Download Geometry Lesson 5 - 3rd year HL MATH`S

Synthetic Geometry
Guide to Axioms, Theorems and Constructions for all Levels
Axioms and Theorems
(supported by 46 definitions, 20 propositions)
*proof required for JCHL and LCHL
** proof required for LCHL only
1
Vertically opposite angles are equal in measure.
Axiom 1: There is exactly one line through any two
given points
Axiom 2: [Ruler Axiom]: The properties of the
distance between points.
Axiom 3: [Protractor Axiom]: The number of degrees
in an angle is always between 0 and 360.
Axiom 4: Congruent triangles conditions (SSS, SAS,
ASA)
Axiom 5: Given any line l and a point P, there is
exactly one line through P that is parallel to l.
2
In an isosceles triangle the angles opposite the equal
sides are equal.
Conversely, if two angles are equal, then the triangle is
isosceles.
3
If a transversal makes equal alternate angles on two
lines then the lines are parallel (and converse).
4*
The angles in any triangle add to 180⁰.
5
Two lines are parallel if, and only if, for any
transversal, the corresponding angles are equal.
6*
Each exterior angle of a triangle is equal to the sum of
the interior opposite angles.
7
The angle opposite the greater of two sides is greater
than the angles opposite the lesser. Conversely, the
side opposite the greater of two angles is greater than
the side opposite the lesser angle.
8
Two sides of a triangle are together greater than the
third.
9*
In a parallelogram, opposite sides are equal, and
opposite angles are equal.
Corollary 1. A diagonal divides a parallelogram into
two congruent triangles.
10
The diagonals of a parallelogram bisect each other.
11** If three parallel lines cut off equal segments on some
transversal line, then they will cut off equal segments
on any other transversal.
12** Let ABC be a triangle. If a line l is parallel to BC and
cuts [AB] in the ratio m:n, then it also cuts [AC] in the
same ratio.
13** If two triangles are similar, then their sides are
proportional, in order.
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14*
15
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19*
20
21
[Theorem of Pythagoras]
[Converse to Pythagoras]. If the square of one side of a
triangle is the sum of the squares of the other two,
then the angle opposite the first side is a right angle.
For a triangle, base x height does not depend on the
choice of base.
Definition 38: The area of a triangle is half the base by
the height.
A diagonal of a parallelogram bisects the area.
The area of a parallelogram is the base x height.
The angle at the centre of a circle standing on a given
arc is twice the angle at any point of the circle
standing on the same arc.
Corollary 2: All angles at a point of a circle, standing
on the same arc are equal
Corollary 3: Each angle in a semi-circle is a right angle.
Corollary 4: If the angle standing on a chord [BC] at
some point of the circle is a right-angle, then [BC] is a
diameter.
Corollary 5: If ABCD is a cyclic quadrilateral, then
opposite angles sum to 180⁰.
(i)
Each tangent is perpendicular to the
radius that goes to the point of contact.
(ii)
If P lies on the circle S, and a line l is
perpendicular to the radius to P, then l is a
tangent to S.
Corollary 6: If two circles intersect at one point only,
then the two centres and the point of contact are
collinear.
(i)
The perpendicular from the centre to a
chord bisects the chord.
(ii)
The perpendicular bisector of a chord
passes through the centre.
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Constructions
1
2
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4
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10
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12
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17
18
19
20
21
22
(supported by 46 definitions, 20 propositions, 5 axioms
and 21 theorems).
Bisector of an angle, using only compass and straight
edge.
Perpendicular bisector of a segment, using only compass
and straight edge.
Line perpendicular to a given line l, passing through a
given point not on l.
Line perpendicular to a given line l, passing through a
given point on l.
Line parallel to given line, through given point.
Division of a line segment into 2 or 3equal segments
without measuring it.
Division of a line segment into any number of equal
segments, without measuring it.
Line segment of a given length on a given ray.
Angle of a given number of degrees with a given ray as
one arm.
Triangle, given lengths of 3 sides.
Triangle, given SAS data.
Triangle, given ASA data
Right-angled triangle, given length of hypotenuse and one
other side
Right-angled triangle, given one side and one of the acute
angles.
Rectangle given side lengths.
Circumcentre and circumcircle of a given triangle, using
only straight edge and compass.
Incentre and incircle of a triangle of a given triangle, using
only straight edge and compass.
