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Chapter 4
Congruent Triangles
4-1 Objectives
a) _____________________________________________________________
_____________________________________________________________
b) _____________________________________________________________
_____________________________________________________________
Congruent or not congruent?
Congruent or not congruent?
What are congruent polygons?
• Come up with a definition of congruent polygons based on the
previous discussion.
How do we name congruent polygons?
• What would our congruence
statement look like?
• It should be very clear from the
statement which parts of the
two triangles are corresponding.
Find the missing angle measures
• ∆ABC  ∆DEF
• If mA = 34, what is mD?
• If BC = 7, what is EF?
• If mA = 30 and mB = 50, what is mF? Explain how you know.
How could we be sure these are congruent?
• Third Angles Theorem
Proving the
rd
3
Angles Theorem
• Given: 𝐴  𝐷, 𝐵  𝐸
• Prove: 𝐶  𝐹
Proving Triangles Congruent
Practice: P. 223-224 #32, 34, 39, 45
Lazy Lawrence
• Lawrence works for a company that makes roof trusses, triangular
pieces that support simple roofs. His job is to ensure that each and
every roof truss that is made at the company is exactly the same size
as all the others. Because Lawrence is so lazy and likes to avoid as
much work as possible, he wants to find the easiest possible way to
do his job. Help Lawrence find the easiest way to show that two
triangular trusses are equal in measure with as little work as possible.
What is the minimum amount of parts that would
need to be congruent to ensure two triangles are
congruent? (work with a partner)
• 1 side (S)? True or False?
• If two triangles have one side of equal measure, then the triangles are
congruent.
• 1 angle (A)? True or False?
• If two triangles have one angle of equal measure, then the triangles are
congruent.
What is the minimum amount of parts that would
need to be congruent to ensure two triangles are
congruent? (work with a partner)
• 2 sides (SS)? True or False?
• If two triangles have two sides of equal measure, then the triangles are
congruent.
• 2 angles (AA)? True or False?
• If two triangles have two angles of equal measure, then the triangles are
congruent.
What is the minimum amount of parts that would
need to be congruent to ensure two triangles are
congruent? (work with a partner)
• 1 side and 1 angle (SA)? True or False?
• If two triangles have one side and one angle of equal measure, then the
triangles are congruent.
• Find and test the remaining combinations (involving 3 parts of the
triangle) to determine which hypotheses would be true. Record on
the next page
• Things to consider: Is SAS different than SSA? Is AAS different than ASA?
Record findings here
Which hypotheses were true?
What about Angle-Angle-Side (AAS)?
• Proof?
What about right triangles?
• Proof?
4-2 Objectives
a) _____________________________________________________________
_____________________________________________________________
4-3 Objectives
a) _____________________________________________________________
_____________________________________________________________
4-6 Objectives
a) _____________________________________________________________
_____________________________________________________________
Practice: p. 231 #16, 17, 29, 31
p. 239 #15, 19, 20
4-4 Objectives
a) _____________________________________________________________
_____________________________________________________________
How can you prove two parts of a triangle are
congruent?
Complete proofs using CPCTC
• Given: 𝑌𝐴 ≅ 𝐵𝐴
B  Y
• Prove: 𝐴𝑍 ≅ 𝐴𝐶
Practice: p. 247 #9, 18, 19
4-5 Objectives
a) _____________________________________________________________
_____________________________________________________________
List what you know about:
• Isosceles Triangles
• Equilateral Triangles
Can you prove the isosceles triangle
theorem?
• If 2 sides of a triangle are congruent, then the angles opposite those
sides are congruent.
• What would the converse of this theorem be?
Corollaries – proved easily using another
theorem
• Corollary to Isosceles Triangle Theorem (Equilateral Triangles)
• Corollary to Converse of the Isosceles Triangle Theorem (Equiangular
Triangles)
Practice: pp. 254-255 #10-12, 23, 25, 30-32
4-7 Objectives
a) _____________________________________________________________
_____________________________________________________________
b) _____________________________________________________________
_____________________________________________________________
Can we prove these overlapping triangles
congruent?
Given: RE  TC , REP  TCG, PC  GE
Prove: REP  TCG
R
T
P C
G
E
Can you prove these triangles congruent?
Practice: pp. 269-270 #15-18, 21, 22