SSS Congruence

The **SSS (Side-Side-Side) Congruence Criterion** says: if the three sides of one triangle are congruent to the three sides of another triangle, then the two triangles are congruent. $$\text{If } \overline{AB} \cong \overline{DE}, \quad \overline{BC} \cong \overline{EF}, \quad \overline{CA} \cong \overline{FD}$$ $$\text{Then } \triangle ABC \cong \triangle DEF$$ You do not need to measure a single angle. Three side lengths fix a triangle completely: this is called **triangle rigidity**. Once the three sides are chosen, the angles are forced, so every triangle with those sides is the same triangle, only moved, turned or flipped. **Matching the sides:** the order of the letters in $\triangle ABC \cong \triangle DEF$ tells you which parts correspond: $A \leftrightarrow D$, $B \leftrightarrow E$, $C \leftrightarrow F$. Pair the shortest side with the shortest, the middle with the middle and the longest with the longest, then name the vertices in the matching order. On a coordinate grid you find each side with the **distance formula**: $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

Checking SSS with Given Lengths

In $\triangle ABC$: $AB = 5$ cm, $BC = 7$ cm, $CA = 6$ cm. In $\triangle DEF$: $DE = 5$ cm, $EF = 7$ cm, $FD = 6$ cm. Are the triangles congruent?

Compare the first pair of sides: $AB = DE = 5$ cm = $\overline{AB} \cong \overline{DE}$

Compare the second pair: $BC = EF = 7$ cm = $\overline{BC} \cong \overline{EF}$

Compare the third pair: $CA = FD = 6$ cm = $\overline{CA} \cong \overline{FD}$

Apply SSS: All three pairs of sides are congruent = $\triangle ABC \cong \triangle DEF$ by SSS

Answer: Yes, $\triangle ABC \cong \triangle DEF$ by SSS. No angles were needed.

Writing the Congruence Statement in the Right Order

In $\triangle PQR$: $PQ = 8$, $QR = 11$, $RP = 9$. In $\triangle XYZ$: $XY = 9$, $YZ = 8$, $ZX = 11$. Write a correct congruence statement.

Pair the sides by length: $8$: $PQ \leftrightarrow YZ$, $\quad 9$: $RP \leftrightarrow XY$, $\quad 11$: $QR \leftrightarrow ZX$ = Three equal pairs, so SSS applies

Find the partner of P: $P$ is where the sides $8$ and $9$ meet. In $\triangle XYZ$ the sides $8$ ($YZ$) and $9$ ($XY$) meet at $Y$ = $P \leftrightarrow Y$

Find the partners of Q and R: $Q$: sides $8$ and $11$ meet, in $\triangle XYZ$ that is $Z$. $R$: sides $9$ and $11$ meet, that is $X$ = $Q \leftrightarrow Z$, $R \leftrightarrow X$

Write the statement in matching order: $P, Q, R$ go with $Y, Z, X$ = $\triangle PQR \cong \triangle YZX$

Answer: $\triangle PQR \cong \triangle YZX$ by SSS. Writing $\triangle PQR \cong \triangle XYZ$ would be wrong, because $P$ does not correspond to $X$.

SSS on the Coordinate Grid

Prove that $\triangle ABC$ with $A(1, 1)$, $B(5, 3)$, $C(2, 6)$ is congruent to $\triangle DEF$ with $D(-7, -1)$, $E(-5, -5)$, $F(-2, -2)$.

Find AB and DE with the distance formula: $AB = \sqrt{4^2 + 2^2} = \sqrt{20}$, $\quad DE = \sqrt{2^2 + (-4)^2} = \sqrt{20}$ = $AB = DE = \sqrt{20} = 2\sqrt{5}$

Find BC and EF: $BC = \sqrt{(-3)^2 + 3^2} = \sqrt{18}$, $\quad EF = \sqrt{3^2 + 3^2} = \sqrt{18}$ = $BC = EF = \sqrt{18} = 3\sqrt{2}$

Find CA and FD: $CA = \sqrt{(-1)^2 + (-5)^2} = \sqrt{26}$, $\quad FD = \sqrt{(-5)^2 + 1^2} = \sqrt{26}$ = $CA = FD = \sqrt{26}$

Apply SSS: Three pairs of equal sides = $\triangle ABC \cong \triangle DEF$ by SSS

Answer: All three pairs of sides are equal ($\sqrt{20}$, $\sqrt{18}$, $\sqrt{26}$), so $\triangle ABC \cong \triangle DEF$ by SSS. Triangle $DEF$ is triangle $ABC$ turned by $90°$ and moved.

Two-Column Proof with a Shared Side

In the kite $ABCD$, $AB = AD$ and $CB = CD$. The diagonal $AC$ is drawn. Prove that $\triangle ABC \cong \triangle ADC$ and that $\angle B \cong \angle D$.

Statement 1 and its reason: $\overline{AB} \cong \overline{AD}$ = Reason: given

Statement 2 and its reason: $\overline{CB} \cong \overline{CD}$ = Reason: given

Statement 3: the shared side: $\overline{AC} \cong \overline{AC}$ = Reason: reflexive property (a side is congruent to itself)

Statement 4: the triangles: $\triangle ABC \cong \triangle ADC$ = Reason: SSS (statements 1, 2, 3)

Statement 5: the angles: $\angle ABC \cong \angle ADC$ = Reason: CPCTC

Answer: $\triangle ABC \cong \triangle ADC$ by SSS, and then $\angle B \cong \angle D$ by CPCTC. The shared side $AC$ is the third pair of sides.

