a triangle are equal. Also, sides opposite to equal angles are equal. In a triangle, angle opposite to the longer side is greater. Sum of any two sides is greater then the third side. The difference between the lengths of any two sides is smaller than the length of the third side.

Special Triangles

Equilateral Triangles

All three sides are equal and all three angles are equal. Each angle of an equilateral triangle is of 60°.

The line-segment joining the mid-points of any two sides of a triangle is parallel to the third side and is half of it.

Isosceles Triangle

A triangle is said to be isosceles if any two of its sides are of same length. The non-equal side of an isosceles triangle is called its base. The base angles of an isosceles triangle have equal measure.

Right Triangles

In a right angled triangle, the side opposite to the right angle is called the hypotenuse and the other two sides are called its legs.

If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse, then the triangles on both sides of the perpendicular are similar to the whole triangle and also to each other.

In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides (Pythagoras Theorem).

A triplet is a set of numbers which will satisfy the Pythagoras Theorem. The frequently used triplets are (3, 4, 5), (5, 12, 13), (7, 24, 25), (8, 15, 17), (9, 40, 41), (11, 60, 61), (12, 35, 37), (16, 63, 65), (20, 21, 29). The multiples of triplets are also triplets. For example, (6, 8, 10) is a multiple of (3, 4, 5) and a triplet.

Quadrilateral

A quadrilateral has four sides, four angles and four vertices. There are six types of quadrilaterals - trapezium, parallelograms, rectangle, rhombus, square and kite. Square, rectangle and rhombus are parallelograms. A square is a rectangle and also a rhombus.

Angle Sum Property: Sum of the angles of a quadrilateral is 360°.

1. Trapezium

Trapezium is a quadrilateral with a pair of parallel sides. 

2. Kite

Kite is a special type of a quadrilateral in which two adjacent sides are equal. There are exactly two distinct consecutive pairs of sides of equal length.

3. Parallelogram

  1. Opposite sides are equal and parallel
  2. Opposite angles are equal
  3. Diagonals bisect each other. They are not equal.
  4. The sum of any two adjacent interior angles is equal to 180°.

Diagonals of a rectangle bisect each other and are equal. Diagonals of a rhombus bisect each other at right angles. Diagonals of a square bisect each other at right angles and are equal.

Parallelograms on the same base (or equal bases) and between the same parallels are equal in area. Area of a parallelogram is the product of its base and the corresponding altitude.

If a parallelogram and a triangle are on the same base and between the same parallels, then area of the triangle is half the area of the parallelogram.

4. Rhombus

Rhombus is a parallelogram as a special case of kite (which is not a a parallelogram). All the sides of rhombus are all of same length. The diagonals of a rhombus are perpendicular (right angles) bisectors of one another.

Sum of the squares of sides is equal to the sum of the squares of its diagonals.

5. Rectangle

A rectangle is a parallelogram with equal angles, in which every angle is a right angle. The diagonals of a rectangle are of equal length.

6. Square

A square is a rectangle with equal sides. A square has all the properties of a rectangle with an additional requirement that all the sides have equal length. The diagonals of a square are perpendicular bisectors of each other.

Mid-Point Theorem

A line through the mid-point of a side of a triangle parallel to another side bisects the third side. A diagonal of a parallelogram divides it into two congruent triangles. The quadrilateral formed by joining the mid-points of the sides of a quadrilateral, in order, is a parallelogram.

Circles

A circle is the collection of all points in a plane, which are at a constant distance (radius) from a fixed point (centre). The angle in a circle is 360°.

There is one and only one circle passing through three non-collinear points.

Chords, Segment, Sector & Arc

Equal chords of a circle subtend equal angles at the centre. Conversely, if the angles subtended by two chords of a circle at the centre are equal, the chords are equal.

The perpendicular from the centre of a circle to a chord bisects the chord. The line drawn through the centre of a circle to bisect a chord is perpendicular to the chord.

Equal chords of a circle are equidistant from the centre. Chords equidistant from the centre of a circle are equal in length.

If two arcs of a circle are congruent, then their corresponding chords are equal and conversely if two chords of a circle are equal, then their corresponding arcs (minor, major) are congruent.

The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.

Angles in the same segment of a circle are equal. Angle in a semicircle is a right angle.

If a line segment joining two points subtends equal angles at two other points lying on the same side of the line, the four points lie on a circle.

Cyclic Quadrilateral

The sum of either pair of opposite angles of a cyclic quadrilateral is 180°. If the sum of a pair of opposite angles of a quadrilateral is 180°, the quadrilateral is cyclic.

Tangent To A Circle

A tangent to a circle is a line that intersects the circle at only one point. The tangent to a circle is a special case of the secant, when the two end points of its corresponding chord coincide.

  • The tangent at any point of a circle is perpendicular to the radius through the point of contact.
  • The lengths of tangents drawn from an external point to a circle are equal.
  • The angle which a chord makes with a tangent at its point of contact is equal to any angle in the alternate segment.
  • If PT is a tangent (with P being an external point and T being the point of contact) and PAB is a secant to circle (with A and B as the points where the secant cuts the circle), then PT2 = PA × PB.

Pair of Circles

If two circles touch each other, the point of contact of the two circles lies on the straight line through the centres of the circles, ie,. the points A, C, B are collinear.

In a given pair of circles there are two types of tangents. The direct tangents and the cross (or transverse) tangents.

When two circles of radii r1 and r2 have there centres at a distance d, the length of

  • direct common tangent = √[d2 - (r- r2)2]
  • transverse tangent = √[d2 - (r1 + r2)2]

If the two circles touch, then d = r1 + r2