Download Lesson 4.2: Angle Relationships in Triangles Page 223 in text

Lesson 4.2: Angle Relationships in Triangles
Page 223 in text
Learning Objectives:
The learners will be able to measures of interior and exterior angles of triangles
The learners will be able to apply theorems about the interior and exterior angles of triangles.
Common Core Standards: Prove geometric theorems
G-CO.10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of
isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the
length; the medians of a triangle meet at a point.
Continuity:
Previous Lessons
Yesterday we learned how
to classify triangles
according to side length and
angle measure.
This Lesson
Today, we will prove that
the angle measures of a
triangle add up to 180
degrees, and we will prove
that the measure of an
exterior angle of a triangle is
equal to the sum of the two
nonadjacent angles of the
triangle
Next Lesson
Next, we’ll add to our
knowledge by exploring
congruence in the context of
triangles.
Lesson Overview
Opening: Hand back tests from Chapter 3
Launch: Warm Up
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
Review Angle Measures
Review Triangle Classifications
Review Interior and Exterior Angles
Explore:
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Triangle Sum Activity
o Tearing triangles angles to show they add up to 180o
o Folding triangle into a rectangle to show angle sum is 180o
Triangle Sum Proof
o Corollaries
 What is a corollary anyways?
Exterior Angle Proof
o Why does this make sense?
o Reflect back to Triangle Sum Activity
Third Angles Theorem
Reflect: Exit Slip

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How are the three main theorems we learned today connected?
What did you learn?
What are you still confused about?
Homework:

2-5, 6-14E, 24, 28,
Detailed Outline
Warm UP
Objective
Activity
Recall Angle
Acute
Right
Obtuse
Measures to connect 0 < ∢𝐴 < 90
m  A = 90
90 < ∢𝐴 < 180
what learners know
about angles to what
they will learn about
classifying triangles
by angle measure.
Classify Triangles by (By Angles):
Angle Measure and
Acute
Right
Obtuse
Side Length by
3 acute  ’s
1 right 
1 obtuse 
identifying
(2 acute  ’s)
(2 acute  ’s)
properties and
attributes of the
figures.
(By Sides):
Scalene
No  sides
Isosceles
 2  sides
Equilateral
3  sides
Straight
m  A = 180
Equiangular
3 congruent  ’s
(all acute  ’s)
Teacher
Use Effective
Questioning
from PBS sheet.
Timing
2 min
Draw visual on
board that
looks like
picture for
learners who
need
organization
and visual.
2 min
Recall Parallel
Postulate
How many lines can be drawn through N parallel to ̅̅̅̅̅
𝑀𝑃 ?
N
M
P
Answer: Exactly 1 by the Parallel Postulate.
Review Exterior
and Interior Angles
Introduce
auxiliary line. If
learners cannot
recall have
them flip back
in their
textbook.
Have students
label all angle
relationships.
3 min
3 min
Dialogue for
Intro.
Proving
Triangles Sum
is 180 degrees.
This is clearly
something
we’ve known
and accepted
for awhile but
now we are
going to prove
it!
Objective of
Activity
Students will
“discover” or
convince themselves
that the Triangle
Sum Theorem
should in fact be
true.
Students will help
with the proof of the
Triangle Sum
Theorem (sketch of
the proof), thus
working on their
proof skills.
Explore
Part 1: Triangle Sum Theorem
Outline of Activity
Teacher
Have students (with partners) rip
corners of each triangle (each
type of triangle by angle) and line
them up on a straight line on
their paper (straight angle) –
have them make conjectures [the
interior angles in a triangle add
up to 180 degrees].
Sketch the proof for The Triangle
Sum Theorem – refer them to
Page 223 in their book to see the
full proof. Mentions use of
Auxiliary line.
***(SEE SKETCH BELOW)
Mentions corollary to the
Triangle Sum Theorem [in a right
triangle, the acute angles are
complementary] CHECK 2 in text
Sketches proof out loud with
students.
Timing
Gives students a
15
non-precise
minutes
justification of
the Triangle Sum
Theorem. Allows
them to learn in a
concrete, handson way.
Allows students
to see the idea of
the proof, but
puts
responsibility on
them to discover
it. Once they
understand the
purpose, they can
delve into the
formal proof and
really make
sense of the idea
being discussed.
Assessment
Students will work
with partners to make
conjectures. They will
make the correct
conjecture, or if they
make an incorrect
conjecture, this allows
me to target
misconceptions.
Sketch of Proof (with guided questing from teacher)
Triangle Sum Proof: With what we
know about parallel lines and
alternate interior angles, it's pretty
straight forward:
Construct Auxiliary Line: a line that is added to a figure to aid in a proof.




