Same shape, same size — and the three measurements that pin a triangle down for good.
Point 6 of 6 in this lesson: 15.3.6 Using congruence to prove things equal
Translation, back in Stage 14, slid a figure across the page without changing it — and turning it or flipping it doesn't change it either. Two figures you can lay perfectly on top of each other by sliding, turning, and flipping are congruent: identical twins. For triangles there is a wonderful shortcut. You don't have to check all six measurements — three sides and three angles — to be sure two triangles match. Because a triangle is rigid, the right three measurements already force the other three. Those shortcuts — SSS, SAS, ASA, AAS — are the most useful tools in all of geometry, the everyday machinery for proving two segments or two angles equal.
Two figures are congruent if one can be moved onto the other by a rigid motion — a slide (translation), a turn (rotation), or a flip (reflection) — so that they land exactly on top of each other, corner on corner and edge on edge. A rigid motion never stretches, shrinks, or bends; it only moves. So congruent figures have exactly the same size and exactly the same shape. The symbol is ≅: read △ABC ≅ △DEF as “triangle ABC is congruent to triangle DEF.”
That little symbol carries a lot of hidden information. The order of the letters tells you the matching. Writing △ABC ≅ △DEF promises that A lands on D, B on E, and C on F. Get the order wrong and you are claiming the wrong corners match — so always line the letters up carefully.
Congruent = same shape and same size, related by a slide, a turn, or a flip. In △ABC ≅ △DEF the order is the dictionary: A↔D, B↔E, C↔F.
Once you know two triangles are congruent, you know a great deal more for free. Every pair of matching parts is equal. Reading the matches straight off △ABC ≅ △DEF:
| Matching sides (equal) | Matching angles (equal) | ||
|---|---|---|---|
| AB | = DE | ∠A | = ∠D |
| BC | = EF | ∠B | = ∠E |
| CA | = FD | ∠C | = ∠F |
Notice how the side AB matches DE because A matches D and B matches E. The same reading gives every angle pair. This is so useful in proofs that it has a name you'll meet again and again: “matching parts of congruent triangles are equal.” It is the payoff move — once the triangles are congruent, you simply read off the equal part you wanted.
If △PQR ≅ △STU and PQ = 6, then ST = 6 (matching sides). If ∠R = 40°, then ∠U = 40° (matching angles), because R↔U in the order of the symbol.
Here is the rigidity of the triangle cashed in as a test. If the three sides of one triangle equal the three sides of another, the triangles are congruent. Fix three side lengths and there is essentially only one triangle you can build from them — flip it or turn it, but the shape is locked. We write the test as SSS (side–side–side).
You actually saw why in the construction lesson (15.2). To build a triangle from three given lengths, you lay one of them as a base, then swing one compass arc of each remaining length from the two ends. The two arcs cross at exactly one point above the base — so the apex has only one place to be, and the triangle is determined.
Three pairs of equal sides force congruence: AB=DE, BC=EF, CA=FD ⇒ △ABC ≅ △DEF.
You don't always need all three sides. Two sides and the angle between them are enough. If two sides and the included angle of one triangle equal the matching two sides and included angle of another, the triangles are congruent. This is SAS (side–angle–side). The word included is the whole game: the angle must be the one tucked between the two named sides, like the hinge between two arms of a pair of scissors. Fix the two arms and the angle of the hinge, and the far ends have nowhere else to go.
If the angle is not between the two sides, the test fails. With two sides and a non-included angle (“SSA”), the side opposite the angle can swing to two different landing spots — giving two different triangles. Two sides and a stray angle are simply not enough.
If you know two angles and one side, you've also got enough — and there are two flavors, depending on where the side sits.
Three equal angles fix the shape but not the size: you can blow a triangle up or shrink it down and keep all three angles. Same shape, different size, is called similar — a later topic — but it is not congruent. So AAA is not a congruence test.
