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Symmetry and
Congruence

Use symmetry and folding to reason about shapes that match exactly.

ISEE Lower Level: Quantitative Reasoning

What is a Line of Symmetry?

A line of symmetry is a line that splits a shape into two halves that are exact mirror images. If you fold the shape along that line, the two halves land perfectly on top of each other.

fold line halves match

What Does Congruent Mean?

Two shapes are congruent when they are exactly the same size and shape. They may be turned or flipped, but if one could be placed perfectly on top of the other, they are congruent. We write this as \( A \cong B \).

Shape A ≅ Shape B

Why Does This Matter?

On the test, you will see shapes with fold lines or mirror images. Knowing how folding works lets you decide which parts of a shape are congruent without measuring everything.

Folding
Mentally fold a shape to see if two parts overlap exactly.
Matching
If halves overlap, they are congruent.
Reasoning
Use one fact to find another without doing all the work.

The Folding Rule

left half right half line of symmetry

Counting Lines of Symmetry

Some shapes have more than one line of symmetry. A square has 4, a rectangle has 2, and many shapes have just 1 or 0.

Square: 4 lines Rectangle: 2 lines Triangle: 1 line

Symmetry vs. Congruence

These two ideas are connected but not the same. Symmetry is about one shape folding onto itself. Congruence is about two shapes being exactly equal.

Symmetry
A property of one shape. A fold line makes its own halves match.
Congruence
A relationship between two shapes that are the same size and shape, written \( A \cong B \).

How to Use Folding to Reason

1
Find the fold line. Look for the dashed line of symmetry in the figure.
2
Imagine folding. Picture one side flipping over the line onto the other.
3
Match the parts. Points and sides that land on top of each other are congruent.
4
Copy the measurement. A side or angle equals its matching partner on the other half.

Example 1: Finding a Length

A shape folds along the dashed line. The left side measures \( 6 \) cm. How long is the matching right side?

\(6\) cm ?

Example 2: Matching Angles

In this symmetric figure, the bottom-left angle is \( 50^\circ \). What is the bottom-right angle?

\(50^\circ\) ?

Example 3: Half a Shape

A folded card forms a triangle. When opened, the whole shape has area \( 24 \) square cm. What is the area of just one half?

half half

Common Mistakes

✕ Any line works
Not every line splits a shape into equal halves. Only true fold lines do.
✕ Flips break it
Thinking a flipped shape is not congruent. A mirror image is still congruent.
✕ Forgetting to halve
Giving the whole area when the question asks for one half only.

Quick Summary

Symmetry Congruence
AboutOne shapeTwo shapes
Key ideaFolds onto itselfSame size & shape
ToolFold linePlace one on the other
ResultTwo equal halves\( A \cong B \)

Question 1 of 5

How many lines of symmetry does a square have?

Question 2 of 5

A symmetric shape folds along a line. The left half has a side of \( 7 \) cm. The matching right side must be:

Question 3 of 5

Two shapes are congruent. Which statement is always true?

Question 4 of 5

A shape with area \( 30 \) square cm folds into two congruent halves along its line of symmetry. What is the area of one half?

Question 5 of 5

In a symmetric figure, the left angle is \( 65^\circ \). After folding along the line of symmetry, the matching right angle is:

Let's Recap

1
A line of symmetry folds a shape into two matching halves.
2
Those halves are congruent: same size and shape.
3
Use folding to copy a side or angle onto its matching partner.
4
When asked for one half, divide the whole by 2.

Great Work!

You can now use symmetry and folding to reason about congruent shapes.