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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.

tusaale · triangle 3 4 5

Geometry · Triangles

Triangle 3 4 5

3,\ 4,\ 5

tallaabo tallaabo

  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.

Jawaabta muuji
A = 6,\quad P = 12,\quad \angle \approx 36.9^\circ, 53.1^\circ, 90.0^\circ

Haddaad Dooro dhibaato, ama qoraal ama sawirkaaga. Talaab kasta, sawir, jawaabta qarsoon ilaa aad weydiiso.

Ku hay shaqadaada

A xisaab bilaash ah oo ku daray qoraalada waxbarasho kasta, diiwaanka waxa aad dhamaatay, dhibaatooyinka aad xal u tahay meel kaliya, iyo macallinka aad weydiisan kartaa oo ku saabsan bogga. Xisaabta oo keliya waa u furan yahay qof kasta, la diiwaan geliyey ama aan.

Ku soo biir Ku soo gal

Calaamadaha la isticmaalay halkan

Taab calaamad kasta oo loogu talagalay qeexidda buuxda, sawir, iyo waxa xarfaha oo dhan ka mid ah ka dhigan tahay.

Sida loo: 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.

Su'aalaha dadku waydiiyaan

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.

In ka badan Geometry

TrianglesPythagorean theoremCirclesPolygonsSolidsCoordinate geometryEuclid's axioms and the structure of proofCircle theoremsTransformations: translation, rotation, reflection, dilationSolid geometry: prisms, pyramids, spheres