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Constructions, congruence and similarity
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With just a ruler and compasses you can draw exact right angles and halve angles. Congruence and similarity explain when shapes match.
Ruler and compass constructions
Perpendicular bisector of a line segment: cuts it in half at 90°. Every point on it is the same distance from both ends.
Angle bisector: cuts an angle exactly in half. Every point on it is the same distance from both arms.
Perpendicular from a point to a line: the shortest distance from a point to a line is along the perpendicular.
Angle bisector: cuts an angle exactly in half. Every point on it is the same distance from both arms.
Perpendicular from a point to a line: the shortest distance from a point to a line is along the perpendicular.
Labelling and congruence
In triangle ABC, the side opposite angle A is called a, opposite B is b, opposite C is c.
Two triangles are congruent (identical) if they match by:
SSS (three sides), SAS (two sides and the angle between), ASA (two angles and a corresponding side) or RHS (right angle, hypotenuse, side).
AAA is not enough: the triangles could be different sizes.
Two triangles are congruent (identical) if they match by:
SSS (three sides), SAS (two sides and the angle between), ASA (two angles and a corresponding side) or RHS (right angle, hypotenuse, side).
AAA is not enough: the triangles could be different sizes.
Similar shapes
Similar shapes have equal angles and sides in the same ratio.
If a side of 4 cm matches a side of 10 cm, the scale factor is 10 ÷ 4 = 2.5, so every length is multiplied by 2.5.
If a side of 4 cm matches a side of 10 cm, the scale factor is 10 ÷ 4 = 2.5, so every length is multiplied by 2.5.
Triangles ABC and PQR are similar. AB = 4 cm, PQ = 10 cm, BC = 6 cm. Find QR.
- Scale factor = 10 ÷ 4 = 2.5
- QR = 6 × 2.5
Answer: 15 cm
Constructions use equal compass arcs. Congruent triangles match by SSS, SAS, ASA or RHS. Similar shapes have equal angles and lengths multiplied by one scale factor.
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