2 Unit Mathematics – Plane Geometry Work Requirements

Lesson Content Set Exercises

1 Angles & Lines

Notation

Definitions

Properties of angles at a point

Ex 5.1: 1 – 5

Ex 5.2: 1 - 8

2 Parallel Lines

Co-interior (parallel postulate)

Alternate

Corresponding

Ex 5.3: 1 – 5, 7, 9

Ex 5.4: 1, 3, 5, 7

Week 1 (B)

3 Triangles

Types of Triangles

Ex 5.5: 1, 2, 3, 5, 7, 9, 11, 13

Ex 5.6: 1 – 11 (Weekend Homework)

4 Angle in Polygons

Interior Angle Sum

Exterior Angle Sum (proof)

Ex 5.7: 1 – 10

Ensure all previous work is complete

5 Congruent Triangles

SSS ,SAS, AAS, RHS

Setting out of Proofs

Ex 5.8: 1 – 6

Ex 5.9: 1 - 10

6 Two Stage Congruence Proofs Ex 5.10 1 – 3

Week 2 (A)

7 Special Triangles Ex 5.13: 1- 5

8 Review of Assignment 2

More Congruence

Ex 5.14: 1, 2, 4, 5

9 Similar Triangles

Definition

Tests for Similarity

Ex 5.15: 1 - 12

10 Pythagoras Theorem

Ex 5.16: 1, 3, 5, 7, 9, 11, 13, 15

11 Special Quadrilaterals Ex 6.1: 1 - 10

Week 3 (B)

12 Assignment 3/Catchup Lesson

13 Properties of

Square/Rectangle/rhombus/trapezium

Ex 6.3: 2, 4, 6, 8, 10, 12, 14, 16, 18,

20

14 Go over Assignment 3

Tests for Parallelograms

Ex 6.4: 2, 4, 6, 8

15 Transversals & Parallel Lines Ex 6.5: 1, 3, 5, 7, 9, 11 Week 4 (A)

16 Ratio Property of Intercepts Ex 6.6: 2, 4, 6, 8, 10, 12, 14, 16, 18,

17 Area Formula

Surveyor’s drawings

Ex 6.9: 2, 4, 6, 8, 10, 12, 14,16, 18,

20, 22

18 Go over Assignment 4

19 Exam Review

20 Exam Review Week 5 (B)

21 Exam Review

Lesson 1 Angles and Lines

- Notation

is parallel to || is similar to |||

is perpendicular to ⊥ therefore ∴

is congruent to ≡ because ∵

� �, , , , ,ABC ABC B AB AB AB∠

���� ����, AB = CD

- Definitions

� Axiom

� Definitions

� Hypothesis

� Conjecture

� Theorem – statement proved on the basis of an axiom

� Corollary

� Point

� Line

� Ray

� Segment (interval)

� Midpoint

� Concurrent Lines (three or more lines are said to be concurrent if they

intersect at a single point.)

� Angle (arm, vertex)

� Adjacent Angles

� Acute

� Obtuse

� Reflex

� Collinear – “X is a point on AB produced”

� Bisect

- Properties of Angles at a Point

� Complimentary (add to 90°) – right angles

� Supplementary (add to 180°) – straight angles

� Revolution (add to 360°)

� Vertically Opposite Angles – proof

Lesson 2 Parallel Lines

- Definitions: parallel lines are coplanar (on the same plane) that never meet no matter

how far they are extended; lines that are not parallel are skew; transversal is any line

that crosses a pair of lines

- Eclid’s parallel postulate (~300BC)– “if a line segment intersects two straight lines

forming two interior angles on the same side that sum to less than two right angles,

then the two lines if extended indefinitely, meet on that side on which the angles sum

to less than two right angles”

- Properties of parallel lines

� Alternate angles in parallel lines are equal (Proof)

� Corresponding angles in parallel lines are equal (Proof)

� Co-interior angles in parallel lines are supplementary (Parallel

Postulate) - Axiom

� Corollary: If a pair of corresponding angles are equal or if a pair of

alternate angles are equal then the two lines are parallel

� Lines that are parallel to the same straight line are also parallel to each

other

Lesson 3 Triangles

- Angle Sum of a Triangle (Proof)

- Exterior Angle of a Triangle (proof)

- Types of Triangles

� Equilateral Triangles (including angle size)

� Isosceles Triangles (two angles same – why? – congruence of triangles

will prove later)

� Scalene Triangle

� Right Angled Triangle

� Obtuse Angled Triangle

� Acute Angled Triangle

- Investigation of Angles in Polygons

Lesson 4 Angles in Polygons

- Specific case – angle sum of quadrilateral (360 °)

- General Case n sided polygon (2n-4) right angles – can be proved by induction but

pattern is obvious – derive in table form (e.g of Theorem)

- Corollary: Exterior Angles of a Polygon by producing the sides in the same direction

add to 360° - prove

Lesson 5 Congruent Triangles

- Definition of Congruence ≡

- Congruent Triangles Tests for

� SSS (Side, Side, Side) – make point that equal angular are not

necessarily congruent

� SAS (Side, Angle, Side)

� AAS (Angle, Angle, Side)

� RHS (Right Angle, Hypotenuse, Side) – why? (special case SSS with

Pythagoras)

- Setting out Proofs

� Always draw a diagram

� Given

� Aim

� Construction

� Proof (symbols ∴∵)

Lesson 6 Two Stage Congruence Proofs

- Refresh proof set-out

- Examples

Lesson 7 Special Triangles

- Properties of Special Triangles

� Isosceles

� Equilateral

� Scalene

� Right Angled

- Proofs

Lesson 8 Review of Assignment 2

Lesson 9 Similar Triangles

- Definition: Two triangles are similar if they have two angles the same

ABC DEF△ ∼△ or ABC△ ||| DEF△ .

