Download Unit 8: Congruent and Similar Triangles Lesson 8.1 Apply

Unit 8: Congruent and Similar Triangles
Lesson 8.1 Apply Congruence and Triangles
Lesson 4.2 from textbook
Objectives
•
• Identify congruent figures and corresponding parts of closed plane figures.
Prove that two triangles are congruent using definitions, properties, theorems, and postulates.
Congruent
Not Congruent
*If two figures are congruent, then their corresponding parts are _______________________.
In the diagram, ∆ABC ≅ ∆FED . Label the two triangles accordingly and mark all corresponding parts
that are congruent.
Congruence Statements: ________________________________________________________________
Example 1
Write a congruence statement for the triangles. Identify
all pairs of corresponding congruent parts.
Triangles _____________________________________
Corresponding Angles ____________________________________
Corresponding Sides _______________________________________
Example 2
In the diagram, DEFG ≅ SPQR .
Find the value of x. _________________
Find the value of y. _________________
Example 3
In the diagram, a rectangular wall is divided
into two sections. Are the sections congruent?
Explain.
_____________________________________________
_____________________________________________
Third Angles Theorem
If two angles of one triangle are congruent to two angles
of another triangles, then the third angles are
_____________________________________________
Example 4
Find m<BDC.
Example 5
Graph the triangle with vertices D(1, 2), E(7, 2), and F(5, 4).
Then, graph a triangle congruent to ∆DEF .
Example 6
Given: AD ≅ CB , DC ≅ BA , ∠ACD ≅ ∠CAB ,
∠CAD ≅ ∠ACB
Prove: ∆ACD ≅ ∆CAB
Statements
Reasons
1.
1. Given
2.
2.
3.
3. Given
4.
4.
5. ∆ACD ≅ ∆CAB
5.
Properties of Congruent Triangles
Reflexive Property
For any triangle ABC, ∆ABC ≅ ________________.
Symmetric Property
If ∆ABC ≅ ∆DEF , then _______________________.
Transitive Property
If ∆ABC ≅ ∆DEF and ∆DEF ≅ ∆JKL , then _____________________.
Unit 8: Congruent and Similar Triangles
Lesson 8.2 Prove Triangles Congruent by SSS
Lesson 4.3 from textbook
Objectives
•
•
Use the Side-Side-Side (SSS) Congruence Postulate to prove that two triangles are congruent,
along with other definitions, properties, theorems, and postulates.
Prove that two triangles are congruent in the coordinate plane using the Distance Formula and
the SSS postulate.
Side-Side-Side (SSS) Congruence Postulate
If three sides of one triangle are congruent to three sides of a
second triangle, then
_________________________________________________
Example 1
Determine whether the congruence statement is true. Explain your reasoning.
∆DFG ≅ ∆HJK
∆ACB ≅ ∆CAD
Example 2
Use the given coordinates to determine if ∆ABC ≅ ∆DEF .
A(-3, -2), B(0, -2), C(-3, -8), D(10, 0), E(10, -3), F(4, 0)
AB = __________
BC = __________
CA = __________
DE = __________
EF = __________
FD = __________
Example 3
Explanation:
____________________________________________________________________________________
____________________________________________________________________________________
Example 4
Example 5
Statements
Reasons
1.
1.
2.
2.
3.
3.
Unit 8: Congruent and Similar Triangles
Lesson 8.3 Prove Triangles Congruent by SAS and HL
Lesson 4.4 from textbook
Objectives
•
Use the Side-Angle-Side (SAS) and Hypotenuse-Leg (HL) Congruence Postulate to prove that
two triangles are congruent, along with other definitions, properties, theorems, and postulates.
• Use two-column proofs to justify statements about congruent triangles.
Side-Angle-Side (SAS) Congruence Postulate
If two sides and the included angle of one triangle are congruent to
the corresponding to sides and corresponding and the corresponding
included angle of a second triangle,
then ___________________________________________________.
Example 1
Decide whether enough information is given to prove that the triangles are congruent using the SAS
Congruence Postulate.
Hypotenuse-Leg Congruence Theorem
If the leg and hypotenuse of a right triangle are congruent to the
corresponding leg and hypotenuse of a second triangle,
then ____________________________________________________.
