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, FEDABC ∆≅∆ . 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, SPQRDEFG ≅ .

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: CBAD ≅ , BADC ≅ , CABACD ∠≅∠ ,

ACBCAD ∠≅∠

Prove: CABACD ∆≅∆

Statements Reasons

1. 1. Given

2. 2.

3. 3. Given

4. 4.

5. CABACD ∆≅∆ 5.

Properties of Congruent Triangles

Reflexive Property

For any triangle ABC, ≅∆ABC ________________.

Symmetric Property

If DEFABC ∆≅∆ , then _______________________.

Transitive Property

If DEFABC ∆≅∆ and JKLDEF ∆≅∆ , 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.

HJKDFG ∆≅∆ CADACB ∆≅∆

Example 2

Use the given coordinates to determine if DEFABC ∆≅∆ .

A(-3, -2), B(0, -2), C(-3, -8), D(10, 0), E(10, -3), F(4, 0)

AB = __________ CA = __________ DE = __________ FD = __________

BC = __________ EF = __________

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 DEFABC ∆≅∆ using indicated postulate.

a) Given: ____________,, ≅≅≅ FECBDEAB (SSS Congruence Postulate)

b) Given: _____________,, ≅≅∠≅∠ FDCADA (SAS Congruence Postulate)

c) Given: _____________,, ≅≅∠≅∠ DEABEB (SAS Congruence Postulate)

Example 3

Example 4

Statements Reasons

1. 1.

2. 2.

3. 3.

4. 4.

5. 5.

Statements Reasons

1. 1.

2. 2. Definition of perpendicular lines.

3. 3. Definition of a right triangle.

4. 4.

5. 5

6. 6.

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

DEFABC ∆≅∆ using indicated postulate.

a) Given: ____________,, ≅∠≅∠≅ DADEAB (AAS Congruence Postulate)

b) Given: _____________,, ≅≅∠≅∠ FDCADA (ASA Congruence Postulate)

Example 3

Tell whether you can use the given information to determine whether DEFABC ∆≅∆ . Explain your

reasoning.

DFACDEABDA ≅≅∠≅∠ ,, __________________________

DEACFCEB ≅∠≅∠∠≅∠ ,, __________________________

Example 4

Given: X is the midpoint of VY and .WZ

Prove: YZXVWX ∆≅∆

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: ________________________ DEFABC ∆≅∆ 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?

DA ∠≅∠ HJGK ≅

____________________ ____________________

____________________ ____________________

Example 2

_____________________ ________________________

Example 3

Given: RTSRTQSQ ∠≅∠∠≅∠ ,

Prove: STQT ≅

*FIRST PROVE TRIANGLES ARE CONGRUENT*

Example 4

Given: KMNM ≅

Prove: MPNMLK ∠≅∠

Statements Reasons

1. 1.

2. 2.

3. 3.

4. 4.

Example 5

Use the diagram to write a plan for a proof:

Prove: CA ∠≅∠

Statements Reasons

1. 1.

2. 2.

3. 3.

4. 4.

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 2. Draw RST∆ so that <R = 40

o and m<T = 50

o,

and m<G = 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 ~ _________ ???

MOONMN== ___________________

10

?

12

16= _________

y

?

16

12= _______________

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 Side-

Angle-Side (SAS) Similarity Theorem to determine whether two triangles are similar.

• Find measures of similar triangles using proportional reasoning.

Side-Side-Side (SSS) Side-Angle-Side (SAS)

Similarity Theorem Similarity Theorem

If the corresponding side lengths of two If two sides of one triangle are proportional to

triangles are proportional, then the triangles two sides of another triangle and their included

are similar 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 Triangle Proportionality Converse

Theorem Theorem

If a line parallel to one side of a triangle If a line divides two sides of a triangle

intersects the other two sides, then it divides proportionally, then it is parallel to the third side

the two sides proportionally

If TU //QS , then _____________________ If US

RU

TQ

RT= , then _________________________

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 = _______________

Example 4

Find the value of x.

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