- Corollary: If two triangles have all three pairs of corresponding sides in proportion

then they are similar

- Corollary: If two triangles have one angle equal to an angle of the other and the sides

about these angles proportional then the triangles are similar.

- Tests for Similarity

� Angles are all equal

� The corresponding sides are proportional

� Two pairs of corresponding sides are proportional and their sides are

similar

Lesson 10 Pythagoras Theorem

- Pythagoras Theorem “The square on the hypotenuse of a right angled triangle is equal

to the sum of the squares on the other two sides” – Proof

- Converse of Pythagoras Theorem “If the square on a side of a triangle is equal to the

sum of the squares on the other two sides, the angle contained by these sides must be a

right angle”

Lesson 11 Special Quadrilaterals

- Properties of Parallelograms (proof first 3 – 2 similar triangles)

� Opposite sides are equal

� Opposite Angles are equal

� Each diagonal bisects a parallelogram into two congruent triangles

� The diagonals of a parallelogram bisect each other

Lesson 12 Assignment 3/Catchup Lesson

Lesson 13 Properties of Square/Rectangle/Rhombus/Trapezium

- Properties of Square (derived as homework)

� Diagonals are equal

� Diagonals meet at right angles

� Diagonals make angles of 45° with the sides – i.e. diagonals are

bisectors of the angles of the square

- Properties of a Rhombus

� Diagonals bisect each other at right angles

� Diagonals bisect the angles through which they pass

- Rectangle Diagonals

� In any rectangle the diagonals are equal

Lesson 14 Tests for Parallelograms

- Properties of Parallelograms (prove 1st only in class – others as handout/vodcast)

� Both pairs of opposite sides are equal, or

� Both pairs of opposite angles are equal, or

� One pair of sides is both equal and parallel, or

� The diagonals bisect each other.

- Properties of a rhombus – a quadrilateral is a rhombus if either of the following is true

(proof as part of exercise):

� all diagonals are equal, or

� the diagonals bisect each other at right angles

Lesson 15 Transversals & Parallel Lines

- If three parallel lines cut off equal intercepts on one transversal then they cut of equal

intercepts on any other transversal

Lesson 16 Ratio Property for Intercepts

- A family of parallel lines will cut all transversals in the same ratio

S

So in the diagram AC BD

CE DF=

- Consequences of the ratio property

� If a line is drawn parallel to the base of a triangle then it cuts the other

sides in the same ratio

Lesson 17 Area Formula

- Areas of

� Parallelogram (base ×perpendicular height)

� Triangle (1

2base × perpendicular height)

� Rhombus (1

2product of diagonals)

� Trapezium (1

2(sum of parallel sides) × distance between them

Name_________________________ Mark /32

2 Unit Mathematics Geometry Weekly Assignment 1 – due 4/5/2009

Answer all questions on this sheet unless otherwise indicated

1. Find the value of the pronumerals in each of the following

a)

b)

c)

d)

1+2+2+2 = 7 marks

2. Find the value of the pronumerals in each of the following

a)

b)

3.5+ 1.5 = 5 marks

2x° (x+60)° a°

b°

c°

120°

(6b-70)° (a+28)°

(4b-10)° (2a-14)°

(2a+25)° a°

b° c°

35°

a° d°

b° c°

65° g°

e° f°

c°

a° b°

65° 70°

3. Find the pronumerals

a)

b)

c) The three angles in a triangle are in the

ratio 3:5:7. Find the magnitude of each

angle.

d) ABC is a triangle in which AB = AC

AB is produced to D so that BD = BC.

Prove that 2ACB DCB∠ = ∠

(Complete this on a separate piece of

paper)

1 + 2 + 2 + 3 = 8 marks

4. For each of the following (complete on a separate piece of paper):

i. State the domain

ii. State the range

iii. Sketch labelling all intercepts

a. 2( ) 9f x x= −

b. ( ) 2 5h x x= −

c. ( ) ( 2)( 1)( 3)g x x x x= − + −

4+4+4=12 marks

72°

x° 30°

x°

35°

25°

50°

Name_________________________ Mark /32

2 Unit Mathematics Geometry Weekly Assignment 2 – due 8/5/2009

Answer all questions on clearly labelled loose leaf paper

a) Two line segments, AD and BC bisect

each other at O. Prove that AB = CD and

that AB || CD by showing that △AOB and

△COD are congruent

b) P is a point inside a square ABCD such that

triangle PDC is equilateral. Prove that

i) △APD ≡ △BPC

ii) △APB is isosceles

c) E, F, G and H are the midpoints of the sides AB, BC, CD, and DA respectively of the

parallelogram ABCD. Assuming that the opposite sides and angles of a parallelogram are

equal, prove that:

i) △AEH≡ △CFG

ii) △APD ≡ △BPC

iii) EFGH is a parallelogram

3+3+5 = 11 marks

Ex 5.11

Ex 5.12

Couple of Trig Proofs

2 Unit Mathematics Geometry Weekly Assignment 3 – due Monday Week 4

Similar Triangles

Complete Investigation Exercise 6.2

Algebra – simplifying & factorising

2 Unit Mathematics Geometry Weekly Assignment 4 – due Monday Week 5

Ex 6.7 1 – 7

Ex 6.8 1- 6

Bearing Question

O

A

B

C

D D C

B A

P