Triangles [Congruency in Triangles]

209. The ASA Match · Two angles and the included side lock the triangle

Dragging A or B changes the base, but the triangles stay perfectly congruent because the angles are locked.

CDEFα = 60°α = 60°β = 45°β = 45°α = 60°α = 60°β = 45°β = 45°AB = 200AB = 200DE = 200DE = 200AB
By the ASA rule (Angle–Side–Angle), if two angles and the included side (the side between them) of one triangle equal those of another, the triangles are congruent. The two angles fix the directions of the other sides and the included side fixes where they meet, so the whole triangle is determined.

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Selina ICSE: Triangles [Congruency in Triangles]

What this lesson covers

Try to break it

Drag A or B along the base line. The base AB stretches, but the angles at A and B are pinned — C slides automatically to keep them at their fixed values. With matching angles and the included side, the triangles stay congruent at every base length.

How you build it

Build two triangles sharing the same two angles and the included side to see the ASA match.

  • Mark point A as the left endpoint of the first base.
  • Mark point B as the right endpoint of the first base.
  • Draw the base segment AB of the first triangle.
  • Mark point D as the left endpoint of the second base.
  • Mark point E as the right endpoint of the second base.
  • Draw the base segment DE equal in length to AB.
  • Construct angle A at vertex A with the given measure.
  • Construct angle D at vertex D with the same measure as angle A.
  • Construct angle B at vertex B with the given measure.
  • Construct angle E at vertex E with the same measure as angle B.
  • Mark point C where the sides of angles A and B meet.
  • Mark point F where the sides of angles D and E meet.

The proof, step by step

Prove that two angles and the included side make the triangles congruent (ASA).

  • Given: In ΔABC and ΔDEF, ∠A = ∠D, ∠B = ∠E, and AB = DE.
  • By the angle sum property of triangles, ∠C = 180° - (∠A + ∠B) and ∠F = 180° - (∠D + ∠E).
  • Since ∠A = ∠D and ∠B = ∠E, it follows that ∠C = ∠F.
  • We now have ∠A = ∠D, AB = DE, and ∠B = ∠E. Therefore, ΔABC ≅ ΔDEF by ASA criterion.

Worked example

In ΔPQR and ΔXYZ, ∠P = ∠X = 50°, ∠Q = ∠Y = 70°, and PQ = XY = 8 cm. Are the triangles congruent? If so, by which criterion?

Two angles and the included side are equal, so by ASA criterion, the triangles are congruent.

  • SAS (Side-Angle-Side)
  • ASA (Angle-Side-Angle) — correct
  • AAS (Angle-Angle-Side)
  • SSS (Side-Side-Side)
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