This chapter is about congruence — figures that are exact copies of each other, having the same shape and size, so that one can be superimposed exactly on the other (after rotating or flipping if needed). The focus is on congruence of triangles and the minimum measurements needed to guarantee it. You discover four sufficient conditions for general triangles — SSS (three sides), SAS (two sides and the included angle), ASA (two angles and the included side) and AAS (two angles and a non-included side) — and one special condition for right triangles, RHS (right angle, hypotenuse and one side). Importantly, SSA (two sides and a non-included angle) does not guarantee congruence, since two different triangles can fit those measurements. The correct correspondence of vertices is written carefully, e.g. \(\triangle ABC \cong \triangle XYZ\). Congruence is then used as a tool to prove properties: in an isosceles triangle the angles opposite the equal sides are equal, and every angle of an equilateral triangle is \(60^\circ\).
Congruent figures and superimpositionSSS and SAS conditionsASA and AAS conditionsRHS condition and the SSA cautionAngles of isosceles and equilateral triangles
Key terms
Congruent figures
Figures of the same shape and size that can be superimposed exactly on one another.
SSS condition
Two triangles are congruent if their three pairs of sides are equal.
SAS condition
Congruence from two sides and the included angle being equal.
ASA condition
Congruence from two angles and the included side being equal.
AAS condition
Congruence from two angles and a non-included side being equal.
RHS condition
For right triangles: congruence from the right angle, hypotenuse and one side.
Hypotenuse
The side opposite the right angle in a right-angled triangle.
Extra questions & answers
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Two figures are congruent if they have:
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