Example 2
State the third congruence that must be given to prove ∆ABC ≅ ∆DEF using indicated postulate.
a) Given: AB ≅ DE , CB ≅ FE , ______ ≅ ______ (SSS Congruence Postulate)
b) Given: ∠A ≅ ∠D, CA ≅ FD, ______ ≅ _______ (SAS Congruence Postulate)
c) Given: ∠B ≅ ∠E , AB ≅ DE , ______ ≅ _______ (SAS Congruence Postulate)
Example 3
Statements
Reasons
1.
1.
2.
2. Definition of perpendicular lines.
3.
3. Definition of a right triangle.
4.
4.
5.
5
6.
6.
Example 4
Statements
Reasons
1.
1.
2.
2.
3.
3.
4.
4.
5.
5.
Unit 8: Congruent and Similar Triangles
Lesson 8.4 Prove Triangles Congruent by ASA and AAS
Lesson 4.5 from textbook
Objectives
•
Use the Angle-Side-Angle (ASA) and Angle-Angle-Side (AAS) Congruence Postulates to prove
that two triangles are congruent, along with other definitions, properties, theorems, and
postulates.
• Use two-column proofs to justify statements about congruent triangles.
Angle-Side-Angle (ASA) Congruence Postulate
Angle-Angle-Side (AAS) Congruence Theorem
Example 1
Is it possible to prove that the two triangles are congruent? If so, state the postulate or theorem you
would use.
________________________
_______________________
Example 2
State the third congruence that must be given to prove
∆ABC ≅ ∆DEF using indicated postulate.
a) Given: AB ≅ DE , ∠A ≅ ∠D, ______ ≅ ______ (AAS Congruence Postulate)
b) Given: ∠A ≅ ∠D, CA ≅ FD, ______ ≅ _______ (ASA Congruence Postulate)
Example 3
Tell whether you can use the given information to determine whether ∆ABC ≅ ∆DEF . Explain your
reasoning.
∠A ≅ ∠D, AB ≅ DE , AC ≅ DF
__________________________
∠B ≅ ∠E , ∠C ≅ ∠F , AC ≅ DE
__________________________
Example 4
Given: X is the midpoint of VY and WZ .
Prove: ∆VWX ≅ ∆YZX
Statements
Reasons
1.
1.
2.
2.
3.
3.
4.
4.
Unit 8: Congruent and Similar Triangles
Lesson 8.5 Using Congruent Triangles
Lesson 4.6 from textbook
Objectives
•
Use congruent triangles to plan and write proofs about their corresponding parts.
Corresponding Parts of Congruent Triangles are Congruent
Theorem (CPCTC)
If ___________________________ are congruent then
the ____________________________ of the congruent
triangles are also _____________________.
Given congruent parts: ________________________
∆ABC ≅ ∆DEF by the ________________
Other corresponding congruent parts: __________________________________________
Example 1
Tell which triangles you can show are congruent in order to prove the statement. What postulate or
theorem would you use?
∠A ≅ ∠D
GK ≅ HJ
____________________
____________________
____________________
____________________
Example 2
_____________________
________________________
Example 3
Given: ∠Q ≅ ∠S , ∠RTQ ≅ ∠RTS
Prove: QT ≅ ST
*FIRST PROVE TRIANGLES ARE CONGRUENT*
Statements
Reasons
1.
1.
2.
2.
3.
3.
4.
4.
Example 4
Given: NM ≅ KM
Prove: ∠MLK ≅ ∠MPN
Statements
Reasons
1.
1.
2.
2.
3.
3.
4.
4.
Example 5
Use the diagram to write a plan for a proof:
Prove: ∠A ≅ ∠C
PLAN:
Unit 8: Congruent and Similar Triangles
Lesson 8.6 Prove Triangles Similar by AA
Lesson 6.4 from textbook
Objectives
•
Identify similar triangles using the Angle-Angle (AA) Similarity Postulate.
• Find measures of similar triangles using proportional reasoning.
ACTIVITY:
Question: What can you conclude about two triangles if you know two pairs of corresponding
angles are congruent?
