seegongsik
Grade 5 (age 10-11)

grade 5 Congruence & Symmetry

Two shapes are congruent when they have exactly the same shape and size, so one can be laid on the other with nothing left over. In the first picture the orange triangle is a copy of the blue one: slide it away or turn it, and it still fits perfectly once you bring it back. A shape has line symmetry when folding it along a line makes the two halves match, and that fold line is called the line of symmetry. In the second picture you can move the line and check that matching points always stay the same distance from it.

What is Congruence?

🧩 Slide it, turn it: same shape, same size
①The orange triangle has the same shape and size as the blue one. Set the slide distance to 0 and the turn angle to 0° or 360°, and the two triangles fit exactly
②Sliding, turning or flipping a shape never changes its side lengths or its angles, so the moved shape is always congruent to the original
③Vertices that land on each other are corresponding vertices, and the same goes for corresponding sides and corresponding angles. In congruent shapes, corresponding sides have equal lengths and corresponding angles have equal sizes
50
30°

What is Line Symmetry?

🪞 Fold it and the halves match
①A shape has line symmetry if folding it along a line makes the two halves match exactly, and that line is the line of symmetry. In the picture, the blue part and the orange part together make one symmetric shape
②Each dotted segment joins a pair of matching points. It meets the line of symmetry at a right angle, and the line cuts it into two equal halves
③Moving the line simply carries the whole shape along with it, and matching points still sit the same distance from the line
④A circle folds onto itself along any diameter, so it has infinitely many lines of symmetry
50%

Summary

Congruent shapes
corresponding sides equal, corresponding angles equal
Sliding, turning or flipping a shape leaves it congruent to the original
Line symmetry
matching points are the same distance from the line of symmetry
The segment joining two matching points is perpendicular to the line of symmetry, and the line cuts it in half
🎯 At a glance
①Congruent: the same shape and the same size, so one fits exactly on the other
②In congruent shapes, corresponding sides are equal and corresponding angles are equal
③Line symmetry: folding along a line makes the two halves match, and that line is the line of symmetry
④Matching points are equally far from the line, and the segment joining them is perpendicular to it
⑤Some shapes have one line of symmetry, some have several, and a circle has infinitely many

Examples and Unit Test

Example 1
Triangle ABC is congruent to triangle DEF. Side AB is 5 cm long. How long is the corresponding side DE?
1
If you lay one triangle on the other, side AB lands exactly on side DE.
2
Sides that match up have the same length, so DE is 5 cm as well.
▸ 5 cm
Once you know two shapes are congruent, you can read off corresponding sides and angles without measuring.
Example 2
In a shape with line symmetry, point A is 3 cm from the line of symmetry. How far from the line is the point that matches A?
1
Fold along the line of symmetry and point A lands right on its matching point.
2
So both points are the same distance from the line, and the matching point is 3 cm away too.
▸ 3 cm
The two points are 6 cm apart, twice 3 cm, because the line of symmetry cuts the segment between them in half.
Unit Test
Which shape has infinitely many lines of symmetry?
①equilateral triangle
②rectangle
③circle
④square
▸ ③ circle
1
An equilateral triangle has 3 lines of symmetry, a rectangle has 2 and a square has 4.
2
A circle folds onto itself along any line through its center, so it has infinitely many lines of symmetry.
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