MATHEMATICS 20-2
Properties of Angles and Triangles
Lessons 3 and 4 not completed
High School collaborative venture with
Harry Ainlay, McNally, M. E LaZerte, Ross Sheppard, Scona,
and W.P. Wagner
Harry Ainlay: Colin Veldkamp
Harry Ainlay: Debby Sumantry
Harry Ainlay: Mathias Stewart
Harry Ainlay: Meriel Hughes
McNally: Enchantra Gramlich
M. E. LaZerte: Monique Merchant
Ross Sheppard: Jeremy Klassen
Ross Sheppard: Tim Gartke
Scona: Joe Johnston
W. P. Wagner: Kiki Brisebois
Facilitator: John Scammell (Consulting Services)
Editor: Jim Reed (Contracted)
2010 – 2011
Mathematics 20-2
Properties of Angles and Triangles
Page 2 of 39
TABLE OF CONTENTS
STAGE 1
DESIRED RESULTS
PAGE
Big Idea
4
Enduring Understandings
4
Essential Questions
4
Knowledge
5
Skills
6
STAGE 2
ASSESSMENT EVIDENCE
Transfer Task (on a separate page which could be photocopied & handed out to
students)
Carpentry Short Cuts
Teacher Notes for Transfer Task
Transfer Task
Rubric
Possible Solution
7
9
15
17
STAGE 3 LEARNING PLANS
Lesson #1
Parallel Lines
21
Lesson #2
Triangles and Polygons
27
Lesson #3
Congruent Triangles
31
Lesson #4
Geometric Proofs
34
Appendix – Worksheets/Keys
Mathematics 20-2
37
Properties of Angles and Triangles
Page 3 of 39
Mathematics 20-2
Properties of Angles and Triangles
STAGE 1
Desired Results
Big Idea:
Angles and lines are encountered in many places in everyday life. The capacity to
describe and relate angles and lines allows us to design and describe objects in many
contexts. The angle and line theorems are a context in which students can apply their
knowledge of logic and reasoning.
Implementation note:
Post the BIG IDEA in a prominent
place in your classroom and refer to
it often.
Enduring Understandings:
Students will understand …





When a transversal intersects parallel lines, there are numerous pairs of equivalent
angles.
A proof follows a logical set of linked steps.
The sum of the angles in a triangle is 180 degrees.
Congruent means same size and shape.
It is possible to use logic to determine whether two triangles are congruent based on
incomplete information.
Essential Questions:





How can you construct parallel lines using only a compass and straight edge?
What conditions are necessary to prove that two triangles are congruent?
How can you tell if an argument is invalid?
Is it possible to draw a triangle whose angles do not have a sum of 180 degrees?
What is so important about proofs?
Implementation note:
Ask students to consider one of the
essential questions every lesson or two.
Has their thinking changed or evolved?
Mathematics 20-2
Properties of Angles and Triangles
Page 4 of 39
Knowledge:
Enduring
Understanding
List enduring
understandings (the
fewer the better)
Specific
Outcomes
List the reference
# from the
Alberta Program
of Studies
Students will know …
Students will understand…

When a transversal
intersects parallel
lines, there are
numerous pairs of
equivalent angles.
*G1, G2
A proof follows a
logical set of linked
steps.
The sum of the angles
in a triangle is 180
degrees.
G1, G2

Congruent means
same size and shape.
It is possible to use
logic to determine
whether two triangles
are congruent based
on incomplete
information.

How to construct a proof.
Students will know …
G2

G1, G2

Students will understand…

Which angle pairs are equal when a
transversal intersects parallel lines.
Which pairs are supplementary.
Students will know …
Students will understand…



Students will understand…

Description of
Knowledge
The paraphrased outcome that the group is
targeting
That the sum of the angles in a triangle is 180
degrees.
 The sum of the interior angles of a polygon is
only dependent on the number of sides.
Students will know …

The necessary and sufficient conditions for
congruency.
Congruent triangles have the same size and
shape.
888888
I*G = Geometry
Mathematics 20-2
Properties of Angles and Triangles
Page 5 of 39
Skills:
Enduring
Understanding
List enduring
understandings (the
fewer the better)
Specific
Outcomes
List the reference
# from the
Alberta Program
of Studies
Students will be able to…
Students will understand…

