Similarity (With Applications to Maps and Models)

315. The Angle-Angle Match · When two angles align, triangles scale perfectly

Ratios of corresponding sides remain equal as you drag A, B, C.

DEFDrag A, B, C to reshape. DEF scales automatically.AB = 200AB = 200BC = 223.6BC = 223.6CA = 223.6CA = 223.6DE = 120DE = 120EF = 134.2EF = 134.2FD = 134.2FD = 134.2∠A = 63°∠A = 63°∠B = 63°∠B = 63°∠C = 53°∠C = 53°∠D = 63°∠D = 63°∠E = 63°∠E = 63°∠F = 53°∠F = 53°ABC
AA (Angle–Angle) similarity: if two angles of one triangle equal two angles of another, the triangles are similar. The third angles then match too, so all corresponding sides share one ratio.

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Selina ICSE: Similarity (With Applications to Maps and Models)

What this lesson covers

Try to break it

Drag A, B, or C to reshape △ABC. △DEF auto-scales to keep all three angles matched (only two are needed, since the third is forced by the 180° angle sum). Once the angles agree, the side ratios fall in line for free — AA is the leanest similarity test.

How you build it

Build two triangles that share the same two angles, and see why AA makes them similar.

  • Point tool: mark A near the lower-left.
  • Point tool: mark B to the right of A — AB is the base of the first triangle.
  • Segment tool: join A to B.
  • Angle tool: click A, click along AB toward B, then set ∠A = 50° with the −/+ buttons — the arm rises toward C and becomes side AC.
  • Angle tool: click B, click toward A, then set ∠B = 60° — its arm rises toward C (side BC) and meets the arm from A.
  • Point tool: mark C where the two arms cross. Triangle ABC is complete (its third angle is 70°).
  • Point tool: mark D somewhere below — the start of a second, different-sized triangle.
  • Point tool: mark E to the right of D — make DE a clearly different length from AB (longer or shorter).
  • Segment tool: join D to E — the base of the second triangle.
  • Angle tool: click D, click along DE toward E, then set ∠D = 50° — exactly matching ∠A. The arm becomes side DF.
  • Angle tool: click E, click toward D, then set ∠E = 60° — matching ∠B. Its arm (side EF) meets the arm from D at F.
  • Point tool: mark F where the two arms cross. △ABC and △DEF share two equal angles (50° and 60°), so by AA they are similar — corresponding sides share one ratio.

The proof, step by step

Prove that two triangles with two equal angles are similar (AA).

  • Given: In ΔABC and ΔDEF, ∠A = ∠D and ∠B = ∠E.
  • The sum of angles in any triangle is 180°. So, ∠C = 180° - (∠A + ∠B) and ∠F = 180° - (∠D + ∠E).
  • Since ∠A=∠D and ∠B=∠E, it follows that ∠C = ∠F. All three corresponding angles are equal.
  • By the AA Similarity Criterion, if two angles of one triangle equal two angles of another, the triangles are similar. Hence, ΔABC ~ ΔDEF.

Worked example

In ΔABC and ΔDEF, ∠A = 50° and ∠B = 60°. In ΔDEF, ∠D = 50° and ∠E = 60°. If AB = 6 cm, DE = 9 cm, and BC = 8 cm, find EF.

Since two angles are equal (∠A=∠D, ∠B=∠E), ΔABC ~ ΔDEF by AA similarity. Thus, AB/DE = BC/EF. Substituting values: 6/9 = 8/EF → EF = (9 × 8)/6 = 12 cm.

  • 10 cm
  • 12 cm — correct
  • 15 cm
  • 6 cm
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