Using Algebra with SSS

$\triangle ABC \cong \triangle DEF$ by SSS. $AB = 2x + 1$, $DE = 11$, $BC = 3y - 2$, $EF = 16$ and $CA = FD = 9$. Find $x$, $y$ and the perimeter of $\triangle ABC$.

Corresponding sides are equal: AB = DE: $2x + 1 = 11 \Rightarrow 2x = 10$ = $x = 5$

BC = EF: $3y - 2 = 16 \Rightarrow 3y = 18$ = $y = 6$

Write the three sides of ABC: $AB = 11$, $BC = 16$, $CA = 9$ = Sides 11, 16 and 9

Add the sides: $11 + 16 + 9$ = Perimeter $= 36$

Answer: $x = 5$, $y = 6$, and the perimeter of $\triangle ABC$ is $36$.

SSS is the criterion you can check with nothing but a ruler, and it explains why triangles are everywhere in building: - **Bridges and roofs**: a triangle made of three fixed beams cannot change shape, so trusses are built from triangles. A square frame with hinged corners collapses into a rhombus, a triangle never does - **Manufacturing**: to check that a triangular part matches the template, workers measure the three sides only - **Maps and surveying**: with coordinates you can prove two triangular plots are identical just by computing distances - **Proofs**: SSS is often the first step of a proof, and then **CPCTC** (Corresponding Parts of Congruent Triangles are Congruent) gives you equal angles for free In the lessons on congruent triangles and on ASA and AAS you met the other criteria. SSS is the only one that needs no angle at all.

Mistake: Writing the vertices in the wrong order, like $\triangle PQR \cong \triangle XYZ$ when $P$ matches $Y$

Why: A congruence statement is a promise that the first letters correspond, the second letters correspond and the third letters correspond. Alphabetical order is not the same as matching order.

Correct: Pair the sides by length first, then find which vertices the matching sides meet at. Write the vertices in that order: $\triangle PQR \cong \triangle YZX$.

Mistake: Thinking SSS also works for quadrilaterals

Why: A square with sides 4 and a thin rhombus with sides 4 have four equal pairs of sides, but they are not congruent. Four sides do not fix the angles.

Correct: Side lengths alone only fix the shape of a triangle. For quadrilaterals you also need angles or a diagonal.

Mistake: Forgetting the shared side in a proof

Why: When two triangles share a side, students often find only two pairs of equal sides and stop, or they use the side without a reason.

Correct: Write the shared side as its own statement, $\overline{AC} \cong \overline{AC}$, with the reason reflexive property. It counts as the third pair.

Mistake: Deciding congruence from equal perimeters

Why: Triangles with sides 3, 4, 5 and 4, 4, 4 both have perimeter 12, but they have different shapes.

Correct: SSS needs three separate pairs of equal sides, not just the same total.

Roof Trusses

Carpenters build roofs from identical triangles because a triangle with fixed sides cannot be pushed out of shape.

Every truss of a garden shed is made from beams of 3 m, 4 m and 5 m. By SSS all trusses are congruent, so the roof is the same along its whole length.

Fields on a Map Grid

Planners use coordinates and the distance formula to compare plots of land without visiting them.

Field 1 has corners $(0, 0)$, $(6, 0)$ and $(0, 8)$, field 2 has corners $(10, 2)$, $(10, 8)$ and $(18, 2)$. Both have sides 6, 8 and 10 grid units, so they are congruent by SSS.

Quality Control of Tiles

A tile factory checks triangular tiles against a template by measuring the three sides.

The template has sides 12 cm, 15 cm and 20 cm. A tile with sides 20 cm, 12 cm and 15 cm passes: the order of measuring does not matter, only the three pairs.

**SSS**: if three sides of one triangle are congruent to three sides of another, the triangles are congruent

No angles are needed: three side lengths fix a triangle completely (triangle rigidity)

Pair shortest with shortest, middle with middle, longest with longest, then write the vertices in matching order

On a coordinate grid, find each side with the distance formula $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

A side shared by two triangles is congruent to itself (reflexive property) and counts as one pair

After SSS, CPCTC gives the matching angles; SSS does not work for quadrilaterals

Q: Why does SSS work without any angles?

A: Try to build a triangle from three sticks of fixed length. Once the ends are joined there is only one shape, you cannot open or close the angles. Every triangle with those three sides is the same triangle, maybe turned or flipped. That is why three sides are enough.

Q: Why does SSS not work for quadrilaterals?

A: Four sticks joined at the corners can still move: a square with sides 4 can be pushed into a rhombus with sides 4. The sides stay the same, the angles change. Adding one diagonal splits the shape into two triangles and makes it rigid.

Q: Does the order of the sides matter?

A: Not for deciding congruence: sides 5, 6, 8 and 8, 5, 6 give congruent triangles. It does matter for the congruence statement: the vertex letters must be written in the matching order.

Q: What is the difference between SSS congruence and SSS similarity?

A: SSS congruence needs three pairs of equal sides, so the triangles have the same size. SSS similarity only needs the three ratios of corresponding sides to be equal, so one triangle can be a scaled copy of the other.

Exit Presentation

SSS Congruence

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Geometry / Congruence & Similarity

SSS Congruence

Learn to prove triangles congruent with the SSS (Side-Side-Side) criterion, match corresponding sides and check SSS on a coordinate grid.

Intermediate25 min

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