When a transversal
intersects parallel
lines, there are
numerous pairs of
equivalent angles.
*G1,G2

G1, G2


Students will understand…

A proof follows a
logical set of linked
steps.
The sum of the angles
in a triangle is 180
degrees.
Determine if lines are parallel, given the
measure of an angle at each intersection
formed by the lines and a transversal.
 Determine particular values of unknown angles
in a diagram of parallel lines intersected by a
transversal.
 Construct parallel lines given a compass or
protractor.
 Solve a contextual problem involving angles.
Students will be able to…
G2




Congruent means
same size and shape.
It is possible to use
logic to determine
whether two triangles
are congruent based
on incomplete
information.
*G = Geometry
Mathematics 20-2
Identify the necessary pieces of information
from a diagram.
Generalize a rule for the relationship between
teh sum of the interior angles and the number
of sides of a polygon.
Determine the sum of the angles in an n-sided
polygon.
Students will be able to…
Students will understand…

Construct a valid proof.
Identify and correct errors in a proof.
Students will be able to…
Students will understand…

Description of
Skills
The paraphrased outcome that the group is
targeting
G1, G2


Prove that two triangles are congruent.
Solve a contextual problem that involves
congruent triangles.
Implementation note:
Teachers need to continually ask
themselves, if their students are
acquiring the knowledge and skills
needed for the unit.
Properties of Angles and Triangles
Page 6 of 39
STAGE 2
1
Assessment Evidence
Desired Results Desired Results
Carpentry Short Cuts
Teacher Notes
There is one transfer task to evaluate student understanding of the concepts relating to
properties of angles and triangles. A photocopy-ready version of the transfer task is
included in this section.
Implementation note:
Students must be given the transfer task & rubric at
the beginning of the unit. They need to know how
they will be assessed and what they are working
toward.
Each student will:



