Congruent triangles are identical in shape and size; similar triangles are enlargements of one another. A proof quotes one of the tests below, names the triangles with vertices in corresponding order, then reads off the remaining equal sides, equal angles or side ratios.
Tests for congruent triangles: \(\triangle ABC\equiv\triangle XYZ\)
| Test | What must match | Sketch |
|---|---|---|
| SSS | \(AB=XY\) \(BC=YZ\) \(AC=XZ\) | |
| SAS | \(AB=XY\) \(AC=XZ\) \(\angle BAC=\angle YXZ\) (included angle) | |
| ASA | \(\angle BAC=\angle YXZ\) \(\angle ABC=\angle XYZ\) \(AB=XY\) (included side) | |
| RHS | \(\angle ABC=\angle XYZ=90^\circ\) \(AC=XZ\) (hypotenuse) \(BC=YZ\) |
Tests for similar triangles: \(\triangle ABC\sim\triangle XYZ\)
| Test | What must match |
|---|---|
| AA similarity | \(\angle BAC=\angle YXZ\) \(\angle ABC=\angle XYZ\) |
| SSS similarity | \(\dfrac{XY}{AB}=\dfrac{YZ}{BC}=\dfrac{XZ}{AC}\) |
| SAS similarity | \(\dfrac{XY}{AB}=\dfrac{XZ}{AC}\) \(\angle BAC=\angle YXZ\) (included angle) |
Congruence transfers every remaining side and angle; similarity transfers every remaining angle and the common ratio of corresponding sides.
Midpoint theorem
If \(M\) and \(N\) are the midpoints of \(AB\) and \(AC\) respectively, then \(MN//BC\) and \(MN=\dfrac{1}{2}BC\) (Midpoint Theorem).
Write the vertices in corresponding order — \(\triangle PTA\sim\triangle PBT\) already tells the reader that \(\angle PTA=\angle PBT\) and \(\dfrac{PT}{PB}=\dfrac{PA}{PT}\). Quote the test in brackets: (SAS), (RHS), (AA similarity).
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