Download The learners will use properties of congruent triangles

Lesson 4.3: Congruent Triangles
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Learning Objectives:
The learners will use properties of congruent triangles
The learners will prove triangles congruent by using the definition of congruence.
Common Core Standards: Prove triangles congruent by using the definition of congruence
G-CO.7. Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if
corresponding pairs of sides and corresponding pairs of angles are congruent.
Continuity:
Previous Lessons
Yesterday we proved several
angle relationships in
triangles using interior and
exterior angles.
This Lesson
Today, we will prove
triangles (polygons)
congruent by examining
their corresponding sides
and angles.
Next Lesson
Next, we will examine more
specific triangle congruence
theorems.
Opening:


Give Homework Quiz on 4.1
Go over homework from 4.2 (2-5, 6-14E, 24, 28)
Launch: Warm Up
Look at the figure, what do we know about ∆𝐴𝐵𝐶 and ∆𝐷𝐸𝐹?


∠𝐵 ≅ ∠𝐸
∠𝐴 ≅ ∠𝐷
How do we know this?

Because of the arc marks. We use these to show angles are congruent.
What can we conclude about ∠𝐶 & ∠𝐹?

∠𝐶 ≅ ∠𝐹
How do we know this?

Third Angles Theorem
Review Congruence
What do we know?
Congruent_________
Definition
Segment
Two line segments are
congruent if they have
the same length.
Picture
Symbols
̅̅̅̅
𝐴𝐵 ≅ ̅̅̅̅
𝑃𝑄
is read as "The line
segment AB is congruent to
the line segment PQ".
They need not lie at the same
angle or position on the plane
Note the single 'tic' marks on
the lines. These are a
graphical way to show that
the two line segments are
congruent.
Angle
Two angles are
congruent if they have
the same measure.
∠ABC ≅ ∠PQR
is read as "The angle ABC
is congruent to the angle
PQR"
Congruent angles may lie
in different orientations or
positions.
Rays and lines cannot be
congruent because they do
not have both end points
defined, and so have no
definite length.
Explore:

Naming Congruent Corresponding Parts
o How do you name a polygon?
 You must write the vertices in consecutive order.
 What does consecutive mean?
 You can write them in counterclockwise or clockwise order
 Using the triangles to the right we can name them:
o ∆𝐴𝐵𝐶 and ∆𝐷𝐸𝐹
o ∆𝐶𝐵𝐴 and ∆𝐹𝐸𝐷
 Give a four sided polygon example to show how you can NOT label
A
B
C
D
o Polygon ABCD
o NOT polygon ADCB
o Not consecutive
o In a congruence statement, the order of the vertices indicates the corresponding parts
 Example: Congruent triangles have the same size and the same shape. The corresponding
sides and the corresponding angles of congruent triangles are equal.
Note:

Using Corresponding Parts of Congruent Triangles
If two figures have the same shape and the same size, then they are said to be congruent figures.
For example, rectangle ABCD and rectangle PQRS are congruent rectangles as they have the same shape and the same size.
Side AB and side PQ are in the same relative position in each of the figures.
We say that the side AB and side PQ are corresponding sides.
Congruent figures are exact duplicates of each other. One could be fitted over the other so that their corresponding parts coincide.
The concept of congruent figures applies to figures of any type.
Just to recap the ongoing discussion:


Angles and sides of two plane figures are said to be corresponding if they are in the same relative positions in each of the figures.
If one figure coincides with another after a transformation (i.e. a translation, reflection or rotation) that moves the points on the figure
but does not alter its angles or side-lengths, then the figures are said to be congruent.

Proving Triangles are congruent
o You must show all three pairs of sides and all three pairs of angles are congruent
o Geometric figures are congruent if they are the same sized and shape.
o Corresponding angles and corresponding sides are in the same position in polygons with an equal number of
sides.
o Two polygons are congruent if and only if their corresponding sides and angles are congruent.
 Triangles that are the same size and shape are congruent.
Reflect: Exit Slip


What did you learn?
What are you still confused about?
Homework:

