Congruent and Similar Figures

In short: Congruent figures are the same shape and size, proved with the tests SSS, SAS, AAS or RHS. Similar figures are the same shape but a different size: with scale factor $k$, lengths are multiplied by $k$, areas by $k^2$ and volumes by $k^3$.

Starting point: Years 9 and 10

Angle facts to know

• Angles on a straight line add to $180^{\circ }$. Angles around a point add to $360^{\circ }$.
• Vertically opposite angles are equal.
• Parallel lines: alternate angles are equal (Z shape), corresponding angles are equal (F shape), co-interior angles add to $180^{\circ }$ (C shape).
• Angles in a triangle add to $180^{\circ }$. The exterior angle equals the sum of the two opposite interior angles.
• Angles in a polygon with $n$ sides add to $(n-2)\times 180^{\circ }$.

Congruent triangles

Congruent figures are exactly the same shape and size. We write $△ABC\equiv △DEF$, listing matching vertices in order.

Writing a congruence proof

State each fact with a reason, name the test, then use "matching sides (or angles) of congruent triangles are equal" to finish.

Congruence proves the familiar quadrilateral facts: a rhombus has perpendicular diagonals, a rectangle has equal diagonals, and the diagonals of a parallelogram bisect each other.

Similar figures

Similar figures have the same shape. Matching angles are equal and matching sides are in the same ratio, the scale factor $k$.

$$k=\frac{\text{image length}}{\text{original length}}$$

Quick summary

• Know the angle facts: straight line $180^{\circ }$, triangle $180^{\circ }$, polygon $(n-2)\times 180^{\circ }$, and the parallel line rules.
• Congruent: same shape and size. Tests: SSS, SAS, AAS, RHS.
• Similar: same shape. Tests: AAA, SSS, SAS, RHS (ratios).
• Scale factor $k$: lengths $\times k$, areas $\times k^2$, volumes $\times k^3$.
• Proofs give a reason for every statement.

Quiz