Use properties of angles and triangles to prove that a drawer is divided into thirds.
Use properties of angles and triangles to find the error in a proof showing two
triangles are congruent.
Use properties of angles and triangles to show 4 triangles within a larger triangle are
congruent.
Mathematics 20-2
Properties of Angles and Triangles
Page 7 of 39
Teacher Notes for Carpentry Short Cuts Transfer Task
Glossary
adjacent angles – Angles with a common vertex and a common arm
alternate exterior angles – Angles that are in opposite positions relative to a transversal
intersecting two lines. If the alternate angles are outside the two lines intersected by the
transversal, they are called alternate exterior angles.
alternate exterior angles – Angles that are in opposite positions relative to a transversal
intersecting two lines. If the alternate angles are inside the two lines intersected by the
transversal, they are called alternate interior angles.
congruent – Have the same shape and size
converse – A conditional statement formed by interchanging the if and then clauses of
another conditional statement
convex polygon – A polygon with all interior angles smaller in measure than a straight
angle (180°).
corresponding angles – Angles that are in the same position relative to lines intersected
by a transversal
equilateral triangle – A triangle with three congruent sides (and three congruent angles)
exterior angle of a polygon – An angle at a vertex of the polygon, outside the polygon,
formed by one side and the extension of an adjacent side
isosceles triangle - A triangle with two congruent sides (and two congruent angles)
non-adjacent interior angles – The two angles in a triangle that do not have the same
vertex as the exterior angle [Math 20-2 (Nelson: page 516)]
scalene triangle – A triangle with no congruent sides (and no congruent angles)
transversal – A line that intersects two (or more) lines
Glossary hyperlinks redirect you to the Learn Alberta Mathematics Glossary
(http://www.learnalberta.ca/content/memg/index.html). Some terms can be found in more
than one division. Some terms have animations to illustrate meanings.
Implementation note:
Teachers need to consider what performances and
products will reveal evidence of understanding?
What other evidence will be collected to reflect
the desired results?
Mathematics 20-2
Properties of Angles and Triangles
Page 8 of 39
Carpentry Short Cuts - Student Assessment Task
1.
Part A: Your friend is a carpenter and he is making a chest of
drawers. He would like each drawer to have a handle centered in the
face of the drawer. The problem he faces is that he has a twelve-inch
ruler, but the drawer is only ten inches wide. What he needs to do is
to divide the drawer perfectly into thirds horizontally so that he can put
the handle on correctly, but he is concerned that dividing ten by three
will mean that he is slightly off when he goes to measure. What he
does instead is to rotate the ruler slightly so that he is measuring
twelve inches from edge to edge as shown below. It is easy to mark
off a third of 12: he simply makes his line from a corner to the opposite
side and marks every 4 inches.
Source: http://www.smallscalechemistry.colostate.edu/equipment.html
Carpentry Short Cuts
He then slides the ruler down a few inches and repeats the steps above.
Now if he connects the marks he made, he will have cut the drawer
into thirds.
Carpentry Short Cuts
As a math student you find this interesting, but you are concerned that
what he is doing is actually correct. Prove that your friend has actually
divided the drawer into thirds. Assume that the two vertical lines you
drew are perpendicular to the top and bottom of the drawer, the
corners of the drawer are square, and each oblique line segment is
equal. What you are proving is that the horizontal line segments
between the vertical lines and the edges are all equal.
Carpentry Short Cuts
Part B: Your friend is building a rafter in the shape of an isosceles triangle.
He knows the rafter will be strongest when the two smaller triangles created
by the support beam are congruent. He draws an isosceles triangle with a
blue line segment representing the support beam, as shown in the diagram
below. The following proof is used to show that no matter where the line
segment representing the support beam meets the opposite side, that the
two triangles formed are congruent.
A
C
Statement
is isosceles
D
B
Reason
Given
Definition of Isosceles
Definition of Isosceles
Reflexive property / Shared Side
SAS
Identify and explain the error in the above proof. Under what circumstances is
the above proof valid? In other words what other condition(s) would be
necessary for the two triangles to be congruent?
s
Carpentry Short Cuts
Part C: In the shaded space provided, draw points A, B and C. Construct a
parallel line to line segment AB, through C. Construct another parallel line
to line segment BC though A, and finally repeat this process drawing a line
parallel to AC through B. These lines should intersect to form a large
triangle. Now connect points A, B and C. Prove that all four of the resulting
triangles are congruent.
Carpentry Short Cuts
Put your proof here.
Statement
Reason
Glossary
adjacent angles – Angles with a common vertex and a common arm
alternate exterior angles – Angles that are in opposite positions relative to a
transversal intersecting two lines. If the alternate angles are outside the two lines
intersected by the transversal, they are called alternate exterior angles.
alternate exterior angles – Angles that are in opposite positions relative to a
transversal intersecting two lines. If the alternate angles are inside the two lines
intersected by the transversal, they are called alternate interior angles.
congruent – Have the same shape and size
converse – A conditional statement formed by interchanging the if and then clauses
of another conditional statement
convex polygon – A polygon with all interior angles smaller in measure than a
straight angle (180°).
corresponding angles – Angles that are in the same position relative to lines
intersected by a transversal