Angle of 60⁰ without using a protractor or set square.
Tangent to a given circle at a given point on it.
Parallelogram, given the length of the sides and the
measure of the angles.
Centroid of a triangle.
Orthocentre of a circle.
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First Year Common Course:
Synthetic Geometry
Concepts:
Set, plane, point, line, ray, angle, real number, length, degree. Triangle, right-angle, parallel
lines, area, segment, collinear points, distance, midpoint of a line segment, reflex angle, ordinary angle,
straight angle, null angle, supplementary angles, vertically-opposite angles , acute angle, obtuse angle, angle
bisector, perpendicular lines, perpendicular bisector of a line segment, isosceles triangle, equilateral triangle,
scalene triangle, right-angled triangle, exterior angles of a triangle, interior opposite angles, alternate angles,
corresponding angles, , transversal line, circle,
First Year Common Course
Constructions
1
2
3
4
5
6
8
9
10
11
1
2
3
4
5
6
Bisector of an angle, using only
compass and straight edge
Perpendicular bisector of a segment,
using only compass and straight
edge.
Line perpendicular to a given line l,
passing through a given point not on
l
Line perpendicular to a given line l,
passing through a given point on l.
Line parallel to given line, through
given point
Division of a line segment into 2
equal segments without measuring
it;
Division of a line segment into 3
equal segments without measuring
it.
Line segment of a given length on a
given ray
Angle of a given number of degrees
with a given ray as one arm
Triangle, given lengths of 3 sides
Triangle, given SAS data
First Year Common Course Theorems
Vertically opposite angles are equal in
measure
In an isosceles triangle the angles
opposite the equal sides are equal
Conversely, if two angles are equal,
then the triangle is isosceles
Alternate Angles
The angles in any triangle add to 180⁰
Corresponding angles
Exterior Angle
Lesson 7
Lesson 7
Lesson 9
Lesson 9
Lesson 10
Lesson 10
Lesson 8
Lesson 8
Lesson 11
Lesson 11
Lesson 4
Lesson 13
Lesson 5
Lesson 12
Lesson 5
Lesson 12
Geometry Lesson 1
Introduction
Plane; points, lines, line segments, rays; collinear;
Length of a line segment;
2 Student Activities;
Interactive web-page.
Geometry Lesson 2
Introduction to angles
Angle as a rotation; forearm: bend it; no equipment needed; leg ; acute and reflex; its all the
way you look at it;
Use of geo-strips to get acute, right, straight, obtuse, reflex and ordinary angles
Orientation;
Perpendicular, parallel, vertical, horizontal.
Degrees
Snooker;
Shooting football angle; narrow the angle; sharp angle; look at from a different angle (rotation)
Geometry Lesson 3
Measuring angles; introduction to the protractor
Use of geo-strips + large protractor to demonstrate
Misconceptions; longer arm;
Arc longer further out on the arm of the angle;
3 right angles in different orientations;
Estimate 70 as 110
Draw 2 triangles one larger than the other: compare angles;
Swing;
Common errors;
NB 60 and 120 etc. angles and other supplementary angles can cause problems in reading from the
correct side of the protractor.
Measure of angles from left or right side.
Measure of angles in different orientations.
Use of interactive protractor on web-site (http://www.keymath.com/x3311.xml ).
Static v. rotation;
Student Activity:
Plenty of diagrams given. Have them estimate, then measure.
Addition of angles ( not necessary for 1st year?)
Geometry Lesson 4
A straight angle has 180⁰ (supplementary angles)
use of geo-strips
Activity Sheet 1
Vertically opposite angles: use of geo-strips
Theorem 1:
Vertically opposite angles are equal in measure:
Have them discover this by drawing diagrams in pairs and comparing results around the class.
“Theorems are full of potential for surprise and delight. Every theorem can be taught by considering
the unexpected matter which theorems claim to be true. Rather than simply telling students what
theorems, it would be helpful if we assumed we didn’t know it… it is the mathematics teacher
responsibility to recover the surprise embedded in the theorem and convey it to the pupils. The
method is simple: just imagine you do not know the fact. This is where the teacher meets the
students”.
More practice with the protractor.
Real-life situations: reflections in mirrors, angle of incidence;
Activity sheet 2
Geometry Lesson 5
Corresponding Angles
Alternate Angles
Theorem 3:
(i) If a transversal makes equal alternate angles on two lines, then the lines are
parallel
(ii) (Conversely)
If two lines are parallel, then any transversal will make equal
alternate angles with them.
