Key takeaway: Congruent triangles are triangles with exactly the same size and shape, so their three pairs of corresponding sides and three pairs of corresponding angles are all equal.
Grade 10 – GeometryG-CO.B.7Geometry and Measurement
Definition of Congruent Triangles
Two triangles are congruent triangles when one can be moved onto the other by a slide, flip or turn so that every part fits exactly. Matching parts, called corresponding parts, then have equal measures: three sides and three angles.
The order of letters tells you which parts match. If \(\triangle JKL \cong \triangle MNP\), then \(J\) matches \(M\), \(K\) matches \(N\) and \(L\) matches \(P\). You rarely need all six facts; shortcuts such as SSS and SAS prove congruence with fewer.
Formula and notation
The symbol \(\cong\) means “is congruent to.” Write the vertices in matching order:
\[ \triangle JKL \cong \triangle MNP \]
This statement says \(JK = MN\), \(KL = NP\), \(JL = MP\), \(\angle J = \angle M\), \(\angle K = \angle N\) and \(\angle L = \angle P\).
Key facts
- Congruent triangles have equal perimeters and equal areas.
- The order of the letters in the congruence statement shows which parts match.
- Tick marks on sides and arcs on angles show which parts are equal.
- Equal angles alone do not give congruence; that only gives similar shapes.
Congruent Triangles example
Given \(\triangle JKL \cong \triangle MNP\), with \(JK = 5y – 4\), \(MN = 2y + 11\), \(m\angle K = (8z)^{\circ}\) and \(m\angle N = (3z + 35)^{\circ}\). Find \(JK\) and \(m\angle K\).
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Corresponding sides are equal: \(5y – 4 = 2y + 11\), so \(3y = 15\) and \(y = 5\).
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Then \(JK = 5(5) – 4 = 21\); check \(MN = 2(5) + 11 = 21\).
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Corresponding angles are equal: \(8z = 3z + 35\), so \(z = 7\) and \(m\angle K = 56^{\circ}\).
\(JK = \) 21 units and \(m\angle K = \) \(56^{\circ}\).
Common mistake
Watch out
Given \(\triangle JKL \cong \triangle MNP\), Aria writes \(JK = NP\) because those are the first side of one and the last side of the other.
Why it is wrong
Match by position in the statement: \(J\) to \(M\), \(K\) to \(N\), \(L\) to \(P\). Side \(JK\) corresponds to side \(MN\), and \(KL\) corresponds to \(NP\). Mixing the letters pairs the wrong sides.
Related terms and lessons
- Triangle Congruence (SSS, SAS, ASA, AAS)
- CPCTC
- Congruent Figures
- Rigid Transformation
- Similar Triangles
- HL Theorem (Hypotenuse-Leg)
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