Slide and turn △ABC and it lands perfectly on △DEF. Matching sides carry matching ticks; matching angles carry matching arcs — △ABC ≅ △DEF, a perfect overlay.
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.
15.3.1 What congruence means
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.
Key idea
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.
Drag the slider to slide-and-turn △DEF onto △ABC. At the end the two coincide and the ≅ banner lights up.
Try it Overlay — lay one triangle on the other
Move the slider from 0% to 100% and watch the moving copy slide and turn until it lands on the fixed triangle.
Move the copy
15.3.2 Matching sides and matching angles are equal
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.
Example
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.
15.3.3 Side-Side-Side (SSS)
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.
SSS: matching single, double, and triple ticks show all three sides equal, so the two triangles are congruent. The dashed compass arcs show the apex has only one possible spot.
Key idea — SSS
Three pairs of equal sides force congruence: AB=DE, BC=EF, CA=FD ⇒ △ABC ≅ △DEF.
15.3.4 Side-Angle-Side (SAS)
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.
SAS: the marked angle sits between the two marked sides. That included angle is the hinge — open it the same amount with the same two arms and the triangles must match.
Watch out — SSA is not a test
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.
The SSA trap: same angle at the left, same long side, same swinging side — but it can close at two different points. SSA does not pin the triangle down.
15.3.5 ASA and AAS
If you know two angles and one side, you've also got enough — and there are two flavors, depending on where the side sits.
ASA (angle–side–angle): the side lies between the two angles. Fix the base and the two angles rising from its ends, and the two arms must meet at exactly one point.
AAS (angle–angle–side): the side is not between the two angles. This still works, because the third angle is forced: once two angles are fixed, the third is 180° − (the other two) by the angle-sum rule (15.1). So AAS quietly becomes ASA, and the triangle is determined.
ASA (left): the known side is between the two known angles. AAS (right): the side is off to the side — but the third angle is forced, so it works too.
Watch out — AAA is not congruence
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.
Both triangles have the same three angles, yet one is bigger. AAA only guarantees the same shape — they are similar, not congruent.
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 ✗
Try it Which test proves it?
Look at which parts are marked equal, then pick the test that pins the triangle down. Two of the cards are traps.
Case1
15.3.6 Using congruence to prove things equal
Now the payoff. Congruence is rarely the goal of a proof — it is the tool. The workhorse move of school geometry is this:
The master move
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.
Step the proof forward one line at a time. Each part lights up in the kite as its line appears, ending with the green conclusion ∠B = ∠C.
Try it A guided congruence proof
Advance the step counter to reveal each “Statement — Reason” line and see which part of the kite it justifies.
Reveal step0
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.
★ Recap
What to carry away
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.
✎ Exercises
Given △ABC ≅ △DEF with AB = 5. Which side of △DEF must also equal 5, and why?
Answer
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.
Answer
∠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.
Answer
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?
Answer
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?
Answer
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.
Answer
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).
🎯 Quick check
Six questions to lock it in. Tap the answer you think is right.
§ For teachers and parents
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).
eastmath.com · Stage 15 · 15.3 Congruent Triangles · Reasoning, one step at a time
eastmath.com · 15.3 Congruent Triangles · 15.3.5 ASA and AAS