Download Geometry 10 March - Andrew Craig – Maths homepage

Geometry 1: parallel lines and triangles
Dr Andrew Craig
Department of Pure and Applied Mathematics, University of Johannesburg
[email protected]
10 March 2017
Acknowledgement
The content presented here is based on the
Everything Maths textbooks produced by Siyavula.
They are available for free download (including
Teacher’s Guides) from:
www.everythingmaths.co.za
Today we will cover the following topics:
I Angles and parallel lines
I Triangles
Angles and parallel lines
It all begins with lines
I
Acute angle: 0◦ < angle < 90◦
(â,ĉ)
90◦
I
Right angle: angle =
I
Obtuse angle: 90◦ < angle < 180◦
I
Straight angle: angle = 180◦
I
Reflex angle: 180◦ < angle < 360◦
ˆ
(b̂,d)
(â + b̂ + ĉ)
It all begins with lines
I
I
I
I
I
Adjacent angles: share a vertex and a common side
ˆ dˆ& ĉ)
(e.g. â & d,
Vertically opposite angles: angles opposite each other when
two lines intersect. They share a vertex and are equal.
ˆ
(â = ĉ, b̂ = d)
Supplementary angles: two angles that add up to 180◦
(â + b̂ = 180◦ , b̂ + ĉ = 180◦ , ĉ + dˆ = 180◦ , dˆ + â = 180◦ )
Complementary angles: two angles that add up to 90◦
Revolution: the sum of all angles around a vertex
(â + b̂ + ĉ + dˆ = 360◦ )
Parallel lines
Two lines are parallel if the perpendicular distance between them is
constant. We indicate on a diagram that two lines are parallel with
a pair of arrows. We also write
AB||CD
and
M N ||OP
(We write AB for the line through A and B and AB for the line
segment from point A to point B.)
Def.: a transversal is a line that intersects two lines at two distinct
points.
Def.: interior angles lie between parallel lines
ˆ
(â, b̂, ĉ, d)
Def.: exterior angles lie outside parallel lines
(ê, fˆ, ĝ, ĥ)
Def.: corresponding angles same side of the lines
and same side of the transversal (â & ê, b̂ & fˆ,
ĉ & ĝ,dˆ& ĥ) – F-shape
If the lines are parallel, then corresponding angles
are equal.
Def.: co-interior angles lie between the lines and on
ˆ b̂ & ĉ) –
the same side of the transversal (â & d,
C-shape
If the lines are parallel, co-interior angles are
supplementary.
Def.: alternate interior angles lie inside the lines and
ˆ –
on opposite sides of the transversal (â, b̂, ĉ, d)
Z-shape
If the lines are parallel, the alternate interior angles
are equal.
From equal angles to parallel lines
If two lines are intersected by a transversal such
that
I corresponding angles are equal; or
I alternate interior angles are equal; or
I co-interior angles are supplementary
then the two lines are parallel.
Exercise 1: parallel lines
Determine the unknown angles. Is EF ||CG?
Justify your answer.
Exercise 2: parallel lines
If AB||CD and AB||EF , explain why CD must be
parallel to EF .
Exercise 3: parallel lines
Find the value of x.
Exercise 3: parallel lines
Better wording: Find the value of x without using
properties of triangles.
Assessment:
Create a question that involves two sets of parallel
lines.
Discussion:
I
I
Which definitions and concepts from angles and
parallel lines do learners find easy to grasp?
What are the most common misconceptions
that learners have or most common errors that
learners make with angles and parallel lines?
Triangles
Triangles
A triangle is a three-sided polygon. They can be
classified according the their sides and/or angles.
A triangle with vertices A, B and C is denoted by
4ABC.
Types of triangles
I
I
I
I
I
I
Scalene triangle: all sides and all angles are
different.
Isosceles triangle: two sides have equal length;
the angles opposite the equal sides are equal.
Equilateral: all sides are equal in length; all
angles are equal.
Acute: all angles are less than 90◦ .
Obtuse: one interior angle is greater than 90◦ .
Right-angled: one interior angle is equal to 90◦ .
Activity: show that interior angles of a ∆ add up to 180◦
Congruent triangles
Def.: two triangles are congruent if they have equal
corresponding angles and sides.
Informally one can say that two triangles are
congruent if one fits exactly over the other.
We denote congruency by 4ABC ≡ 4P QR.
You might sometimes see the notation
4ABC ∼
= 4P QR.
Note: ordering of vertices is important.
Requirements for congruency
I
I
I
I
RHS (or 90◦ HS): the hypotenuse and one
corresponding side of two right-angled triangles
are equal.
SSS: all three corresponding sides of two
triangles are equal in length.
SAS (S∠S): two sides and the included angle
are equal to the corresponding two sides and
corresponding angle.
AAS (∠∠S): one side and two angles are equal
to the corresponding side and corresponding
angles.
Congruency in pictures
RHS:
SSS:
Congruency in pictures
SAS:
AAS:
Similar triangles
Def.: two triangles are similar if their corresponding
angles are equal and their corresponding sides are in
proportion.
Informally, two triangles are similar if they have the
same shape but possibly different sizes.
If 4ABC and 4DEF are similar we write
4ABC|||4DEF .
Note 1: if two triangles are congruent they are also
similar. However, not all pairs of similar triangles
are a pair of congruent triangles.
Note 2: the ordering of the vertices is important.
Requirements for similarity
I
AAA: all three corresponding angles in two triangles are equal.
I
SSS: all three pairs of corresponding sides are in proportion, i.e.
The theorem of Pythagoras
If 4ABC is a right-angled triangle with
B̂ = 90◦ , then
b2 = a2 + c2
Converse: If b2 = a2 + c2 then 4ABC is a
right-angled triangle with B̂ = 90◦ .
Exercise 1: similarity
Calculate the unknown variables:
Exercise 2: similarity
Calculate the unknown variables:
Discussion:
I
I
I
I
Which concept do students find easiest to
understand: congruency or similarity of
triangles?
Is it the same concept on which they find it
easiest to answer questions?
What are the most common misconceptions
that learners have or most common errors that
learners make with congruent/similar triangles?
Are there specific questions or specific types of
questions that we can get them to attempt in
order to challenge their misconceptions?