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Congruent and similar triangles

SSS, SAS, ASA congruence, and similarity by scaling.

Two triangles are congruent when they match exactly (SSS, SAS, ASA or AAS decide it), and similar when one is a scaled copy of the other: equal angles, proportional sides. Picture it: the 3-4-5 and 6-8-10 triangles are the same shape at two sizes; every angle agrees. Think it: similarity is what makes trigonometry possible, the ratios of sides depend only on the angles.

Worked example · triangle 3 4 5

Geometry · Triangles

Triangle 3 4 5

3,\ 4,\ 5

Step by step

  1. a = 3,\ b = 4,\ c = 5

    Three sides (SSS). Check the triangle inequality: each side is less than the sum of the other two. ✓

  2. P = a + b + c = 12

    Perimeter.

  3. s = \tfrac{P}{2} = 6,\quad A = \sqrt{s(s-a)(s-b)(s-c)} = 6

    Heron's formula for the area.

  4. \cos A = \frac{b^2 + c^2 - a^2}{2bc} = \frac{4}{5} \Rightarrow A \approx 36.870^\circ

    Law of cosines for angle A (opposite side a).

  5. \cos B = \frac{a^2 + c^2 - b^2}{2ac} = \frac{3}{5} \Rightarrow B \approx 53.130^\circ

    Law of cosines for angle B (opposite side b).

  6. \cos C = \frac{a^2 + b^2 - c^2}{2ab} = 0 \Rightarrow C \approx 90.000^\circ

    Law of cosines for angle C (opposite side c).

  7. A + B + C = 180.0^\circ

    The angles add to 180°: a right, scalene triangle.

Reveal the answer
A = 6,\quad P = 12,\quad \angle \approx 36.9^\circ, 53.1^\circ, 90.0^\circ

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How to: Congruent and similar triangles

  1. Match the three pieces you know to a congruence rule: SSS, SAS, ASA or AAS (never SSA).
  2. For similarity, check two angles agree, or that all three side ratios are equal.
  3. Write the correspondence of vertices in order (A↔D, B↔E, C↔F) before using any equal parts.

Questions people ask

Why does every triangle have angles adding to 180°?

Draw a line through one vertex parallel to the opposite side: the two alternate angles equal the other two angles of the triangle, and the three angles at the vertex lie on a straight line. That is Euclid's proof, and it is why the fact holds only on a flat plane.

When do I use the law of sines versus the law of cosines?

Cosines when you know three sides, or two sides and the angle between them. Sines when you know an angle and the side opposite it, plus one more piece.

What is the difference between area and perimeter?

Perimeter is the length of the boundary (one dimension, measured in metres); area is the amount of surface inside (two dimensions, square metres). Doubling every side doubles the perimeter but quadruples the area.

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