Here are the four good tests at a glance — and the two famous traps.
| Test | What's equal | Works? |
|---|---|---|
| SSS | all three sides | yes ✓ |
| SAS | two sides + the included angle | yes ✓ |
| ASA | two angles + the side between them | yes ✓ |
| AAS | two angles + a side not between | yes ✓ |
| SSA | two sides + a non-included angle | no ✗ |
| AAA | all three angles | no ✗ |
Now the payoff. Congruence is rarely the goal of a proof — it is the tool. The workhorse move of school geometry is this:
To prove two segments (or two angles) are equal, find a pair of congruent triangles that contain them, prove the congruence with SSS / SAS / ASA / AAS, then read off the matching parts.
Watch it work on a kite. In the figure, △ABD and △ACD share the slanted side AD down the middle, with AB = AC and BD = CD given. We want to prove ∠B = ∠C. A proof is just the given, step by step, each line backed by a reason — so we write it as “Statement — Reason” and step through it below.
The same three moves — given, name the test, conclude, read off the matching part — drive almost every triangle proof you will ever write. They are the reason congruence is the centerpiece of this whole stage.
Congruent (≅) = same shape and size, joined by a slide, turn, or flip. The order of the letters names the matching corners, and once triangles are congruent, all six matching parts are equal.
You don't need all six to prove congruence. The four valid tests are SSS, SAS (included angle!), ASA, and AAS. The two famous traps are SSA (the swinging side gives two triangles) and AAA (same shape, any size — only similar).
The master move: to prove two segments or angles equal, find congruent triangles, prove them congruent, then read off the matching parts.
Given △ABC ≅ △DEF with AB = 5. Which side of △DEF must also equal 5, and why?
DE = 5. The order of the symbol matches A↔D and B↔E, so side AB matches side DE — and matching sides of congruent triangles are equal.
In △ABC, name the angle included between sides AB and BC.
∠B. The two sides AB and BC meet at vertex B, so ∠B is the angle tucked between them — exactly the angle you'd use in an SAS argument.
Is SSA (two sides and a non-included angle) a valid congruence test? Explain.
No. The side opposite the given angle can swing to two different landing spots, producing two different triangles. Two sides and a stray angle do not pin a triangle down.
Which congruence test uses two angles and the side between them? And which uses two angles and a side not between them?
The side between the angles is ASA; a side not between is AAS. Both work, because the third angle is forced by the 180° angle-sum rule.
Does AAA prove two triangles congruent? If not, what does it prove?
No. Equal angles fix the shape but not the size — you can scale the triangle up or down. AAA proves the triangles are similar, not congruent.
In a figure, AB = AC, ∠BAD = ∠CAD, and AD is shared by △ABD and △ACD. Prove the two triangles are congruent and hence that BD = CD.
AB = AC (given); ∠BAD = ∠CAD (given — the included angle); AD = AD (common side). By SAS, △ABD ≅ △ACD. Therefore BD = CD (matching parts of congruent triangles).
Six questions to lock it in. Tap the answer you think is right.
This lesson turns rigid motion into a working tool. The big idea is that a triangle is so rigid that the right three measurements — not all six — already determine it, which is why SSS, SAS, ASA, and AAS are congruence tests. Encourage your learner to say the proof out loud: state the given, name the test, write the one-line conclusion, then read off the matching part. That “Statement — Reason” habit is the heart of school geometry.
The misconception to watch for has two heads. First, treating SSA or AAA as valid tests — they are not, and the “swinging side” and the “same shape, different size” figures above are worth lingering on. Second, ignoring the correspondence order when reading matching parts: △ABC ≅ △DEF means AB = DE, never AB = DF. Insisting that the letters line up is the cheapest way to prevent later errors.
This material aligns with Common Core 8.G.A.2 (understanding congruence through sequences of rigid motions) and the high-school standards HS G-CO.B.7 and HS G-CO.B.8 (showing that the SSS, SAS, and ASA criteria for triangle congruence follow from the definition of congruence in terms of rigid motions).