Understanding Elementary Shape

This chapter provides you with an outline of "Understanding Elementary Shapes" which is typically taught in mathematics. When we observe objects in our surroundings, we can identify various elementary shapes, including two-dimensional ones like squares, rectangles, and ovals, as well as three-dimensional shapes like rectangular prisms, cylinders, and spheres. These geometric forms are used in everyday objects such as houses, windows, books, skyscrapers, flower pots, toy trains and balloons.

Understanding Elementary Shape

Point: A point represents an exact position on a flat surface. It is denoted by a dot (.). It has no dimensions, such as length, width or depth. Typically, uppercase letters like A, B, P, and T are employed to identify points.

Line Segment: A line segment is a portion of a line that has two endpoints.

Ray: A ray is a portion of a line with a single endpoint that extends infinitely in one direction.

Line: A line denotes an unending straight path that extends infinitely in both directions.

Intersecting Lines: When two lines meet at a single point, they are referred to as intersecting lines. The point where these lines intersect is known as the point of intersection.

Perpendicular Lines: Perpendicular lines are two lines that intersect each other at a right angle, exactly 90°.

Concurrent Lines: Concurrent lines are lines that come together at a shared point within a plane. This meeting point is referred to as the point of concurrency.

Parallel Lines: Parallel lines are two lines in the same plane that never intersect and they run alongside each other without ever meeting.

Polygons

A polygon is a closed shape enclosed by three or more line segments. The sides of a polygon are the line segments that make up its boundary.

The point where two adjacent sides of a polygon meet is called a vertex.

A diagonal is a line segment that connects two non-adjacent vertices of the polygon.

For example, in the diagram below, AB, BC, CD, DE and EA represent the sides of the polygon while A, B, C, D and E denote the vertices. The diagonals of the polygon are AC, AD, BD, BE, CA and CE.

Types of Polygons

There are two primary types of polygons: Regular polygons and Irregular polygons.

Regular Polygon: A regular polygon is a polygon in which all of its sides have the same length and all of its angles are of equal measure.

Irregular Polygon: An irregular polygon is a polygon that has sides of varying lengths and angles of differing measures.

Sum of Interior Angles in a Polygon

To find the total sum of the interior angles in a polygon, you can use the formula:

Sum of Interior Angles in a Polygon = (n − 2) × 180°

where 'n' represents the number of sides of the polygon.

Angles

An angle is a geometric shape created by two lines or rays that originate from a common point. This common point is known as the vertex of the angle. These lines or rays are referred to as the arms of the angle.

Exterior Angle: An exterior angle is an angle that exists outside a geometric shape. It forms by extending one of the shape's lines beyond the point where they intersect.

Interior Angle: An interior angle is an angle located within the confines of a geometric shape. The sum of the interior and exterior angles is 180°.

Types of Angles

a. Zero Angle: A zero angle is an angle with a measurement of 0°.

b. Acute Angle: An acute angle falls within the range of 0° to 90°. Examples of acute angles are 5°, 15°, 20°, 50°, 55°, 67 ½°, 89°, etc.

c. Right Angle: A right angle is exactly 90° in measurement.

d. Obtuse Angle: An obtuse angle falls within the measurement range of 90° to 180°. Examples of obtuse angles are 95°, 105°, 120°, 150°, 167 ½°, 189°, etc.

e. Straight Angle: A straight angle is exactly 180° in measurement.

f. Reflex Angle: A reflex angle lies within the measurement range of 180° to 360°. Examples of reflex angles are 185°, 200°, 240°, 250°, 267 ½°, 359°, etc.

g. Complete Angle: A complete angle is exactly 360° in measurement. It is known as a full angle.

h. Complementary Angles: Complementary angles are two or more angles whose sum equals 90°.

Example:  What is the complement of 37°?

a) 43°
b) 53°
c) 63°
d) 73°

Answer: b) 53°

Explanation: Complement of 37° = 90° − 37° = 53°

i. Supplementary Angles: Supplementary angles are two or more angles whose sum equals 180°.

Example: What is the supplement of 37.4°?

a) 112.6°
b) 122.6°
c) 132.6°
d) 142.6°

Answer: d) 142.6°

Explanation: Supplement of 37° = 180° − 37.4° = 142.6°

j. Adjacent Angles: Adjacent angles are two angles that share a common vertex and a common side but do not overlap.

k. Vertically Opposite Angles: When two lines intersect each other, the angles directly across from one another are known as vertically opposite angles. Pairs of vertically opposite angles are always equal to each other.

Transversal: A transversal is a line that intersects two or more lines at different points.

l. Co-Interior Angles: Co-interior angles are the interior angles located on the same side of a transversal.

m. Alternate Angles: When two straight lines are intersected by a transversal, the angles created on the opposite side of the transversal concerning both lines are known as alternate angles.

n. Linear Pair: A linear pair consists of adjacent angles formed when two lines intersect, and these angles are supplementary, meaning their sum equals 180°.