A carpenter building two identical stable roof trusses needs to be sure they're the same. A seamstress cutting two identical saddle pads must guarantee each piece is exact. But you can't always lay one on top of the other to check. Mathematicians found a shortcut: three tests that guarantee two triangles are congruent — perfectly identical in every way.
Congruent triangles (≅) have the same shape AND the same size. All corresponding sides and angles are equal. If one triangle is a perfect copy of the other (even if rotated or flipped), they're congruent.
The Three Congruence Tests
SSS
Side–Side–Side
All three sides of one triangle equal all three sides of the other.
SAS
Side–Angle–Side
Two sides and the angle between them are equal. (Angle must be included.)
ASA
Angle–Side–Angle
Two angles and the side between them are equal.
Worked Example 1 — Identify the congruence rule
Triangle A has sides 5 cm, 7 cm, 9 cm. Triangle B has sides 9 cm, 5 cm, 7 cm (same three lengths, different order listed).
All three sides match → SSS ≅. They are congruent.
Worked Example 2 — Find a missing value using congruence
Triangles △PQR ≅ △XYZ (by SAS). In △PQR: $PQ = 8$ cm, angle $Q = 54°$, $QR = (2x+1)$ cm. In △XYZ: $YZ = 13$ cm. Since △PQR ≅ △XYZ, corresponding sides are equal: $QR = YZ$.
$$2x + 1 = 13 \quad \mid -1$$
$$2x = 12 \quad \mid \div 2$$
$$x = 6$$
So $QR = 13$ cm.
Warm-Up — Which congruence rule applies? (or none?)
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Problem 1
Two triangular saddle pads: Pad A has sides 30 cm, 45 cm, 50 cm. Pad B has sides 45 cm, 30 cm, 50 cm. Are they congruent? Which rule?
name the rule (or "cannot determine") →
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Problem 2
Two roof trusses: Truss A has a base of 4 m, the angle at the left corner is $62°$, and the left side is 2.8 m. Truss B has a base of 4 m, the angle at the left corner is $62°$, and the left side is 2.8 m. Congruent? Which rule?
name the rule →
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Problem 3
Two art stencils: Stencil A has angles $40°$, $60°$, $80°$. Stencil B has angles $40°$, $60°$, $80°$. Are they congruent? Explain carefully — what's missing?
congruent or not? why? →
From here — write the equation yourself, then solve
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Problem 4
△ABC ≅ △DEF (SSS). In △ABC: $AB = 12$ cm, $BC = (3x - 2)$ cm, $AC = 9$ cm. In △DEF: $EF = 19$ cm. Find $x$ and the length $BC$.
write equation → solve → state BC →
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Problem 5
△KLM ≅ △PQR (ASA). The angle at $L$ in △KLM is $(4x + 8)°$. The corresponding angle at $Q$ in △PQR is $68°$. Find $x$. What is the angle at $L$?
write equation → solve → check →
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Problem 6
Two congruent triangular fence panels (SAS): the included angle in Panel A is $(5x - 10)°$ and in Panel B is $(2x + 20)°$. Find $x$ and the angle. Is this a right angle? An obtuse angle?
write equation → solve → classify →
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Problem 7 Challenge
△ABC ≅ △DEF. You know: $AB = (2x + 3)$ cm, $DE = (x + 10)$ cm, $BC = (y + 5)$ cm, $EF = (3y - 7)$ cm. Find $x$ and $y$, then find all six side lengths (assuming $AC = DF = 8$ cm).
two equations → solve each → list all sides →
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Problem 8 Open
A carpenter has three wooden sticks: 60 cm, 80 cm, and 100 cm. She builds a triangular brace. Can she build a second congruent brace? What's the minimum information a second carpenter would need to guarantee an identical brace? Write your answer as a clear rule.
think → write rule → justify →
Show answers
Problem 1
Yes, congruent by SSS — same three side lengths (order listed doesn't matter)
Problem 2
Yes, congruent by SAS — two sides (4 m, 2.8 m) and the included angle ($62°$) match
Problem 3
Not necessarily congruent — same angles means same shape (similar) but could be different sizes. We have AAA, not a congruence case. We need at least one side length.
Problem 4
$3x - 2 = 19$ → $3x = 21$ → $x = 7$ · $BC = 19$ cm
Problem 5
$4x + 8 = 68$ → $4x = 60$ → $x = 15$ · Angle at $L = 68°$
Problem 6
$5x - 10 = 2x + 20$ → $3x = 30$ → $x = 10$ · Angle $= 5(10)-10 = 40°$ · Acute angle
Problem 7
$x$: $2x+3 = x+10$ → $x = 7$ · $y$: $y+5 = 3y-7$ → $12 = 2y$ → $y = 6$ · Sides: $AB=DE=17$ cm, $BC=EF=11$ cm, $AC=DF=8$ cm
Problem 8
Yes — by SSS: the three lengths 60, 80, 100 cm uniquely determine the triangle. The minimum info needed is all three side lengths (SSS), OR two sides and the included angle (SAS), OR two angles and the included side (ASA).
Coming up next → Lesson 15: Similar Triangles
Congruent = same shape, same size. But what if you only need the same shape? Triangles can be scaled up or down while keeping their proportions perfectly intact. These are similar triangles — and they're the key to understanding why slope never changes along a straight line.