HSC Extension 1 and 2 Mathematics/Plane geometry; geometrical properties
- 1 Points, lines and angles
- 2 Plane Geometry terms
- 3 Angles on a straight line
- 4 Angles at a point
- 5 Vertically opposite angles
- 6 Parallel lines
- 7 Angle sum of triangles
- 8 Exterior angle properties
- 9 Angles sum of a quadrilateral and of a general polygon
- 10 The size of the angles of a general polygon
- 11 Sum of the exterior angles of a general polygon
- 12 Congruence of triangles. Tests for congruence
- 13 Properties of special triangles and quadrilaterals. Tests for special quadrilaterals
- 14 Properties of transversals to parallel lines
- 15 Similarity of triangles. Tests for similarity
- 16 Equal intercepts
- 17 Pythagoras’ theorem and its converse
- 18 Area formulae
Points, lines and angles
Plane Geometry terms
A line : a straight line of infinite length.
Interval: a part of the line between two points.
Collinear points: lie on the same line.
Concurrent points: pass through the same point
Produced: to extend, continue an interval.
Altitude: perpendicular line from base to highest point of a figure.
Angles on a straight line
Add to 180°
Angles at a point
Add to 360º
Vertically opposite angles
Vertically opposite angles are equal.
A pair of lines in the same plane that never meet.
A transversal is a line drawn across a pair (or more) of straight lines.
Special types of angles are formed when parallel lines are cut by a transversal. These include:
- Corresponding angles
- Alternate angles
- Co-interior angles (supplementary)
Angle sum of triangles
The interior angle sum of a triangle is 180 degrees.
Exterior angle properties
An exterior angle is the angle formed on the outside of the triangle. The exterior angle of a triangle is equal to the sum of its interior opposite angles.
Angles sum of a quadrilateral and of a general polygon
The angle sum of a quadrilateral is 360 degrees.
The angle sum of any general polygon = (n-2) x 180 degrees (where 'n' is the number of sides)
The size of the angles of a general polygon
Sum of the exterior angles of a general polygon
The sum of the exterior angles of any general polygon is 360 degrees.
Congruence of triangles. Tests for congruence
Congruent triangles are triangles that have exactly the same shape and same size. There are four tests to test congruency of two triangles:
- SSS three sides equal to three sides of other triangle
- SAS two sides and one angle is equal to the other *Angle must be the angle between the two sides
- AAS two angles and a side are equal to the other *Side must be corresponding side i.e. same position on both triangles (this can be seen clearly by marking matching angles)
- RHS the hypotenuse and one side of a right-angled triangle are equal to the other right-angled triangle
Properties of special triangles and quadrilaterals. Tests for special quadrilaterals
Parallelogram: Opposite sides are parallel and equal, diagonals bisect one another.
Rhombus: All sides are equal and opposite sides are parallel and equal, opposite angles are equal, diagonals bisect one another and are perpendicular, diagonals bisect angles through which they pass.
Kite: Diagonals perpendicular
Rectangle: Opposite sides parallel and equal, diagonals are equal and bisect one another, all angles are right angles
Tests for special quadrilaterals
Tests are true when any ONE of the points listed are true.
Parallelogram: - opposite sides are equal - opposite angles are equal - opposite sides are parallel & equal - diagonals bisect one another
Rectangle: (a special type of parallelogram) - diagonals are equal
Rhombus: - all sides are equal - diagonals bisect each other at right angles
Properties of transversals to parallel lines
Ratio of Intercepts
Parallel lines preserve the ratio of intercepts on transversals. The property derived from this is the ratio of intercepts which is proved using similar triangles.
Rule: When two (or more) transversals cut a series of parallel lines, the ratios of their intercepts are equal.
Similarity of triangles. Tests for similarity
Definition of similarity of triangles. Statement of tests for similarity of triangles (equality of corresponding angles or of two such pairs, corresponding sides proportional, two pairs of corresponding sides proportional and equality of the included angles). Parallel lines preserve ratios of intercepts on transversals. Line parallel to one side of a triangle divides the other two sides in proportion*.
A line joining the midpoints of two sides of a triangle is parallel to the third side and half its length.*
Pythagoras’ theorem and its converse
Pythagoras’ Theorem. Proof using similar triangles. Converse.
Triangle: A = 1/2bh
Paralellogram: A = bh = lb
Trapezium A= h/2(a+b)