1. Draw ∆EFG so that m<E = 40o
and m<G = 50o.
2. Draw ∆RST so that <R = 40o and m<T = 50o,
and is not congruent to ∆EFG .
3. Calculate m<F and m<S using Triangle Sum Theorem. ___________________________________
4. Measure and record the side lengths of both triangles. (to the nearest mm).
___________________________________________
5. Are the triangles similar? Explain. ____________________________________________________
6. If all we know is that two angles in two different triangles
are congruent, can we conclude that the triangles are similar? ______________________________
Angle-Angle (AA) Similarity Postulate
If two angles of one triangle are congruent to two angles of
another triangle, then the two triangles are similar.
_______________________________________________
Example 1
Determine whether the triangles are similar. If they are, write a similarity statement. Explain your reasoning.
______________________________________
___________________________________
Example 2
Use the diagram to complete the information.
∆MON ~ _________
MN ON MO
=
=
___________________
?
?
?
16 ?
=
_________
12 10
12 ?
= _______________
16 y
x = ____________
y = ____________
Example 3
The A-frame building shown in the figure has a balcony that
is 16 feet long, 16 feet high, and parallel to the ground. The
building is 28 feet wide at its base. How tall is the A-frame building?
Height = ________________
Unit 8: Congruent and Similar Triangles
Lesson 8.7 Prove Triangles Similar by SSS and SAS
Lesson 6.5 from textbook
Objectives
•
Use the similarity theorems such as the Side-Side-Side (SSS) Similarity Theorem and the SideAngle-Side (SAS) Similarity Theorem to determine whether two triangles are similar.
• Find measures of similar triangles using proportional reasoning.
Side-Side-Side (SSS)
Similarity Theorem
Side-Angle-Side (SAS)
Similarity Theorem
If the corresponding side lengths of two
triangles are proportional, then the triangles
are similar
If two sides of one triangle are proportional to
two sides of another triangle and their included
angles are congruent, then the triangles are
similar.
If ___________________________________
If ___________________________________
___________________, then ∆ABC ~ ∆RST .
___________________ , then ∆ABC ~ ∆RST .
Example 1
Determine which two of the three triangles are similar. Find the scale factor for the pair. State
which theorem was used to support your answer.
Similar Triangles ________________________
Scale factor
_______________
Theorem
___________________________
Example 2
Are the triangles similar? If so, state the similarity and the
postulate or theorem that justifies your answer.
____________________________________
Example 3
Find the values of x that makes ∆ABC ~ ∆DEF .
x = ________________________
Example 4
A large tree has fallen against another tree and rests at an angle
as shown in the figure. To estimate the length of the tree from
the ground you make the measurements shown in the figure.
What theorem or postulate can be used to show that the
triangles in the figure are similar?
____________________________________
Explain how you can use similar triangles to estimate the length
of the tree. Then estimate the length.
___________________________________
Example 5
Unit 8: Congruent and Similar Triangles
Lesson 8.8 Use Proportionality Theorems
Lesson 6.6 from textbook
Objectives
•
•
Use proportionality theorems to calculate segments lengths and to determine parallel lines.
Apply proportions to solve problems involving missing lengths and angle measures in similar
figures.
Triangle Proportionality
Theorem
Triangle Proportionality Converse
Theorem
If a line parallel to one side of a triangle
intersects the other two sides, then it divides
the two sides proportionally
If a line divides two sides of a triangle
proportionally, then it is parallel to the third side
If TU // QS , then _____________________
If
Example 1
In the diagram, TU // QS , RS = 4, ST = 6, and QU = 9.
What is the length of RQ ?
RQ = ___________________
Example 2
Determine whether PS // QR . Explain.
________________________________________
Example 3
Use the figure to find the length of each segment.
GF = ______________
FC = _______________
ED = ______________
FE = _______________
RT RU
=
, then _________________________
TQ US
Example 4
Find the value of x.
x = ______________
Example 5
The figure is a diagram of a cross section of the
attic of a house. A vent pipe comes through the
floor 6 feet from the edge of the house. What is
the distance x on the roof, from the edge of the
roof to the vent pipe?
______________________________
Example 6
x = ______________