Congruent Triangles

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Congruent Triangles: meaning, method and solved example (Geometry) – Maths PDF

In short: Congruent triangles have exactly the same shape and size, so their three pairs of matching sides and three pairs of matching angles are all equal.

GeometryGeometryG-CO.B.7

What is Congruent Triangles?

Congruent triangles are identical twins in geometry: one can be moved onto the other by sliding, flipping or turning, and it fits perfectly. Matching parts are called corresponding, and they are equal.

The notation \(\triangle ABC \cong \triangle DEF\) lists vertices in matching order, so \(A\) pairs with \(D\), \(B\) with \(E\) and \(C\) with \(F\). From that single statement you can read off \(AB = DE\), \(BC = EF\), \(AC = DF\), and \(\angle A = \angle D\), \(\angle B = \angle E\), \(\angle C = \angle F\).

How to spot it

  • A statement uses the symbol \(\cong\) between two triangle names, such as \(\triangle ABC \cong \triangle DEF\).
  • A figure has tick marks and arcs that pair up sides and angles of two triangles.
  • An unknown side or angle in one triangle must be found from the matching part of the other.
  • Two roof frames, bridge panels or tiles must be shown to be congruent triangles before a part is ordered.

The method

  1. Read the vertex order in the statement to pair up the corresponding sides and angles.
  2. Write an equation for each pair that involves an unknown, such as \(5x – 4 = 3x + 8\).
  3. Solve each equation, keeping the work for every variable separate.
  4. Substitute back to find the requested lengths or angle measures, and test that the matching pairs agree.
ABCDEFAB = DEBC = EFAC = DF
Same color means corresponding sides: each pair has equal length.

Congruent Triangles: solved example

Problem

It is known that \(\triangle ABC \cong \triangle DEF\), with \(AB = 5x – 4\), \(DE = 3x + 8\), \(m\angle C = (6y + 2)^{\circ}\) and \(m\angle F = 98^{\circ}\). Find \(x\), \(y\) and \(AB\).

Solution

\(AB\) corresponds to \(DE\), so \(5x – 4 = 3x + 8\), which gives \(2x = 12\) and \(x = 6\). Then \(AB = 5(6) – 4 = 26\), and \(DE = 3(6) + 8 = 26\) agrees.

\(\angle C\) corresponds to \(\angle F\), so \(6y + 2 = 98\) and \(y = 16\). Results: \(x = 6\), \(y = 16\) and \(AB = 26\).

Try it

Question 1

If \(\triangle JKL \cong \triangle MNP\), which side matches \(\overline{KL}\)?

Show the answer

\(K\) pairs with \(N\) and \(L\) with \(P\), so \(\overline{KL}\) matches \(\overline{NP}\).

Question 2

Triangle \(RST\) is congruent to triangle \(UVW\), and \(RS = 15\) cm, \(ST = 11\) cm, \(RT = 9\) cm. What is the perimeter of \(UVW\)?

Show the answer

Matching sides are equal, so the perimeter is \(15 + 11 + 9 = 35\) cm.

Congruent Triangles vs. Similar Triangles

Similar triangles share angle measures but may differ in size; their sides are in proportion. These triangles share angles and side lengths, so the scale factor is exactly 1. A 3-4-5 triangle and a 6-8-10 triangle are similar, yet only copies of the same 3-4-5 triangle are congruent.

Similar Triangles

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