equilateral triangle – A triangle with three congruent sides (and three congruent
angles)
exterior angle of a polygon – An angle at a vertex of the polygon, outside the
polygon, formed by one side and the extension of an adjacent side
isosceles triangle - A triangle with two congruent sides (and two congruent angles)
non-adjacent interior angles – The two angles in a triangle that do not have the
same vertex as the exterior angle [Math 20-2 (Nelson: page 516)]
scalene triangle – A triangle with no congruent sides (and no congruent angles)
transversal – A line that intersects two (or more) lines
Glossary hyperlinks redirect you to the Learn Alberta Mathematics Glossary
(http://www.learnalberta.ca/content/memg/index.html). Some terms can be found in more
than one division. Some terms have animations to illustrate meanings.
Assessment
Mathematics 20-2
Properties of Angles and Triangles
Rubric
Level
Excellent
Criteria
4
Math
All required
Content
elements are
Part A
present and
correct
Math
Content
Part B
All required
elements are
present and
correct
Math
Content
Part C
All required
elements are
present and
correct
Presents
Data
Presentation
of data is
clear,
precise and
accurate
Explains
Choices
Provides
insightful
explanations
Proficient
3
All required
elements are
present but
may contain
minor errors
Adequate
2
Some
required
elements are
missing, or
contain
major errors
All required
Some
elements are required
present but
elements are
may contain missing, or
minor errors contain
major errors
All required
Some
elements are required
present but
elements are
may contain missing, or
minor errors contain
major errors
Presentation Presentation
of data is
of data is
complete
simplistic
and
and
unambiguo plausible
us
Provides
Provides
logical
explanations
explanations that are
complete
but vague
Limited
1
Most
required
elements are
missing or
incorrect
Insufficient
Blank
No score is
awarded as
there is no
evidence
given
Most
required
elements are
missing or
incorrect
No score is
awarded as
there is no
evidence
given
Most
required
elements are
missing or
incorrect
No score is
awarded as
there is no
evidence
given
Presentation
of data is
vague and
inaccurate
Presentation
of data is
incomprehe
nsible
Provides
explanations
that are
incomplete
or
confusing.
No
explanation
is provided
When work is judged to be limited or insufficient, the teacher makes decisions
about appropriate intervention to help the student improve.
Possible Solution to Carpentry Short Cuts
As a math student you find this interesting, but you are concerned that
what he is doing is actually correct. Prove that your friend has actually
divided the drawer into thirds. Assume that the two vertical lines you
drew are perpendicular to the top and bottom of the drawer, the
corners of the drawer are square, and each oblique line segment is
equal. What you are proving is that the horizontal line segments
between the vertical lines and the edges are all equal.
Use corresponding angles to show that all of the A1’s and B1’s
are equal. Since the sum of interior angles in a triangle is 180o,
the missing angle in the lightest grey triangles is 90 – B1. This
establishes that
(ASA). Within each of these
large triangles there are 2 similar triangles (lightest grey triangle
and the triangle formed by the two lightest grey areas. Since the
original diagonal line was divided into 3 equal regions, the
corresponding lines in each similar triangle are also equal.
Carpentry Short Cuts
Part B: Your friend is building a rafter in the shape of an isosceles triangle.
He knows the rafter will be strongest when the two smaller triangles created
by the support beam are congruent. He draws an isosceles triangle with a
blue line segment representing the support beam, as shown in the diagram
below. The following proof is used to show that no matter where the line
segment representing the support beam meets the opposite side, that the
two triangles formed are congruent.
A
C
Statement
is isosceles
D
B
Reason
Given
Definition of Isosceles
Definition of Isosceles
Reflexive property / Shared Side
SAS
Identify and explain the error in the above proof. Under what circumstances is
the above proof valid? In other words what other condition(s) would be
necessary for the two triangles to be congruent?
The error is in the property used to show the congruence of the
s triangles. The reasons given actually demonstrate the SSA property,
which cannot be used to show congruence of triangles, because
under certain conditions two triangles can be constructed that are
not congruent.
The proof would be valid if the support beam is a perpendicular
bisector. This would make
and now we can say
(SAS).
We could also establish congruence by ASA if we establish
.
Carpentry Short Cuts
Part C: In the shaded space provided, draw points A, B and C. Construct a
parallel line to line segment AB, through C. Construct another parallel line
to line segment BC though A, and finally repeat this process drawing a line
parallel to AC through B. These lines should intersect to form a large
triangle. Now connect points A, B and C. Prove that all four of the resulting
triangles are congruent.
A1 and B1 are placed using corresponding angles. The third angle in the
outer triangles must be 180 – (A1 + B1).
The innermost triangle angles can be determined because the sum of
angles that form a line is 180o. This establishes the four smaller triangles
are similar. Equal angles with a common side proves congruence (ASA).
Carpentry Short Cuts
Put your proof here.
Statement
A1 angles are equal
B1 angles are equal
Third angle in outer triangles is
180 – (A1 + B1)
Adjacent triangles share a
common side.
Reason
corresponding angles
corresponding angles
sum of interior angles is 180o
These triangles are congruent (ASA)
STAGE 3
Learning Plans
Lesson 1
Parallel Lines
STAGE 1
BIG IDEA: Angles and lines are encountered in many places in everyday life. The capacity to describe
and relate angles and lines allows us to design and describe objects in many contexts. The angle and
line theorems are a context in which students can apply their knowledge of logic and reasoning.
ENDURING UNDERSTANDINGS:
ESSENTIAL QUESTIONS:
Students will understand …



