Download Unit 4 Lesson 5 Congruent Triangles

KITE ASSEMBLY Tanya is following directions for
making a kite. She has two congruent triangular
pieces of fabric that need to be sewn together along
their longest side. The directions say to begin sewing
the two pieces of fabric together at their smallest
angles. At which two angles should she begin sewing?
Answer: A and D
Honors Geometry
Unit 4 Lesson 5
Proving Triangles Congruent
Objectives
• I can identify corresponding parts of congruent
triangles
• I can use the definition of congruent triangles
• I can discover and apply theorems about triangles
Recall
• Congruent polygons
– All sides are congruent (same length)
– All angles are congruent (same measure)
• Congruent triangles
– 3 congruent sides
– 3 congruent angles
– Total of 6 congruent parts
Corresponding Parts
• When two (or more) polygons are congruent, it is
necessary to identify their corresponding parts
– Use dash marks on sides and arcs on angles!
• Corresponding parts – between congruent
polygons, corresponding parts have the same
measure and are found in the same position
– Corresponding parts = matching parts
– CPCF (Corresponding parts of congruent figures)
Locate Corresponding Parts
Identify Corresponding Parts
Identify all of the
congruent
corresponding parts.
Mark them on the
diagram using the
appropriate symbols.
Angles:
Sides:
Congruence Statements
• Write a valid congruence statement by listing
corresponding parts in the same order
– Corresponding parts will appear in the same position
– Also, you can read a congruence statement and
determine which parts are corresponding
Example – Congruence Statement
• Same example:
• Corresponding parts:
• Congruence Statement:
Example - Congruence Statement
• Statement:
• Notice: Since angles A and H were congruent,
they appear first (same as B & J, C & K)
• Notice: Segments AB and HJ are congruent, and
those two letters appear first…
• There are multiple correct answers!
Write a congruence statement
for the triangles.
A. ΔLMN  ΔRTS
B. ΔLMN  ΔSTR
C. ΔLMN  ΔRST
D. ΔLMN  ΔTRS
A.
B.
C.
D.
A
B
C
D
Name the corresponding congruent
angles for the congruent triangles.
A. L  R, N  T, M  S
B. L  R, M  S, N  T
C. L  T, M  R, N  S
D. L  R, N  S, M  T
A.
B.
C.
D.
A
B
C
D
List all the congruent parts and
write a congruence statement.
Algebra
In the diagram, ΔITP  ΔNGO. Find the values of
x and y.
O  P
mO = mP
6y – 14 = 40
6y = 54
y= 9
CPCF
Definition of congruence
Substitution
Add 14 to each side.
Divide each side by 6.
Algebra
In the diagram, ΔITP  ΔNGO. Find the values of
x and y.
NG = IT
x – 2y = 7.5
x – 2(9) = 7.5
x – 18 = 7.5
x = 25.5
CPCF
Definition of congruence
Substitution
y=9
Simplify.
Add 18 to each side.
Algebra
In the diagram, ΔFHJ  ΔHFG. Find the values of
x and y.
A. x = 4.5, y = 2.75
B. x = 2.75, y = 4.5
C. x = 1.8, y = 19
D. x = 4.5, y = 5.5
A.
B.
C.
D.
A
B
C
D
Congruent Triangles
• For the rest of the lesson, we will focus on
proving that triangles are congruent
• Recall that congruent triangles share 6
corresponding, congruent parts
• There are 4 shortcuts – ways to show that all six
parts are congruent by only using 3
– You only have to do half the work!
The Four Congruence Postulates
• You must follow the specific order named by the
postulate
• A – a pair of congruent angles
• S – a pair of congruent sides
• Included angle – an angle found between two
congruent sides
• Included side – a side found between two
congruent angles
SSS Congruence
SAS Congruence
ASA Congruence
AAS Congruence
Notice…
• There is a combination of letters that we did NOT
use – because it will NOT prove that all 6 parts
are congruent
• One angle and the next two sides… or two sides
and the next angle
Hints…
• To determine WHICH of the 4 postulates is
illustrated in a particular example
– Read all given information
– Use the information to find congruent parts
– Find any other congruent parts
• Vertical pairs, shared side, etc
– Mark congruent parts on the diagram
– Determine in which order the parts appear
• Is a side between two marked angles?
• Is an angle between two marked sides?
Determine which postulate can be used to
prove that the triangles are congruent. If it
is not possible to prove congruence, choose
not possible.
A. SSS
B. ASA
C. SAS
D. not possible
A.
B.
C.
D.
A
B
C
D
Determine which postulate can be used to
prove that the triangles are congruent. If it
is not possible to prove congruence, choose
not possible.
A. SSS
B. ASA
C. SAS
D. not possible
A.
B.
C.
D.
A
B
C
D
Determine which postulate can be used to
prove that the triangles are congruent. If it
is not possible to prove congruence, choose
not possible.
A. SSA
B. ASA
C. SSS
D. not possible
A.
B.
C.
D.
A
B
C
D
Use SSS congruence…
EXTENDED RESPONSE Triangle DVW has vertices D(–5, –1),
V(–1, –2), and W(–7, –4). Triangle LPM has vertices L(1, –5),
P(2, –1), and M(4, –7).
a. Graph both triangles on the same coordinate
plane.
b. Use the distance formula to make a conjecture as to
whether the triangles are congruent. Explain your
reasoning.
Find the side lengths of the two
triangles
Triangle DVW
Triangle LPM
CONCLUSION: DVW  LPM by SSS Congruence
Use SSS Congruence
Determine whether ΔABC  ΔDEF for A(–5, 5),
B(0, 3), C(–4, 1),
D(6, –3), E(1, –1), and F(5, 1).
A. yes
B. no
C. cannot be determined
1.
2.
3.
A
B
C
Video Clip: Proof
• http://ed.ted.com/lessons/scott-kennedy-howto-prove-a-mathematical-theory
• 4:39
• Open browser first
Use SAS to Prove Triangles are Congruent
Draw a picture
Given:EI  HF; G is the midpoint of both EI and HF.
Prove: ΔFEG  ΔHIG
Statements
Reasons
1. EI  HF; G is the midpoint of
EI; G is the midpoint of HF.
1. Given
2.
2. Midpoint Theorem
3. FGE  HGI
3. Vertical Angles
Theorem
4. ΔFEG  ΔHIG
4. SAS
Statements
1.
Reasons
2.
2. _________
? Property
3. ΔABG ΔCGB
3. SSS
1. Given
A. Reflexive
B. Symmetric
C. Transitive
D. Substitution
A.
B.
C.
D.
A
B
C
D
Recap