Page 234: 2-11, 17-19
Warm UP
Recall Angle
Measures
Acute
0 < ∢𝐴 < 90
Right
m  A = 90
Obtuse
90 < ∢𝐴 < 180
Straight
m  A = 180
Classifying
Triangles by Angle
Measure and Side
Length
(By Angles):
Acute
3 acute  ’s
Right
1 right 
(2 acute  ’s)
Obtuse
1 obtuse 
(2 acute  ’s)
Isosceles
 2  sides
Equilateral
3  sides
(By Sides):
Scalene
No  sides
Recall Parallel
Postulate
How many lines can be drawn through N parallel to ̅̅̅̅̅
𝑴𝑷?
N
M
P
Exactly 1 by the Parallel Postulate.
Equiangular
3 congruent  ’s
(all acute  ’s)
Review Exterior
and Interior Angles
Triangle
Students will
Have students (with partners) rip
Gives students a 10
Students will work with
Sum
Theorem:
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!
“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.
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.
Mentions corollary to the
Triangle Sum Theorem [in a right
triangle, the acute angles are
complementary]
Sketches proof out loud with
students.
non-precise
justification of
the Triangle
Sum Theorem.
Allows them to
learn in a
concrete,
hands-on way.
Allows students
to see the idea
of the proof, but
puts
responsibility
on them to look
it up later.
minutes
partners to make
conjectures. They will
make the correct
conjecture, or if they
make an incorrect
conjecture, this allows
me to target
misconceptions.
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?
CHECK 2
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.
Interior and Exterior Angles
in Triangle
CHECK 3
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 extrension of the
adjacent side. (outside the figure)
 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 Theorem
Exterior Angle: A better visual
of why the proof make sense
Exterior angle – an angle created by extending one side of a figure
CHECK 3
Third Angles Theorem
CHECK 4
Explore Check 4
Must use third angles theorem
How is the third angles theorem related to the Triangle Sum Theorem?
Exterior angle – an angle created by extending one side of a figure
Theorem: The measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles.
…..
In the figures above the exterior angle ∠ABP is equal to the sum of the remote interior angles ∠BAC and ∠ACB.
.
Proof:
………
.
.
If the equivalent angle is taken at each vertex, the exterior angles always add to 360°
Since the measure of an exterior angle equals the sum of its two remote interior angles, the exterior angle is greater than the measure of either
individual remote interior angle.
1
Hands –on
Classification
based on
angles
Students will recall
angle types and will
use the cut-out
triangles to classify
them by angles and by
side lengths with a
partner.
Gives student pairs several
different numbered triangles.
Asks students to remember what
acute, right and obtuse angles
are.
Writes on Board: (see Board
Content; attached)
Talks about classifications of
triangles based on types of
angles. (See board content;
attached)
Allows students
to see visual
representation
of the material
and also to
manipulate the
triangles and
learn in a
kinesthetic and
hands-on way.
The partner
practice allows
them to classify
together.
15
minutes
Students will correctly
identify triangles
according to angle. If
they incorrectly identify
triangles, I have a
greater sense of their
misconceptions, and so
can target them right
away or later on in the
lesson.
Says that these are the interior
angles of the triangle.
Identifies what are the exterior
angles of triangles (see Board
Content; attached)
5
Triangle Sum Students will
Theorem
“discover” or convince
Instructs students to classify the
angles they have (construction
paper) with their partners (based
on angles.)
Confirms which triangles are
which types, using the number on
the back of each triangle for
identification purposes.
Have students (with partners) rip Gives students a 10
non-precise
minutes
corners of each triangle (each
Students will work with
partners to make
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.
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 (in a cloud) the proof for
The Triangle Sum Theorem –
refer them to Page 219 in their
book to see the full proof.
Mentions use of Auxiliary line.
justification of
the Triangle
Sum Theorem.
Allows them to
learn in a
concrete,
hands-on way.
Allows students
to see the idea
of the proof, but
puts
responsibility
on them to look
it up later.
conjectures. They will
make the correct
conjecture, or if they
make an incorrect
conjecture, this allows
me to target
misconceptions.
Mentions corollary to the
Triangle Sum Theorem [in a right
triangle, the acute angles are
complementary]
Sketches proof out loud with
students.
6
Exterior
Angle
Theorem
Students will help
sketch the Exterior
angle theorem.
Students will see why
this theorem seems
logical (Geogebra).
Sketches the Exterior Angle
Theorem-- tells them this is part
of their homework.
Uses GEOGEBRA to “convince
them” in a non-precise sense that
Includes
technology in
the classroom
for the purpose
of making a
conjecture.
10
minutes
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
7
Closing
Statements
this is true (see attached)
Conclusion Statement (see above) Wraps up what
Exit Ticket (see attached)
was covered.
Homework Worksheet
Assesses
student
learning.
10
minutes
true.
Students will complete
the Exit Ticket and will
work on homework.