Congruent Triangles

What this move does

Two triangles are congruent if one of them can be moved by translation, rotation and reflection until it lies exactly on the other. Vertices that land on each other are called corresponding vertices; corresponding sides are equal and corresponding angles are equal.

A triangle has six measurements: three sides and three angles. The congruence criteria say you do not need to check all six. The right three are enough:

SSS SAS ASA AAS HL (right triangles)

To prove two triangles congruent, follow three steps:

Find the correspondence: imagine moving one triangle onto the other. Which vertex lands on which? Write corresponding vertices in the same positions.
Collect three conditions: what you are given, a shared side, vertical angles, equal angles from parallel lines… Put together one of the five criteria above. Watch out: in SAS the angle must be the included angle, between the two sides.
Read off the conclusion in order: the corresponding sides and angles you get from the congruence are read off by letter position.

The workbench above is practice for step 1. Look at the “Direction A→B→C” reading: does $A \to B \to C$ go clockwise or counterclockwise? Translation and rotation never change the direction; only a reflection does.

Predict first

Two triangles are congruent. In one, the vertices $A \to B \to C$ go counterclockwise; in the other, the corresponding vertices $A^{'} \to B^{'} \to C^{'}$ go clockwise. Can you make them coincide using only translations and rotations (no reflection)?

Yes, just rotate enough timesNo, you need one reflectionIt depends on the shape of the triangle

Choose an answer first; then you can see the result.

Work through one

Move △ABC onto △A′B′C′

Step 1 / 4

Check the direction first
The solid triangle $A \to B \to C$ goes counterclockwise; the dashed triangle $A^{'} \to B^{'} \to C^{'}$ goes clockwise. The directions are opposite, so this move needs one reflection.
Reflect
Translate
Rotate

Now an example on paper. Watch the pattern “find the correspondence → collect conditions → read off the conclusion”.

Example: AB = AD and CB = CD. Prove that AC bisects ∠BAD

Step 1 / 3

Find the correspondence
The diagonal $A C$ splits quadrilateral $A B C D$ into $△ A B C$ and $△ A D C$ . Fold along $A C$ and $B$ should land on $D$ : the correspondence is $A \leftrightarrow A$ , $B \leftrightarrow D$ , $C \leftrightarrow C$ .

$△ A B C and △ A D C$
Collect three conditions
Read off the conclusion in order

When it fails

Break it

Counterexample lab: remove one condition

Conclusion: Two triangles with two sides and one angle equal are congruent

Fix $∠ A = 3 0^{\circ}$ and $A B = 6$ , and then choose the other side $A C$ of this angle. Drag the slider: each value of $A C$ gives exactly one triangle.

Practice

Exercise 1Basic

$△ A B C ≅ △ D E F$ , $∠ A = 5 0^{\circ}$ , $∠ E = 7 0^{\circ}$ . Find $∠ C$ in degrees.

Exercise 2Basic

We know $∠ A = 3 0^{\circ}$ , $A B = 6$ and $B C = a$ , where $a$ is a positive integer. For how many values of $a$ are there exactly two (non-congruent) triangles that fit?

Exercise 3Advanced

In $△ A B C$ , $A B = 5$ , $A C = 9$ , and $A D$ is the median to side $B C$ . If the length of $A D$ is an integer, how many values can it take?

Exercise 4Challenge

In $△ A B C$ , $A B = A C$ , and the points $D$ , $E$ lie on side $B C$ with $B D = C E$ . Prove that $A D = A E$ .

What this move does

Work through one

Move △ABC onto △A′B′C′

Example: AB = AD and CB = CD. Prove that AC bisects ∠BAD

When it fails

Counterexample lab: remove one condition

Practice

Mistake Museum · Exhibits from this stop

Treating “side-side-angle” as a congruence criterion

Corresponding vertices not lined up