Download Geo Unit 4

Bell Work for 11/2
Plot the Following Points on a Coordinate Plane:
A(-4,6) B(-2,1) C(-3,-2) D(1,5) E(3,0) F(2,-3)
Connect the points to form two triangles, ABC and DEF
What are some similarities you notice between the two
triangles? What are some differences?
Looking at the points, how can we form triangle DEF using
what we know about the points that form ABC?
Congruent Triangles
Geometry Unit 4
Objective for Today
• Identify congruent polygons based on properties of correspondence.
Congruency
• What did we mean when we said two segments or angles were
congruent?
• What might we mean when we say polygons are congruent?
Congruency
• Congruent Figures have the same size and shape
• If two figures can be manipulated in one of three ways to occupy the
same exact space, then the figures are congruent
• Three Ways to manipulate and keep congruent
• Translation
• Reflection
• Rotation
Translation
Translation—Creating an image of a figure by plotting a
duplicate of each point moved a certain number of
points along the x and/or y axis
Example: Translate the points A(5,6);B(-2,4); and C(3,-3):
• Up Two
• Left Three
• Up Two and Left Three
Reflection
Reflection—Creating an image of a figure by plotting a
duplicate of each point on the opposite side of a specific
line, the same distance away from the line
Example: Reflect the points A(5,6);B(-2,4); and C(3,-3):
• Across the X-Axis
• Across the Y-Axis
Rotation
Rotation—Creating an image of a figure by plotting a
duplicate of each point rotated in a clockwise or
counterclockwise direction a certain number of degrees
around a certain point
Example: Translate the points A(5,6);B(-2,4); and C(3,-3):
• 90 degrees clockwise around the origin
• 90 degrees counter clockwise around the origin
• 180 degrees around the origin
Correspondence
• Each point created using these three methods is said to correspond
with the point used to make it
• Similarly, segments and angles can correspond too
• If all corresponding parts of a figure are the same as the original, than
the two are congruent
Objective for Today
• Identify congruent polygons based on properties of correspondence.
Congruence
Practice
Practice
Practice
Proving Congruency
Congruency
Bell Work for 11/3
Are ABC and DEF congruent in this case?
• We cannot know, we need 3 sides and 3 angles to prove
congruency
Proving Figures Congruent
• What do we need in order to prove that two figures are congruent?
Practice
Bell Work for 11/5
Quiz Today
Everything off of your desks except a pencil and scratch paper.
SSS Postulate
Bell
Work
for
11/9
• Would you use SSS or SAS to prove the following pairs of triangles congruent? If there is not
enough information, write not enough information
SSS Postulate
Using SSS and SAS
Using SSS and SAS
Two New Triplets
Using ASA
Bell Work for 11/10
Recap of Triplets to Prove Triangles
• SSS
• SAS
• ASA
• AAS
What is not enough to prove triangles congruent?
• AAA
• SSA
Bell Work for 11/11
Bell Work for 11/13
CPCTC
• Corresponding Parts of Congruent Triangles are Congruent
• If we know that two triangles are congruent, we know all the pairs of
corresponding parts are also congruent
• CPCTC is simply the converse of the above statement
• In order to use CPCTC, we must always prove triangles congruent first
Bell Work for 11/16
Types of Triangles
Isosceles Triangles
•
Isosceles Triangles are triangles that
have two congruent sides
• We call the congruent sides the
legs and the other the base
•
Isosceles Triangles also have two
congruent angles
• The congruent angles are called the
Base angles and the other is the
Vertex angle
Using Isosceles Triangles
Using Isosceles Triangles
Equilateral Triangles
Bell Work for 11/17
Practice
Practice
• Given the Angle Measures, fill in the all the other remaining angle measures
Find the value of each variable
Bell Work for 11/18
Types of triangles
• By Angles
Right Triangles
Right Triangles
• Properties of Right Triangles
Practice
Bell Work for 11/19
Bell Work for 11/30
• Write Which Postulate or Theorem you would use to prove each pair
of triangles congruent. If there is not enough information, state so.
Steps for All Triangle Proofs
1. Identify the corresponding parts of the triangles
2. Find which ones you can prove are congruent using the ways we
have learned
•
•
•
•
Midpoints and Bisectors
Parallel lines and transversals
Vertical Angles
Reflexive Property
3. Once we have congruent parts, figure out which triangle
congruence theorem to use
• SSS, SAS, ASA, AAS, HL (These are the ONLY ones you can use)
4. Optional: After step 3, use CPCTC to find other congruent parts
Bell Work for 12/1
• Copy down the figure to the right.
• How many triangles can you find in the
figure?
Overlapping Triangles
• Overlapping triangles are triangles that share a common part
• Either a side or an angle
• Common parts can help us prove congruence because they provide a
part that we know is congruent
• What property tells us something is congruent to itself?
Common Parts
• To identify common parts, draw each pair of overlapping triangles as
the two individual triangles and see what is the same in both of them
• Whatever side or angle shares the EXACT same letters is the common
part
Finding Common Parts
Finding Common Parts
Finding Common Parts
Finding Common Parts
Bell Work for 12/2
Find the common parts for each pair of triangles
Using Congruent Triangles
• Because overlapping triangles contain multiple triangles, we can use
the information we find about certain ones to prove things about
other ones in the figure
• We can use CPCTC and the triangle congruence Theorems to do this
Bell Work for 12/3
• Name a pair of overlapping triangles in each figure, and state which postulate we
can use to prove them congruent