< Class 6 | Exercise 3.2

Class 6 • Chapter 4: Basic Geometrical Ideas

Introduction
Geometry studies shapes, sizes, positions and figures made of points and lines. This page gives short, exam-friendly definitions with simple notation using MathJax.
Learning goals
  • Recognize and name points, line segments, lines, and rays.
  • Tell the difference between intersecting and parallel lines.
  • Understand curves: simple, open, and closed.
  • Identify sides, vertices, diagonals of polygons.
  • Read and write angle notation like \(\angle ABC\).
1) Points
A point shows an exact position. It has no length, width or thickness. We name points with capital letters like \(A, B, C\).
\(\text{Example of : Points } A, B,C\)
points
2) Line Segment
A line segment is the shortest path joining two points. It has two endpoints.
\(\overline{AB}\) means the segment from \(A\) to \(B\).
line segment
3) Line
A line extends forever in both directions. It has no endpoints.
We write \( \overleftrightarrow{AB} \) for the line through points \(A\) and \(B\).
line
4) Ray
A ray has a starting point and extends forever in one direction.
Ray \( \overrightarrow{AB} \) starts at \(A\) and passes through \(B\).
ray
5) Intersecting Lines
Intersecting lines cross at a common point called the point of intersection.
If lines \(l_1\) and \(l_2\) meet at \(P\), we write \(l_1 , l_2 \) intersect at P.
intersecting lines
6) Parallel Lines
Parallel lines are in the same plane and never meet, no matter how far extended.
We write \( m \parallel n \).
parallel lines
7) Curves
A curve is any drawing that is not made up of straight line segments only. It can be smooth or zig-zag.
curves
8) Simple Curve
A simple curve does not cross itself.
simple curve
9) Open and Closed Curve
An open curve has different start and end points. A closed curve joins back to its start point and encloses a region.
Examples: an open arc vs a closed circle.
open and closed curve
10) Sides and Vertices
In a figure made of segments, each segment is a side. The meeting point of two sides is a vertex (plural: vertices).
Triangle \( \triangle ABC \) has sides \(\overline{AB}, \overline{BC}, \overline{CA}\) and vertices \(A, B, C\).
open and closed curve
11) Diagonals
A diagonal of a polygon joins two non-adjacent vertices.
In quadrilateral \(ABCD\), \(\overline{AC}\) and \(\overline{BD}\) are diagonals.
diagonals
12) Polygon
A polygon is a closed figure made only of line segments. The segments do not cross except at their endpoints.
Examples: triangle, quadrilateral, pentagon, hexagon.
polygons
13) Angles
An angle is formed by two rays with a common endpoint called the vertex.
\(\angle POQ\) means the angle with vertex at \(O\) and arms \(PO\) and \(OQ\).
angles
Quick recap
  • Use capital letters to name points.
  • Segments have two endpoints. Lines have none. Rays have one.
  • Lines either intersect or stay parallel.
  • Polygons are closed and made only of segments. Diagonals connect non-adjacent vertices.
  • Angles are named with the vertex in the middle, like \(\angle ABC\).
You are ready to start the exercises for Chapter 4.