When a transversal intersects parallel lines,
there are numerous pairs of equivalent
angles.
Congruent means same size and shape.
How can you construct parallel lines using
only a compass and straight edge?
KNOWLEDGE:
SKILLS:
Students will know …
Students will be able to …



Which angle pairs are equal when a
transversal intersects parallel lines.
Which pairs are supplementary.



Determine if lines are parallel, given the
measure of an angle at each intersection
formed by the lines and a transversal.
Determine particular values of unknown
angles in a diagram of parallel lines
intersected by a transversal.
Construct parallel lines given a compass or
protractor.
Solve a contextual problem involving angles.
Implementation note:
Each lesson is a conceptual unit and is not intended to
be taught on a one lesson per block basis. Each
represents a concept to be covered and can take
anywhere from part of a class to several classes to
complete.
Mathematics 20-2
Properties of Angles and Triangles
Page 21 of 39
Lesson Summary








Introduce Terminology related to pairs of angles formed by transversals and parallel
lines.
Use applets to introduce terminology.
Have students generate notes.
Explore properties of transversals and parallel lines.
Provide access to Math Interactives – Exploring Parallel Lines (Explore It #1).
Provide students with Quick Check.
Practice properties involving two parallel lines and transversal(s).
Provide access to Math Interactives – Exploring Parallel Lines (Use It).
Lesson Plan
Introduce Terminology related to pairs of angles formed by transversals and parallel lines.
Use the applets found at http://members.shaw.ca/jreed/math20-2/program2.htm to introduce
the relationships and terminology related to transversals and parallel lines. Text links are
provided for the LearnAlberta resources that are displayed on
http://members.shaw.ca/jreed/math20-2/program2.htm. Screenshots on this page are hot
linked to the original resources.
Discuss the relationships using the applets titled:
Mathematics 20-2
Properties of Angles and Triangles
Page 22 of 39




Opposite Angles
Alternate Angles (Transversal)
Angles on the Same Side of the Transversal,
Corresponding Angles (Transversal)
Have students generate their own notes individually or as a group to help them remember the
properties discussed. Encourage students to draw pictures of the equal or supplementary
angle pairs.
Mathematics 20-2
Properties of Angles and Triangles
Page 23 of 39
Explore properties of transversals and parallel lines.
Math Interactives – Exploring Parallel Lines (Explore It #1) found on the same page as the
previous applets (http://members.shaw.ca/jreed/math20-2/program2.htm)
Allow students to play and explore using this applet. This applet provides a great visual of the
relationships of transversals and parallel lines and will help students reinforce their
understanding. Encourage students to play with all the options within the applet (Angle Type,
Reference Angle, Parallel Line Orientation)
Quick Check
This may be used to assess student’s knowledge. The next section of this lesson involves an
interactive game and this quick check could be used as an entry pass to be able to play the
game.
Quick Check
 copy was added to Appendix
Mathematics 20-2
Properties of Angles and Triangles
Page 24 of 39
Practice properties of transversals
Math Interactives – Exploring Parallel Lines (Use It) found on the same page as the previous
applets (http://members.shaw.ca/jreed/math20-2/program2.htm)
This game is a great way for students to practice their skills. Encourage students to use the
Hint button, the Explore It applet, or their notes to help them win the game.
This would also be a great place to discuss the Essential Question for the lesson and
challenge students to come up with a strategy to draw two parallel lines using a ruler and a
protractor.
Going Beyond
Resources
Math 20-2 (Nelson: sec 2.1 and 2.2, page(s) 70-82)
Supporting
Mathematics 20-2
Properties of Angles and Triangles
Page 25 of 39
Assessment
Glossary
adjacent angles – Angles with a common vertex and a common arm
alternate exterior angles – Angles that are in opposite positions relative to a transversal
intersecting two lines. If the alternate angles are outside the two lines intersected by the
transversal, they are called alternate exterior angles.
alternate interior angles – Angles that are in opposite positions relative to a transversal
intersecting two lines. If the alternate angles are inside the two lines intersected by the
transversal, they are called alternate interior angles.
congruent – Have the same shape and size
corresponding angles – Angles that are in the same position relative to lines intersected by a
transversal
transversal – A line that intersects two (or more) lines
Glossary hyperlinks redirect you to the Learn Alberta Mathematics Glossary
(http://www.learnalberta.ca/content/memg/index.html). Some terms can be found in more
than one division. Some terms have animations to illustrate meanings.
Other
Mathematics 20-2
Properties of Angles and Triangles
Page 26 of 39
Lesson 2
Triangles and Polygons
STAGE 1
BIG IDEA: Angles and lines are encountered in many places in everyday life. The capacity to describe
and relate angles and lines allows us to design and describe objects in many contexts. The angle and
line theorems are a context in which students can apply their knowledge of logic and reasoning.
ENDURING UNDERSTANDINGS:
ESSENTIAL QUESTIONS:
Students will understand …



