Basic Geometrical Ideas

Basic geometric ideas refer to the fundamental concepts and principles in geometry. This chapter provides the foundation for understanding and working with geometric shapes, figures and their properties.

Triangles

A triangle is a fundamental geometric shape that has three sides, three vertices and three angles.

A, B and C are three vertices and a, b and c are the three angles of the given triangle.

Angle Sum Property of a Triangle

The Angle Sum Property of a Triangle states that the total of the angles within a triangle is either 180° or equivalent to two right angles.

In a triangle with vertices A, B, and C, the sum of its three angles always equals 180°, expressed as:

∠A + ∠B + ∠C = 180°

Exterior angle: The measure of the exterior angle of a triangle is equal to the sum of the measure of the two interior angles that are opposite to it.

Classification of Triangles on the Basis of their Sides

There are three types of triangles based on the length of their sides are as follows:

Classification of Triangles on the Basis of their Angles 

There are three types of triangles based on the length of their angles are as follows:

Congruent Triangles

Congruent triangles are defined as triangles where each angle in one triangle is identical to the corresponding angle in the other triangle and each side in one triangle is equivalent in length to the corresponding side in the other triangle.

Quadrilaterals

A quadrilateral is a closed geometric shape formed by four line segments. It has four sides, four angles and four vertices. Types of quadrilaterals are shown as:

Angle Sum Property of a Quadrilateral

The total of the angles in a quadrilateral always amounts to 360° which is referred to as the angle sum property of a quadrilateral.

∠ A + ∠ B + ∠ C + ∠ D = 360°

Adjacent and Opposite Sides

In a quadrilateral, there are two types of side relationships:

1. Adjacent Sides: Adjacent sides are two sides that share a common endpoint or vertex. They are like neighbours. In the given quadrilateral ABCD, the adjacent side pairs are:

→ AB and BC
→ BC and CD
→ CD and DA
→ DA and AB

Opposite Sides: Opposite sides are pairs of sides that do not share a common endpoint or vertex. They are situated on opposite ends of the quadrilateral. In the given quadrilateral ABCD, the opposite side pairs are:

→ AB and DC
→ AD and BC

Convex and Concave Quadrilaterals

A convex quadrilateral is a quadrilateral when all of its angles are smaller than 180°.

A concave quadrilateral is a quadrilateral if it has at least one angle exceeding 180°.

Types of Quadrilaterals and their Properties

Circles

A circle is formed by the set of all points within a plane that are equidistant from a fixed central point. This central point is known as the centre of the circle and the uniform distance from the centre to any point on the circle is referred to as the radius.

Circumference of a circle: The measurement of the outer boundary of a circle is called its circumference.

Where,

r  → radius of the circle

Chord: A chord in a circle is a line segment that connects any two points along the circumference of a circle. AB is a chord in the given figure.

Diameter: The diameter of a circle is a special chord that passes through the centre of the circle. The diameter is the largest chord. It is exactly twice the length of the radius.

Diameter = 2 × Radius And Radius = Diameter/2

PQ is a diameter and OP is a radius which is shown in the given figure.

Secant: A secant of a circle is a line that intersects the circle and touches it at two distinct points. AB is a secant in the given figure.

Arc: An arc of a circle refers to a portion of the circumference of a circle.

AC, CB, BD and DA are arcs which are shown in the figure.

Sector: A sector of a circle is the area enclosed by an arc and two radii that connect the arc's endpoints to the circle's centre.

Segment: When a chord of a circle divides the circular region into two parts, each of these parts is referred to as a segment of the circle.

Semicircle: A semicircle is formed when the diameter of a circle divides it into two equal parts. Each of these equal parts is referred to as a semicircle.

Concentric Circles: Concentric circles are circles that share the same centre point but have varying radii.

Example 1: What is the value of angle q in the given figure?

a) 34°
b) 44°
c) 54°
d) 64°

Answer: c) 54°

Explanation: Let the equal angle be p, shown as:

p + 117° = 180° (Linear property of angles)

⇒ p = 180°− 117°
⇒ p = 63°

Sum of the measure of the two interior angles that are opposite to it is the exterior angle of a triangle. 

⇒ p + q = 117°
⇒ 63° + q = 117°
⇒ q = 117° − 63° 
⇒ q = 54°

Example 2: What are the values of unknown angles?

a) x° = 45°; y° = 85°
b) x° = 45°; y° = 95°
c) x° = 85°; y° = 45°
d) x° = 95°; y° = 45°

Answer: b)  x° = 45°; y° = 95°

Explanation: ABCD is a quadrilateral. ABD and BCD are triangles.

Sum of all angles of a triangle is 180°.

In triangle ABD,

55° + 80° + x° = 180° (Sum of its three angles in a triangle is 180°)

⇒ 135° + x° = 180°
⇒ x° = 180° − 135°
⇒ x° = 45°

In triangle BCD, 

40° + 45° + y° = 180° (Sum of its three angles in a triangle is 180°)
⇒ 85° + y° = 180°
⇒ y° = 180° − 85°
⇒ y° = 95°