Theorem 5:
Two lines are parallel if and only if for any transversal, corresponding angles are
equal.
New words:
transversal, alternate, corresponding
Have the students draw 2 parallel lines, by drawing lines along both edges of a ruler (construction
not done yet) and a line cutting across these. Measure all the angles (protractor
practice) and see what they notice; compare results across the class.
Have the students draw (obvious) non-parallel lines with a transversal, and check the same angles.
Ask them can they spot 2 “Z” shapes in the diagram and 4 “F” shapes.
Plenty of student activities on this.
Geometry Lesson 6
Use of the Compass
Student activities: use of compass to draw circles and various shapes
Draw a 60 angle with just a ruler and compass
Draw a 90 angle with just a ruler and compass
Draw a square within a circle
Draw an athletics track
Psycho-motor skill development
Geometry lesson 7
Construction 1 Use of compass to bisect an angle
Construction 2 Use of compass to draw the perpendicular bisector of a line segment
Geometry lesson 8
Construction 8:
Line segment of a given length on a given ray
Construction 9:
Angle of a given number of degrees with a given ray as one arm
Geometry Lesson 9
Construction 3: Line perpendicular to a given line l, passing through a given point not on l
( 2 methods: ruler and compass or ruler and set-square)
Construction 4: Line perpendicular to a given line l, passing through a given point on l
( 2 methods: ruler and compass or ruler and set-square)
Geometry Lesson 10
Construction 5: Line parallel to a given line, through a given point; (ruler and compass and “sliding”
method or ruler and compass method)
Construction 6: Division of a line segment into 2 or 3 equal segments without measuring it;
Geometry Lesson 11
Introduction to Triangles
Use of geostrips: scalene, isosceles, equilateral, right-angled.
Construction 10:
Triangle given SSS
Construction 11:
Triangle given SAS
Geometry lesson 12
Theorem 4:
The angles in any triangle add to 180⁰.
Have the students draw a number of triangles or use the ones from the previous class and measure
their angles with the protractor. Include some obtuse-angled triangles.
Tear (not cut) off the three angles, having first marked them and put them along the edge of a ruler.
It will be obvious that they add to 180⁰.
Student Activities.
Theorem 6:
angles.
Each exterior angle of a triangle is equal to the sum of the the interior opposite
Don’t tell them the theorem first. Have the students draw several cases and see if they can come
up with the theorem.
Very important to have triangles in various orientations.
Geometry Lesson 13
Theorem 2:
(i) In an isosceles triangle the angles opposite the equal sides are equal
(ii) Conversely, if two angles are equal, then the triangle is isosceles
Word “iso”
isobars
Students draw their own isosceles triangle: (more practice with the compass and ruler ) and then
measure the angles . each pupil with a different triangle; compare results….. cut out and fold…
Use of geo-strips: put two identical triangles on top of each other, compare the equal angles and
then flip the top one over…
Draw an isosceles triangle containing a 90⁰ angle; and discover that the 45 setsquare is one of these.
Draw equilateral triangles inn a variety of orientations and mark in equal parts.
Draw an equilateral triangle: also isosceles.
Real-life:
architecture: pyramids; Maslow’s pyramid of human needs; food-pyramids;
Leaving Cert. Ordinary Level Course: Synthetic Geometry
Lessons 1 -5
Revision of Junior Cert. ordinary material
Lesson 6
Lesson 7
Geometrical Investigations
Investigations of Quadrilaterals
Square
Describe it in
words
Draw 3
examples in
different
orientations
How many
axes of
symmetry
does it have?
Show on a
diagram.
Does it have a
centre of
symmetry?
Show on a
diagram.
Which sides
are equal?
What is the
sum of all the
angles?
Are all angles
equal ?
Which angles
are equal?
What is the
sum of 2
adjacent
angles?
Does a
diagonal bisect
its area?
Does a
diagonal
divide it into
two congruent
triangles?
Do the 2
diagonals
bisect each
other?
Are the
diagonals
Rhombus
Rectangle
Parallelogram
Kite
perpendicular?
Do the
diagonals
bisect the
angles they
pass through?
Do the
diagonals
divide it into 4
triangles of
equal area?
Do the
diagonals
divide it into 4
congruent
triangles?
How do you
calculate its
area?