A proof follows a logical set of linked steps.
The sum of the angles in a triangle is 180
degrees.
Is it possible to draw a triangle whose angles
do not have a sum of 180 degrees?
KNOWLEDGE:
SKILLS:
Students will know …
Students will be able to …



That the sum of the angles in a triangle is 180
degrees.
The sum of the interior angles of a polygon is
only dependent on the number of sides.

Generalize a rule for the relationship between
the sum of the interior angles and the number
of sides of a polygon.
Determine the sum of the angles in an n-sided
polygon.
Implementation note:
Each lesson is a conceptual unit and is not intended to
be taught on a one lesson per block basis. Each
represents a concept to be covered and can take
anywhere from part of a class to several classes to
complete.
Lesson Summary

The students will explore angles in polygons. The will see several ways of illustrating
that the sum of the angles in a triangle is 180°, and will extend this to other closed
polygons.
Mathematics 20-2
Properties of Angles and Triangles
Page 27 of 39
Lesson Plan
Lesson Goal
By the end of the lesson, students should be aware that the sum of the angles in a triangle is
180°, and the sum of the angles (°) in an n-sided polygon is 180(n-2).
Hook/Activate Prior Knowledge
Give students a piece of paper, a straight edge, and a protractor. Tell them that you are going
to give a prize to the student who draws the triangle whose angles have the greatest sum, and
another prize to the student who draws the triangle whose angles have the smallest sum. Let
them draw and measure some triangles, and conclude that all triangles have angles with a
sum of 180°.
Show them a rudimentary proof, by having them cut out one of their triangles, and piece it
together as shown below.
Lesson
Define Exterior Angle
Have students draw a triangle and an exterior angle. Use the protractor to measure the
exterior angle and compare it to the interior angles. Students should notice that the exterior
angle is equal to the sum of the interior and opposite angles. They should also notice that the
exterior angle and its adjacent angle are supplementary.
Give an example like Example 3 on Page 88 of the Nelson resource, and give students time to
work through it in pairs.
Have students use their rulers to draw 4-, 5-, and 6-sided polygons. Ask them to measure the
interior angles of each of them, and find the sum.
Ask students to look for a pattern and use that pattern to predict the sum of the angles in a 25sided polygon. Compare answers. Students should discover that the sum of the angles (°) in
an n-sided polygon have a sum of 180(n-2).
Mathematics 20-2
Properties of Angles and Triangles
Page 28 of 39
Give students an example like #16 on Page 102 of the Nelson resource, and have them work
through it in pairs.
Going Beyond
Students could use Geogebra or Geometer’s Sketchpad to complete the investigations in this
lesson.
Students could use a spreadsheet to determine the sum of the angles in an n-sided polygon.
Ask students whether the 180(n-2) formula works on all polygons. Does it work on convex,
concave, regular, and irregular polygons?
Resources
Math 20-2 (Nelson: sec 2.3 and 2.4, page(s) 86 to 103)
Supporting
Sum of interior angles in a triangle applets:
http://www.walter-fendt.de/m14e/anglesum.htm
http://206.110.20.132/~dhabecker/geogebrahtml/triangle_sum/triangle_sum.html
http://staff.argyll.epsb.ca/jreed/math9/strand3/triangle_angle_sum.htm
Supplementary angles in a triangle applet:
http://206.110.20.132/~dhabecker/geogebrahtml/supplementary_defined/supplementary.html
(Really Nice) Sum of angles in a polygon applet:
http://www.mathopenref.com/polygoninteriorangles.html
Exterior angles applets:
http://www.mathopenref.com/polygonanglerelation.html
http://www.mathopenref.com/triangleextangle.html
Interior/exterior polygon angles:
http://staff.argyll.epsb.ca/jreed/math9/strand3/polygon_angles.htm
Mathematics 20-2
Properties of Angles and Triangles
Page 29 of 39
Assessment
Students could be given an exit slip with a couple questions like:
http://www.onlinemathlearning.com/quadrilateral-worksheets.html
a.
b.
c.
Questions could be assigned for homework from the Nelson resource sections 2.3 and 2.4.
Glossary
adjacent angles – Angles with a common vertex and a common arm
convex polygon – A polygon with all interior angles smaller in measure than a straight angle
(180°).
exterior angle of a polygon – An angle at a vertex of the polygon, outside the polygon,
formed by one side and the extension of an adjacent side
Glossary hyperlinks redirect you to the Learn Alberta Mathematics Glossary
(http://www.learnalberta.ca/content/memg/index.html). Some terms can be found in more
than one division. Some terms have animations to illustrate meanings.
Other
Mathematics 20-2
Properties of Angles and Triangles
Page 30 of 39
Lesson 3
Congruent Triangles
STAGE 1
BIG IDEA: Angles and lines are encountered in many places in everyday life. The capacity to describe
and relate angles and lines allows us to design and describe objects in many contexts. The angle and
line theorems are a context in which students can apply their knowledge of logic and reasoning.
ENDURING UNDERSTANDINGS:
ESSENTIAL QUESTIONS:
Students will understand …




