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:

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

Rotation (Turn)
Turning a shape around a fixed point

Reflection (Flip)
Creating a mirror image across a line

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 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:
- Identify all vertices of the shape
- Apply the rotation transformation to each vertex
- Plot the new vertices to create the rotated shape

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)

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:
- Identify corresponding vertices between the two shapes
- Calculate the angle of rotation needed to align them
- Apply the rotation to one shape and check if it matches the other
- Verify that all corresponding sides and angles are equal

Multiple Examples
Let's examine several examples of shapes and their rotated versions to verify congruence:
Example 1: Triangles

Triangle ABC can be rotated 90° to match triangle DEF. All corresponding sides and angles are equal.
Example 2: Quadrilaterals

Quadrilateral PQRS is a 180° rotation of quadrilateral WXYZ. They are congruent.
Example 3: 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.

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

Question 2
Verify if these triangles are congruent through rotation:

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:

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