## Geometry

### 4.1 – Fundamental Results

o Angles on a straight line add up to 180°
 x + y + z = 180°

o Angles at a point add up to 360°
 a + b + c + d = 360°

o Vertically opposite angles are equal
 x = x

o The sum of all angles in a quadrilateral is 360°
 a + b + c + d = 360°

o The sum of all angles in a triangle is 180°
 x + y + z = 180°

o An isosceles triangle has 2 sides and 2 angles the same

o An equilateral triangle has 3 sides and 3 angles the same

• Polygons:

o The exterior angles of a polygon add up to 360°
 a + b + c + d + e = 360°

o The sum of interior angles of a polygon = (n – 2) × 180
 n = number of sides of the polygon

o A regular polygon has equal angles and sides

• Parallel Lines:

o Corresponding angles are equal
 a = c

o Alternate angles are equal
 c = d

o Allied angles add up to 180°
 b + c = 180°

### 4.2 – Pythagoras’ Theorem

o a² + b² = c²

4.3 – Symmetry

• Line Symmetry:

o A line of symmetry divides a 2D shape into 2 identical shapes.
 E.g. The letter A has one line of symmetry

• Rotational Symmetry:

o The order of rotational symmetry is the number of times a shape
fits its outline during a complete rotation

o Square:
 All sides are equal
 All angles are 90°
 Opposite sides are parallel
 Diagonals bisect at right angles
 4 lines of symmetry
 Order of rotational symmetry is 4

o Rectangle:
 Opposite sides are parallel and equal
 All angles are 90°
 Diagonals bisect each other
 2 lines of symmetry
 Order of rotational symmetry is 2

o Parallelogram:
 Opposite sides are parallel and equal
 Opposite angles are equal
 Diagonals bisect each other
 No lines of symmetry
 Order of rotational symmetry is 2

o Trapezium:
 One pair of sides are parallel
 No lines of symmetry
 No rotational symmetry

o Kite:
 Two pairs of adjacent sides equal
 Diagonals meet at right angles bisecting one of them
 1 line of symmetry
 No rotational symmetry

• Planes of Symmetry:
o A plane of symmetry divides a 3D shape into 2 identical solid shapes.

### 4.4 – Similarity

o When two shapes’ corresponding sides are proportional
and their corresponding angles equal.

o Two triangles are similar if they have the same angles.

 E.g. In triangles ABC and XYZ
A = X and B = Y
∴ The triangles are similar

### 4.5 – Congruence

o When two shapes have exactly the same angles and dimensions.

o To prove that two triangles are congruent, one of the following criteria must be proven:

 SSS (Side Side Side): All three pairs of corresponding sides are equal.

 ASA (Angle Side Angle): Two pairs of corresponding angles are equal, along with one pair of corresponding sides.

 SAS (Side Angle Side): Two pairs of corresponding sides are equal, along with the corresponding angles.

### 4.6 – Circle Theorems

1. The angle subtended at the center of a circle is twice the
angle subtended at the circumference.

 AÔB = 2 × AĈB

1. Angles subtended by an arc in the same segment of a circle
are equal

 AXB = AŶB = AZB

1. The opposite angles in a cyclic quadrilateral add up to 180°

 A + C = 180°
 B + D = 180°

1. The angle in a semicircle is a right angle.

 AĈB = 90°

• Tangents to Circles:

1. The angle between a tangent and a radius drawn to
the point of contact is 90°

 ABO = 90°

1. From any point outside a circle just two tangents
to the circle may be drawn and they are of equal
length.

 TA = TB

• The Alternate Segment Theorem:

o The angle between a tangent and its cord is equal
to the angle in the alternate segment.

 ABC = BDC

### 4.7 – Constructions

o When constructing, the diagram should be drawn using equipment
such as a pair of compasses and a ruler.

o Constructing Triangles:
 Draw the longest side as the base line of the triangle

 Set a pair of compasses to one of other lengths given for a side
and draw an arc centered on one end of the base, above the base line.

 Similarly, set a pair of compasses to the last length given for a side
and draw an arc centered on the other end of the base, above the base line.

 Join this crossing point to each end of the base line.

 The triangle has been constructed.

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