How can we justify the auxiliary lines existence? That is, why are we allowed to
construct this line?
Through any two points there is exactly one line.
̅̅̅̅?
But how can we make our auxiliary parallel to AC
By the parallel postulate!!
What kind of Angles did we create? Label Interior and Exterior


What would the transversal be in each case?
Does this make sense in terms of the activity we did when we tore angles from the
triangles?
COROLLARIES for triangle sum theorem: What is a corollary anyways? A theorem
whose proof follows directly from another theorem. So basically, we get these for free, well
almost free.
CHECK 2 in text
Part 2: Exterior Angles Theorem
Dialogue for
Intro.
Part 2:
Exterior
Angles. In the
first activity we
were able to create
and examine the
features of a
triangle. In doing
this, we learned
how to use
deductive
reasoning to
formally prove the
sum of the
measures of the
interior angles is
equal to the
measure of a
straight angle
(180o) of a line
drawn through one
of the vertices. We
will now use that
information to
examine the
exterior angles of a
triangle.
Objective of Activity
Students will use
what they know to
help them sketch the
Exterior Angle
theorem. Students
will see why this
theorem seems
logical and then they
will formulate a
proof.
Outline of Activity
Recall what we know about
triangles
Define terms: remote interior
angles, exterior angles, and
interior angles.***See Definition
Whiteboard Activity BELOW
Read the Exterior Angles
Theorem from book on page 225.
Do CHECK 3 in book with
partners.
Teacher uses GEOGEBRA to
“convince them” in a non-precise
sense that the theorem is true.
Then sketch the proof for The
Exterior Angle Theorem on the
board– refer them to Page 225 in
their book to examples of the
Theorem. ***See Geogebra
sketch BELOW
Teacher
Allows students
to see visual
representation of
the material and
build off prior
knowledge by
connecting
familiar concepts
with the new
ideas.
Includes
technology in the
classroom for the
purpose of
making a
conjecture.
Timing
20 min
Assessment
Students will be
enthusiastic, or at least
engaged in watching
Geogebra. They will
have ideas of how to
sketch the proof of this
theorem and will
believe it to be true. If
the students aren’t
asking the right kinds
of questions or are
applying the wrong
vocabulary, I can stop
the demonstration and
review the main ideas
we are using to
construct the proof.
Interior and Exterior Angles in
Triangle
Whiteboard Organization
An interior angle is formed by two sides of a triangle.( inside the figure)

In figure: ∠1, ∠2, ∠3
An exterior angle is formed by one side of the triangle and the extension of the
adjacent side. (outside the figure)
CHECK 3




IN figure: ∠4
Each exterior angle has two remote interior angles.
In figure: ∠1 𝑎𝑛𝑑 ∠2
A remote interior angle is an interior angle that is not
adjacent to the exterior angle. (Interior and away from exterior)
Exterior Angle: A better visual of why the proof make sense
Geometers Sketchpad Proof (tying everything back together)
Theorem: The measure of an exterior angle
of a triangle is equal to the sum of the
measures of its two remote interior angles.
Using Geogebra, try 2 examples to see if the
theorem is true.
In the figures above the exterior angle ∠ABP is equal to the sum of the remote interior angles ∠BAC and
∠ACB.
Now that the conjecture is believed to be
true, work through a formal proof with
students.
Dialogue for
Intro.
Next we are
going to look
at the Third
Angles
Theorem. We
will discuss
why it makes
sense and how
we can use it.
Reflection
Objective
Summarize lesson.
Complete Exit Slip
Apply
understanding to
homework.
Objective of
Activity
The students will
understand how to
compare angles
amongst two
triangles and will
understand how to
find the third angle of
a triangle by applying
the previous
theorems.
PART 3: Third Angles Theorem
Outline of Activity
Teacher
Read the Third Angles Theorem
Aloud to the class.
Let them work on CHECK 4 in the
text in partners.
As a class discuss our findings
and connections.
Timing
Gives the
10 min
students freedom
to explore the
third theorem,
which is fairly
straightforward,
with one another.
Asks questions
that force the
learners to use
the exterior
angles theorem
and triangle sum
theorem.
Activity
Today we learned how to classify triangles, we learned that the angle measures of a triangle add up
to 180 degrees, and we learned that the measure of an exterior angle of a triangle is equal to the
sum of the two nonadjacent angles of the triangle. Now, it’s time to try some problems that apply
what you know from today’s lesson and from your previous experience with angles, side lengths,
and triangles, to some homework problems. Tomorrow we’ll add to our knowledge of triangles by
exploring congruence in the context of triangles.
Assessment
Students will be given
the freedom to explore
examples that force
them to apply all of the
theorems they learned
to find the third angle.
The teacher can walk
around and ask
questions to ensure
learners understand
the material.
Teacher
Wraps up what
was covered.
Assesses
student
learning.
Timing
10 min
………
.