Introduction to Congruent Shapes

Congruent shapes are geometric figures that have exactly the same size and shape, regardless of their position or orientation. When two shapes are congruent, one can be transformed into the other through a combination of rotations, reflections, and translations.

To determine if two shapes are congruent, we can compare their corresponding sides and angles. If all corresponding sides are equal in length and all corresponding angles are equal in measure, then the shapes are congruent.

Examples of congruent shapes:

Examples of congruent shapes

Key Characteristics

  • Same shape and size
  • Corresponding sides are equal
  • Corresponding angles are equal
  • Can be transformed into each other through rotation, reflection, or translation

Properties of Congruent Shapes

Side and Angle Relationships

For two shapes to be congruent, their corresponding parts must be equal. This means:

Corresponding Sides

All matching sides must be equal in length.

If shape A ≅ shape B, then:

AB = A'B', BC = B'C', etc.

Corresponding Angles

All matching angles must be equal in measure.

If shape A ≅ shape B, then:

∠A = ∠A', ∠B = ∠B', etc.

Transformations Preserving Congruence

The following transformations preserve congruence (they don't change size or shape):

Translation (Slide)

Moving a shape without rotating or flipping it

Translation example

Rotation (Turn)

Turning a shape around a fixed point

Rotation example

Reflection (Flip)

Creating a mirror image across a line

Reflection example

Rotation on Coordinate Plane

Understanding Rotation

Rotation is a transformation that turns a figure around a fixed point called the center of rotation. On a coordinate plane, we can precisely describe rotations using coordinates and angles.

Rotation on coordinate plane

Rotation Rules

The general rules for rotating a point (x, y) about the origin (0,0):

\[ \begin{array}{|c|c|} \hline \text{Rotation} & \text{New Coordinates} \\ \hline 90^\circ \text{ counterclockwise} & (-y, x) \\ 180^\circ & (-x, -y) \\ 270^\circ \text{ counterclockwise} & (y, -x) \\ 360^\circ & (x, y) \\ \hline \end{array} \]

Rotating Shapes

To rotate an entire shape, apply the rotation rules to each vertex:

  1. Identify all vertices of the shape
  2. Apply the rotation transformation to each vertex
  3. Plot the new vertices to create the rotated shape
Shape rotation example

Practice Exercise

Question 1

Rotate triangle ABC with vertices A(2,3), B(5,3), C(3,6) 90° counterclockwise about the origin. What are the new coordinates?

Applying the 90° rotation rule (x,y) → (-y,x):

A(2,3) → A'(-3,2)

B(5,3) → B'(-3,5)

C(3,6) → C'(-6,3)

Rotation solution

Question 2

A rectangle has vertices at (1,1), (4,1), (4,3), and (1,3). Rotate it 180° about the origin. What are the new coordinates?

Applying the 180° rotation rule (x,y) → (-x,-y):

(1,1) → (-1,-1)

(4,1) → (-4,-1)

(4,3) → (-4,-3)

(1,3) → (-1,-3)

Verifying Congruence Through Rotation

Verification Process

To verify if two shapes are congruent through rotation:

  1. Identify corresponding vertices between the two shapes
  2. Calculate the angle of rotation needed to align them
  3. Apply the rotation to one shape and check if it matches the other
  4. Verify that all corresponding sides and angles are equal
Congruence verification process

Multiple Examples

Let's examine several examples of shapes and their rotated versions to verify congruence:

Example 1: Triangles

Congruent triangles

Triangle ABC can be rotated 90° to match triangle DEF. All corresponding sides and angles are equal.

Example 2: Quadrilaterals

Congruent quadrilaterals

Quadrilateral PQRS is a 180° rotation of quadrilateral WXYZ. They are congruent.

Example 3: Complex Shapes

Congruent complex shapes

These irregular shapes are congruent through a combination of rotation and translation.

Practice Exercise

Question 1

Are these two shapes congruent? If so, describe the rotation that would make them match.

Verification problem 1

Yes, they are congruent. Shape B is a 90° clockwise rotation of Shape A.

Verification solution 1

Question 2

Verify if these triangles are congruent through rotation:

Verification problem 2

They are not congruent. While they appear similar, the side lengths don't match exactly.

Question 3

Determine if these quadrilaterals are congruent and describe the transformation:

Verification problem 3

They are congruent. Quadrilateral EFGH is a 270° counterclockwise rotation of ABCD.

Ready for more challenges?

Full Practice Set

Math Challenge

Timed Test in 7 Levels

Test your understanding of congruent shapes through progressively challenging levels