Euclidean Geometry Key Concepts • Classifying Angles • Parallel Lines and Transversal Lines • Classifying Triangles • Pr

Key Concepts  Classifying Angles  Parallel lines and transversal lines  Classifying Triangles  Properties of Triangles . Relationship between angles . Congruency . Similarity . Pythagoras . Mid-point Theorem  Properties of Quadrilaterals

Terminology

Acute angle: Greater than 00 but less than 900

Right angle: Angle equal to 900

Obtuse angle: Angle greater than 900

Straight angle: Angle equal to 1800

Reflex angle: Angle greater than 1800 but less than 3600

Revolution: Sum of the angles around a point, equal to 3600

Adjacent angles: Angles that share a vertex and a common side.

Vertically opposite angles: Angles opposite each other when two lines intersect. They share a vertex and are equal.

Supplementary angles: Two angles that add up to 1800.

Complementary angles: Two angles that add up to 900.

Parallel lines Lines that are always the same distance apart

A transversal line A line that intersects two or more parallel lines.

Interior angles Angles that lie in between the parallel lines.

Exterior angles Angles that lie outside the parallel lines.

Corresponding angles Angles on the same side of the lines and the same side of the transversal.

Co-interior angles Angles that lie in between the lines and on the same side of the transversal.

Alternate interior angles Interior angles that lie inside the line and on opposite sides of the transversal.

X-planation

Properties of the angles formed by a transversal line intersecting two parallel lines

If the lines are parallel  the corresponding angles will be equal  the co-interior angles are supplementary  the alternate interior angles will be equal.

If the corresponding angles will be equal or the co-interior angles are supplementary or the alternate interior angles will be equal

 the lines are parallel

Classifying Triangles

There are four kinds of triangles: Scalene Triangle Isosceles Triangle No sides are equal in length Two sides are equal Base angles are equal

||| || | ||

Equilateral Triangle Right-angled triangle All three sides are equal One interior angle is 90 All three interior angles are equal A

60

|| ||

60 || 60 B C

Relationship between angles Sum of the angles of a triangle Exterior angle of a triangle

c b

a b a c abc  180  c a b

Congruency of triangles Rule 1

Two triangles are congruent if three sides of one triangle are equal in | || | || length to the three sides of the other triangle. (SSS) ||| |||

Rule 2

Two triangles are congruent if two sides and the included angle are || || equal to two sides and the included | | angle of the other triangle. (SAS)

Rule 3

Two triangles are congruent if two angles and one side are equal to | | two angles and one side of the other  triangle. (SAA) 

Rule 4

Two right-angled triangles are congruent if the hypotenuse and a side of the one

|| || | triangle is equal to the hypotenuse and a | side of the other triangle. (RHS)

Similarity

Rule 1 (AAA) If all three pairs of corresponding angles of two triangles are equal, then the triangles are similar.

Rule 2 (SSS) If all three pairs of corresponding sides of two triangles are in proportion, then the triangles are similar.

The Theorem of Pythagoras A

AC2 AB 2 BC 2 or AB2 AC 2 BC 2 or 2 2 2 BC AC AB B C

Mid-Point Theorem The line joining the mid-points of two sides of a triangle is parallel to the third side and equal to half the length of the third side.

Properties of quadrilaterals

Trapezium A > D  Two sides are parallel. = =

B > C Parallelogram A > D

|||  Opposite sides parallel and equal. E

 Opposite angles equal. = =

> >  Diagonals bisect each other. |||

B > C

Rectangle = A D  Opposite sides parallel and equal in length. ||| |||  Diagonals are equal in length and bisect E each other. ||| |||

 Interior angles are right angles. = B C

Rhombus = A D

 Opposite sides are parallel. |||

 All sides equal in length.

=  Diagonals bisect each other at right angles. =  Diagonals bisect the opposite angles. E |||

= B C

Square = A 45 45 D

 Opposite sides parallel. 45 ||| 45

 All sides equal in length. ||| =  Diagonals are equal in length. = E  Diagonals bisect each other at right angles.  Interior angles are right angles. ||| |||  Diagonals bisect interior angles 45 45

(each bisected angle equals 45 ) 45 = 45 B C Kite A

 Adjacent pairs of sides are equal in length ||| |||  The longer diagonal bisects the opposite angles.

 The longer diagonal bisects the other diagonal.

= =  The diagonals intersect at right angles. B E D

C X-ample Questions

Question 1 Calculate the size of the angles marked with small letters:

(a) (b) 70 x 49 y x

70 100

x x

Question 2: Calculate the size of the angles marked with small letters:

(a) (b) 80 30 y

x x 40

Question 3: Prove that ABC   ADC A

.1 .2

| |

1 2 B C D

Question 4: Consider the diagram below. Is ΔABC ||| ΔDEF? Give reasons for your answer.

Question 5: In ΔMNP, M = 900, S is the mid-point of MN and T is the mid-point of NR.

(a) Prove U is the mid-point of NP.

(b) If ST = 4 cm and the area of ΔSNT is 6 cm2, calculate the area of ΔMNR.

Question 6

(a) Using the information provided on the diagram, prove that AD||BC. 2x

(b) What type of quadrilateral is ABCD? Give a reason. x 120

Question 7

In the diagram, PQRS is a parallelogram. 45 ˆ ˆ P1  45 and PR bisects R Prove that PQRS is a square.

Question 8 Prove that ABCD is a trapezium.

Question 9 ABCD is a parallelogram with diagonal AC. Given that AF = HC, show that:

ΔAFD Ξ ΔCHB

X-ercise 1. Calculate the value of a and b

2. Find the value of x