Year 9 Geometry: revision mind map

1Angles and triangles

Triangle angles add to 180°. Angles on a straight line add to 180°, around a point 360°.

Two angles are 50° and 60°: the third is 70°

Exterior angle = sum of the two opposite interior angles

An isosceles triangle has two equal sides and two equal base angles.

2Parallel lines

Corresponding angles are equal (F), alternate angles are equal (Z), co-interior angles add to 180° (C).

If one angle is 65°, its co-interior angle is 115°

Watch out: these rules only work when the lines are parallel.

3Quadrilaterals and polygons

Interior angle sum of an $n$-sided polygon: $(n - 2) \times 180^\circ$.

Quadrilateral: 360°

Hexagon: $4 \times 180^\circ = 720^\circ$; each angle of a regular hexagon is 120°

4Congruent triangles

Congruent means exactly the same shape and size (≅). Tests: SSS, SAS, AAS, RHS.

SSS: all three sides equal

SAS: two sides and the angle between them

RHS: right angle, hypotenuse and one side

Watch out: AAA is not a congruence test: same angles can be different sizes.

5Using congruence in proof

List the equal parts with a reason for each, name the test, then conclude.

ABCD is a parallelogram. Prove △ABC ≅ △CDA.

AB = CD and BC = DA (opposite sides of a parallelogram)

AC is common, so △ABC ≅ △CDA (SSS)

6Enlargement and similar figures

Scale factor $= \dfrac{\text{image length}}{\text{original length}}$. Angles stay the same.

A 4 cm side enlarged by a scale factor of 2.5 becomes 10 cm

Similar figures: same shape, sides in the same ratio.

7Similar triangles

Tests: AAA, SSS (ratio), SAS (ratio), RHS (ratio). Match the sides in order.

Matching sides 4 and 6: scale factor $\tfrac64 = 1.5$

Side 5 matches $x$: $x = 5 \times 1.5 = 7.5$

8Proving and applying similar triangles

Show the triangles are similar, then use the scale factor.

A 2 m pole casts a 3 m shadow. A tree's shadow is 12 m at the same time.

Height $= 2 \times \tfrac{12}{3} = 8$ m

Watch out: write the vertices in matching order: △ABC ||| △DEF.