Geometry Outline 2009

2 Unit Mathematics – Plane Geometry Work Requirements

Week 1 (B)

Lesson 1

2

3

Week 2 (A)

4

5

6

Week 5 (B)

Week 4 (A)

Week 3 (B)

7 8 9

10 11 12 13 14 15 16 17 18 19 20 21

Content Angles & Lines Notation Definitions Properties of angles at a point Parallel Lines Co-interior (parallel postulate) Alternate Corresponding Triangles Types of Triangles Angle in Polygons Interior Angle Sum Exterior Angle Sum (proof) Congruent Triangles SSS ,SAS, AAS, RHS Setting out of Proofs Two Stage Congruence Proofs

Set Exercises Ex 5.1: 1 – 5 Ex 5.2: 1 - 8

Special Triangles Review of Assignment 2 More Congruence Similar Triangles Definition Tests for Similarity Pythagoras Theorem

Ex 5.13: 1- 5 Ex 5.14: 1, 2, 4, 5

Special Quadrilaterals Assignment 3/Catchup Lesson Properties of Square/Rectangle/rhombus/trapezium Go over Assignment 3 Tests for Parallelograms Transversals & Parallel Lines Ratio Property of Intercepts Area Formula Surveyor’s drawings Go over Assignment 4 Exam Review Exam Review Exam Review

Ex 6.1: 1 - 10

Lesson 1 Angles and Lines - Notation is parallel to || is similar to ||| therefore ∴ is perpendicular to ⊥ is congruent to ≡ because ∵    ,B  , AB, AB, AB , AB = CD ∠ABC , ABC -

Definitions




Axiom Definitions Hypothesis

Ex 5.3: 1 – 5, 7, 9 Ex 5.4: 1, 3, 5, 7

Ex 5.5: 1, 2, 3, 5, 7, 9, 11, 13 Ex 5.6: 1 – 11 (Weekend Homework) Ex 5.7: 1 – 10 Ensure all previous work is complete Ex 5.8: 1 – 6 Ex 5.9: 1 - 10 Ex 5.10 1 – 3

Ex 5.15: 1 - 12

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

Ex 6.3: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20 Ex 6.4: 2, 4, 6, 8 Ex 6.5: 1, 3, 5, 7, 9, 11 Ex 6.6: 2, 4, 6, 8, 10, 12, 14, 16, 18, Ex 6.9: 2, 4, 6, 8, 10, 12, 14,16, 18, 20, 22












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

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) 1
Triangle ( base × perpendicular height) 2 1
Rhombus ( product of diagonals) 2

So in the diagram




1 Trapezium ( (sum of parallel sides) × distance between them 2

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) b° 2x°

a°

(x+60)°

c)

c° 120°

d)

(6b-70)° (2a-14)°

(a+28)°

(2a+25)°

(4b-10)°

a° b°

35° c°

1+2+2+2 = 7 marks 2. Find the value of the pronumerals in each of the following a) b) a° b°

d°

70° c°

c°

g°

65° e°

a°

65°

b°

f°

3.5+ 1.5 = 5 marks

3. Find the pronumerals a)

b) 35° x°

72° 25° 30° x°

c) The three angles in a triangle are in the ratio 3:5:7. Find the magnitude of each angle.

50°

d) ABC is a triangle in which AB = AC AB is produced to D so that BD = BC. Prove that ∠ACB = 2∠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.

f ( x) = 9 − x 2

b. h( x) = 2 x − 5 c. g ( x) = ( x − 2)( x + 1)( x − 3) 4+4+4=12 marks

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 A

B

C

A

P O

D B

C

D

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