A proof follows a logical set of linked steps.
Congruent means same size and shape.
It is possible to use logic to determine
whether two triangles are congruent based
on incomplete information.


What conditions are necessary to prove that
two triangles are congruent?
How can you tell if an argument is invalid?
What is so important about proofs?
KNOWLEDGE:
SKILLS:
Students will know …
Students will be able to …





How to construct a proof.
The necessary and sufficient conditions for
congruency.
Congruent triangles have the same size and
shape.


Construct a valid proof.
Identify the necessary pieces of information
from a diagram.
Prove that two triangles are congruent.
Solve a contextual problem that involves
congruent triangles.
Lesson Summary

Mathematics 20-2
Properties of Angles and Triangles
Page 31 of 39
Lesson Plan
Hook
Lesson Goal
Activate Prior Knowledge
Lesson
Going Beyond
Resources
Math 20-2 (Nelson: sec 2.5, page(s) 104-106)
Supporting
Congruent triangles: http://staff.argyll.epsb.ca/jreed/math9/strand3/3203.htm
Assessment
Glossary
Mathematics 20-2
Properties of Angles and Triangles
Page 32 of 39
adjacent angles – Angles with a common vertex and a common arm
alternate exterior angles – Angles that are in opposite positions relative to a transversal
intersecting two lines. If the alternate angles are outside the two lines intersected by the
transversal, they are called alternate exterior angles.
alternate exterior angles – Angles that are in opposite positions relative to a transversal
intersecting two lines. If the alternate angles are inside the two lines intersected by the
transversal, they are called alternate interior angles.
congruent – Have the same shape and size
converse – A conditional statement formed by interchanging the if and then clauses of another
conditional statement
convex polygon – A polygon with all interior angles smaller in measure than a straight angle
(180°).
corresponding angles – Angles that are in the same position relative to lines intersected by a
transversal
equilateral triangle – A triangle with three congruent sides (and three congruent angles)
exterior angle of a polygon – An angle at a vertex of the polygon, outside the polygon,
formed by one side and the extension of an adjacent side
isosceles triangle - A triangle with two congruent sides (and two congruent angles)
non-adjacent interior angles – The two angles in a triangle that do not have the same vertex
as the exterior angle [Math 20-2 (Nelson: page 516)]
scalene triangle – A triangle with no congruent sides (and no congruent angles)
transversal – A line that intersects two (or more) lines
Glossary hyperlinks redirect you to the Learn Alberta Mathematics Glossary
(http://www.learnalberta.ca/content/memg/index.html). Some terms can be found in more
than one division. Some terms have animations to illustrate meanings.
Other
Mathematics 20-2
Properties of Angles and Triangles
Page 33 of 39
Lesson 4
Geometric Proofs
STAGE 1
BIG IDEA: Angles and lines are encountered in many places in everyday life. The capacity to describe
and relate angles and lines allows us to design and describe objects in many contexts. The angle and
line theorems are a context in which students can apply their knowledge of logic and reasoning.
ENDURING UNDERSTANDINGS:
ESSENTIAL QUESTIONS:
Students will understand …





When a transversal intersects parallel lines,
there are numerous pairs of equivalent
angles.
A proof follows a logical set of linked steps.
The sum of the angles in a triangle is 180
degrees.
Congruent means same size and shape.
It is possible to use logic to determine
whether two triangles are congruent based
on incomplete information.


