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Mathematics Triangles
Class 9 Intermediate About 15 min Premium

SSS, SAS, ASA, RHS or not enough?

Twelve pairs of triangles, from a bridge truss to two ladders against a wall. Check what is given and which parts match, and sort each pair by the test that proves it congruent, or by “not enough”.

Core concept

Two triangles are congruent if three sides match (SSS), two sides and the angle between them match (SAS), two angles and a side match (ASA or AAS), or a right angle, the hypotenuse and a side match (RHS); equal angles alone, or two sides with an angle not between them, do not fix one triangle

Topic: Triangles

Congruent triangles have the same shape and the same size: one fits exactly on the other. We do not need to check all six parts (three sides and three angle: The amount of turn between two lines that meet, measured in degrees (°). All key words). Any of these is enough. SSS: the three sides of one equal the three sides of the other. SAS: two sides and the angle between them. ASA: two angles and the side between them; since the third angles must then match too, two angles and any matching side (AAS) also work. RHS: in right-angled triangles, the hypotenuse and one other side. Equal angles alone (AAA) only make triangles similar, and two sides with an angle not between them (SSA) can fit two different triangles. These tests prove many facts: a diagonal cuts a parallelogram into two triangles congruent by ASA, so its opposite sides and opposite angles are equal.

Key words explained

What you will learn

  • State the SSS, SAS, ASA (AAS) and RHS tests of congruence.
  • Decide which test, if any, proves two triangles congruent.
  • Explain why AAA and SSA are not enough.
  • Use congruent triangles to prove properties of a parallelogram.

Before you start

  • Know that the angles of a triangle add up to 180°, and Pythagoras' theorem.

Materials

  • Paper, a ruler, a protractor and a pair of compasses

Equipment

  • Scissors to cut out triangles and lay one on another

Procedure

  1. Pick a pair's card and check what is given and which parts match.
  2. Do this for at least eight pairs.
  3. Sort every pair by its congruence test.
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