Symmetry and Congruent Figures - Third Grade Mathematics

In geometry, figures can mirror each other in balance and match each other in exact size and shape. When two figures have the identical shape and the identical size, they are called congruent. When a single figure can be folded along a line so that both halves match each other perfectly, it possesses line symmetry. In this chapter, you will master recognizing congruent shapes and identifying lines of symmetry in geometric figures and everyday objects.

What Are Congruent Figures?

Two figures are congruent if they have the exact same shape AND the exact same size.

Congruent Figures:
Figure A:               Figure B:
+-------+               +-------+
|       |               |       |
|       |               |       |
+-------+               +-------+
Same size, same shape! Figure A is congruent to Figure B.

NOT Congruent:
Figure C:               Figure D:
+-------+               +---------------+
|       |               |               |
+-------+               +---------------+
Same shape (rectangles), but DIFFERENT sizes! Not congruent.

Turning, Sliding, and Flipping Congruent Shapes

Two shapes are still congruent even if one of them has been turned (rotated), slid (translated), or flipped over (reflected)! Imagine cutting a triangle out of cardboard. If you slide it across the table, turn it upside down, or flip it over, it is still the exact same piece of cardboard—its size and shape have not changed at all!

Original:         Rotated:            Flipped:
    |\               /\                 /|
    | \             /  \               / |
    +---\           +----\             /---+
All three triangles are completely congruent!

What Is Line Symmetry?

A figure has line symmetry (or reflectional symmetry) if it can be divided into two halves by a line such that one half is the exact mirror image of the other half. The dividing line is called the line of symmetry.

If you fold the shape along the line of symmetry,
the two halves match up completely with zero overhang!

Butterfly Symmetry:
      \  |  /
       \ | /      <- Dotted vertical line is the Line of Symmetry!
      ---*---
       / | \
      /  |  \

Real-World Examples of Line Symmetry:

  • The human face (approximately symmetric down the middle).
  • Many animal bodies (butterflies, beetles, birds with spread wings).
  • Capital letters:
  • Letter "A" has 1 vertical line of symmetry down the middle.
  • Letter "B" has 1 horizontal line of symmetry across the middle.
  • Letter "H" has BOTH a vertical line and a horizontal line of symmetry!
  • Letter "F" has NO lines of symmetry.

Finding Multiple Lines of Symmetry in Polygons

Some geometric shapes have more than one line of symmetry!

Shape                   Number of Lines of Symmetry
Equilateral Triangle    3 lines of symmetry
Rectangle (non-square)  2 lines of symmetry (1 vertical, 1 horizontal)
Square                  4 lines of symmetry (vertical, horizontal, and 2 diagonals!)
Regular Hexagon         6 lines of symmetry
Circle                  Infinite lines of symmetry!

Lines of Symmetry in a Square (4 total):
       |
    \  |  /
   --+-+-+--  <- Horizontal line
     | | |
   --+-+-+--
    /  |  \
       |
Vertical, horizontal, and two corner-to-corner diagonals!

Notice that in a rectangle, a diagonal line is NOT a line of symmetry! If you try to fold a non-square sheet of paper along its diagonal, the corners stick out and do not match up.

Chapter Practice Exercises

  1. Tell whether the pairs of figures are congruent, and explain why: a. Two squares, each with side lengths of 4 inches b. A circle with a width of 2 inches and a circle with a width of 5 inches c. Two triangles of the exact same size, but one is turned upside down
  2. How many lines of symmetry does each shape possess? a. A rectangle that is 4 inches wide and 6 inches long b. A square c. The capital letter "M" d. The capital letter "X" e. The capital letter "L"
  3. Explain why sliding, turning, or flipping a shape does not destroy its congruence.
  4. Draw or describe an everyday object in your classroom that has at least one line of symmetry. Where does the line of symmetry run?
  5. A student says: "Every triangle has 3 lines of symmetry." Is this student correct? Explain using a right triangle with sides of lengths 3, 4, and 5 inches.

Solutions and Step-by-Step Answers

  1. Congruence: a. Congruent. Both are squares and have identical side lengths (4 inches). b. Not congruent. While they share the same shape (circles), their sizes are different. c. Congruent. Rotating or turning a shape does not change its dimensions or shape.
  2. Lines of symmetry: a. Rectangle -> 2 lines of symmetry (1 vertical, 1 horizontal). b. Square -> 4 lines of symmetry (vertical, horizontal, 2 diagonals). c. Capital "M" -> 1 vertical line of symmetry down the middle. d. Capital "X" -> 2 lines of symmetry (vertical and horizontal, or 4 if drawn symmetrically). e. Capital "L" -> 0 lines of symmetry (folding does not match halves).
  3. Congruence depends strictly on size and shape. Moving an object in space (sliding, turning, flipping) does not alter its side lengths, interior angles, or physical dimensions, so the figure remains congruent to its original form.
  4. Everyday object: A standard classroom door has 1 vertical line of symmetry down the center, dividing it into matching left and right halves.
  5. No, the student is incorrect. Only an equilateral triangle (with 3 equal sides) has 3 lines of symmetry. A scalene right triangle with sides of 3, 4, and 5 inches has no equal sides and has 0 lines of symmetry.