How can you tell if an argument is invalid?
What is so important about proofs?
KNOWLEDGE:
SKILLS:
Students will know …
Students will be able to …

How to construct a proof.



Construct a valid proof..
Prove that two triangles are congruent.
Solve a contextual problem that involves
congruent triangles.
Lesson Summary

Mathematics 20-2
Properties of Angles and Triangles
Page 34 of 39
Lesson Plan
Hook
Lesson Goal
Activate Prior Knowledge
Lesson
Going Beyond
Resources
Math 20-2 (Nelson: sec 2.6, page(s) 107-115)
Supporting
Assessment
Glossary
Mathematics 20-2
Properties of Angles and Triangles
Page 35 of 39
adjacent angles – Angles with a common vertex and a common arm
alternate exterior angles – Angles that are in opposite positions relative to a transversal
intersecting two lines. If the alternate angles are outside the two lines intersected by the
transversal, they are called alternate exterior angles.
alternate exterior angles – Angles that are in opposite positions relative to a transversal
intersecting two lines. If the alternate angles are inside the two lines intersected by the
transversal, they are called alternate interior angles.
congruent – Have the same shape and size
converse – A conditional statement formed by interchanging the if and then clauses of another
conditional statement
convex polygon – A polygon with all interior angles smaller in measure than a straight angle
(180°).
corresponding angles – Angles that are in the same position relative to lines intersected by a
transversal
equilateral triangle – A triangle with three congruent sides (and three congruent angles)
exterior angle of a polygon – An angle at a vertex of the polygon, outside the polygon,
formed by one side and the extension of an adjacent side
isosceles triangle - A triangle with two congruent sides (and two congruent angles)
non-adjacent interior angles – The two angles in a triangle that do not have the same vertex
as the exterior angle [Math 20-2 (Nelson: page 516)]
scalene triangle – A triangle with no congruent sides (and no congruent angles)
transversal – A line that intersects two (or more) lines
Glossary hyperlinks redirect you to the Learn Alberta Mathematics Glossary
(http://www.learnalberta.ca/content/memg/index.html). Some terms can be found in more
than one division. Some terms have animations to illustrate meanings.
Other
Mathematics 20-2
Properties of Angles and Triangles
Page 36 of 39
Appendix
Appendix 1: Properties Quick Check
Mathematics 20-2
Properties of Angles and Triangles
Page 37 of 39
Appendix 1: Properties Quick Check
A
C
E
G
B
D
F
H
Alternate Interior Angles have equal measures.
They are:
___ and ___
___ and ___
Alternate Exterior Angles have equal measures.
They are:
___and____
____and___
Corresponding Angles have equal measures.
They are:
___ and ___
___ and ___
___ and ___
___ and ___
Vertically Opposite angles are equal.
They are:
___ and ___
___ and ___
___ and ___
___ and ___
Interior angles on the same side of the transversal are supplementary (add to180°).
They are:
___ and ___
___ and ___
Exterior angles on the same side of the transversal are supplementary (add to180°).
They are:
Mathematics 20-2
___ and ___
___ and ___
Properties of Angles and Triangles
Page 38 of 39
Properties Quick Check Key
A
C
E
G
B
D
F
H
Alternate Interior Angles have equal measures.
They are:
_C_ and _F_
_D_ and _E_
Alternate Exterior Angles have equal measures.
They are:
_A_and_H__
__B_and_G_
Corresponding Angles have equal measures.
They are:
_A_ and _E_
_B_ and _F_
_C_ and _G_
_D_ and _H_
Vertically Opposite angles are equal.
They are:
_A_ and _D_
_E_ and _H_
_B_ and _C_
_F_ and _G_
Interior angles on the same side of the transversal are supplementary (add to180°).
They are:
_C_ and _E_
_D_ and _F_
Exterior angles on the same side of the transversal are supplementary (add to180°).
They are:
Mathematics 20-2
_A_ and _G_
_B_ and _H_
Properties of Angles and Triangles
Page 39 of 39