Congruent and Similar Figures: Fifth Grade Mathematics

Congruent figures have the identical size and shape, whereas similar figures have the identical shape but may differ in size.

Congruent Figures: Same Shape and Same Size

Two figures are congruent if they have the exact same shape and the exact same size. All corresponding side lengths are equal, and all corresponding interior angle measures are equal.
- Symbolic notation: Figure A ~= Figure B.

- If two congruent shapes were placed on top of one another, they would match point-for-point. Rotating or flipping a shape does not change its congruence.

+--------------------------------------------------------------------------+
|                  CONGRUENCE VERSUS SIMILARITY                            |
+---------------------+---------------------+------------------------------+
| Characteristic      | Congruent Figures   | Similar Figures              |
+---------------------+---------------------+------------------------------+
| Shape               | Identical           | Identical                    |
| Size                | Identical           | Proportional (Scaled)        |
| Corresponding Angles| Exactly EQUAL       | Exactly EQUAL                |
| Corresponding Sides | Exactly EQUAL       | PROPORTIONAL (Scale factor)  |
| Symbol              | ~=                  | ~                            |
+---------------------+---------------------+------------------------------+

Similar Figures: Same Shape, Proportional Size

Two figures are similar if they have the exact same shape, but one is an enlargement or reduction of the other:
1. All corresponding angles are equal.

2. All corresponding sides are proportional, meaning every side of the larger shape is multiplied by the same scale factor.

Visualizing Similar Rectangles:
  Rectangle 1:  Width = 3 cm,  Length = 5 cm
  Rectangle 2:  Width = 6 cm,  Length = 10 cm

  Angle Check: Both have four 90° angles (Equal!).
  Side Ratio Check: 6 / 3 = 2, and 10 / 5 = 2.
  All sides are enlarged by a scale factor of 2.
  Therefore, Rectangle 1 is SIMILAR to Rectangle 2.

Are All Congruent Shapes Similar?

Yes! Congruent shapes are similar figures with a scale factor of 1. However, similar shapes are not necessarily congruent, because their sizes can differ.

Exercises and Solutions

Exercise 1: Congruent vs. Similar Analysis. Triangle ABC has side lengths 6 cm, 8 cm, and 10 cm with angles 37°, 53°, and 90°. Triangle DEF has side lengths 12 cm, 16 cm, and 20 cm with angles 37°, 53°, and 90°. Determine whether Triangle ABC and Triangle DEF are congruent, similar, or neither. Justify your answer.

Solution 1: First check angles: corresponding angles are 37° = 37°, 53° = 53°, and 90° = 90°. All corresponding angles are equal. Next check side ratios: 12 / 6 = 2; 16 / 8 = 2; 20 / 10 = 2. All corresponding side lengths are proportional by a constant scale factor of 2. Because corresponding angles are equal and corresponding sides are proportional, the triangles have the same shape. However, their sizes differ (one is twice as large as the other). Therefore, Triangle ABC and Triangle DEF are similar, but not congruent.

Exercise 2: Scale Factor Side Calculation. Rectangle M has a length of 8 inches and a width of 3 inches. Rectangle N is similar to Rectangle M and has a length of 24 inches. Find the width of Rectangle N.

Solution 2: Find the scale factor between corresponding lengths: divide the length of Rectangle N by the length of Rectangle M: 24 / 8 = 3. The scale factor is 3. Because the rectangles are similar, the width must be scaled by the identical factor: multiply the width of Rectangle M by 3: 3 * 3 = 9 inches. The width of Rectangle N is 9 inches.