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# Warm Up - PowerPoint PPT Presentation

Warm Up. Lesson Presentation. Lesson Quiz. FG , GH , FH ,  F ,  G ,  H. Warm Up 1. Name all sides and angles of ∆ FGH . 2. What is true about  K and  L ? Why? 3. What does it mean for two segments to be congruent?.  ;Third  s Thm. They have the same length. Objectives.

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## PowerPoint Slideshow about ' Warm Up' - clare

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Lesson Presentation

Lesson Quiz

FG, GH, FH, F, G, H

Warm Up

1.Name all sides and angles of ∆FGH.

2. What is true about K and L? Why?

3.What does it mean for two segments to be congruent?

 ;Third s Thm.

They have the same length.

Use properties of congruent triangles.

Prove triangles congruent by using the definition of congruence.

corresponding angles

corresponding sides

congruent polygons

Corresponding angles and corresponding sides are in the same position in polygons with an equal number of sides.

Two polygons are congruent polygons if and only if their corresponding sides are congruent. Thus triangles that are the same size and shape are congruent.

Two vertices that are the endpoints of a side are called consecutive vertices.

For example, P and Q are consecutive vertices.

To name a polygon, write the vertices in consecutive order. For example, you can name polygon PQRS as QRSP or SRQP, but not as PRQS.

In a congruence statement, the order of the vertices indicates the corresponding parts.

Helpful Hint For example, you can name polygon

When you write a statement such as ABCDEF, you are also stating which parts are congruent.

Sides: For example, you can name polygon PQ ST, QR  TW, PR  SW

Example 1: Naming Congruent Corresponding Parts

Given: ∆PQR ∆STW

Identify all pairs of corresponding congruent parts.

Angles: P  S, Q  T, R  W

Sides: For example, you can name polygon LM EF, MN  FG, NP  GH, LP  EH

TEACH! Example 1

If polygon LMNP polygon EFGH, identify all pairs of corresponding congruent parts.

Angles: L  E, M  F, N  G, P  H

Ex. 2A: Corresponding Parts of Congruent Triangles For example, you can name polygon

Given: ∆ABC ∆DBC.

Find the value of x.

BCA andBCD are rt. s.

Def. of  lines.

BCA BCD

Rt. Thm.

Def. of  s

mBCA = mBCD

Substitute values for mBCA and mBCD.

(2x – 16)° = 90°

2x = 106

x = 53

Divide both sides by 2.

Ex. 2B: Corresponding Parts of Congruent Triangles For example, you can name polygon

Given: ∆ABC ∆DBC.

Find mDBC.

∆ Sum Thm.

mABC + mBCA + mA = 180°

Substitute values for mBCA and mA.

mABC + 90 + 49.3 = 180

Simplify.

mABC + 139.3 = 180

Subtract 139.3 from both sides.

mABC = 40.7

DBC  ABC

Corr. s of  ∆s are  .

mDBC = mABC

Def. of  s.

mDBC  40.7°

Trans. Prop. of =

AB For example, you can name polygon  DE

TEACH! Example 2a

Given: ∆ABC  ∆DEF

Find the value of x.

Corr. sides of  ∆s are.

AB = DE

Def. of  parts.

Substitute values for AB and DE.

2x – 2 = 6

2x = 8

x = 4

Divide both sides by 2.

TEACH! For example, you can name polygon Example 2b

Given: ∆ABC  ∆DEF

Find mF.

∆ Sum Thm.

mEFD + mDEF + mFDE = 180°

ABC  DEF

Corr. s of  ∆are .

mABC = mDEF

Def. of  s.

mDEF = 53°

Transitive Prop. of =.

Substitute values for mDEF and mFDE.

mEFD + 53 + 90 = 180

mF + 143 = 180

Simplify.

mF = 37°

Subtract 143 from both sides.

Given: For example, you can name polygon YWXandYWZ are right angles.

YW bisects XYZ. W is the midpoint of XZ. XY  YZ.

Prove: ∆XYW  ∆ZYW

Example 3: Proving Triangles Congruent

5. For example, you can name polygon W is mdpt. of XZ

6.XW ZW

7.YW YW

9.XY YZ

1.YWX and YWZ are rt. s.

1. Given

2.YWX  YWZ

2. Rt.   Thm.

3.YW bisects XYZ

3. Given

4.XYW  ZYW

4. Def. of bisector

5. Given

6. Def. of mdpt.

7. Reflex. Prop. of 

8.X  Z

8. Third s Thm.

9. Given

10.∆XYW  ∆ZYW

10. Def. of  ∆

TEACH! For example, you can name polygon Example 3

ABDE, A D

Prove:∆ABC  ∆DEC

4. For example, you can name polygon ABDE

6.BC EC,

AC DC

1.A D

1. Given

2.BCA  DCE

2. Vertical s are .

3.ABC DEC

3. Third s Thm.

4. Given

5. Given

6. Def. of bisector

7.∆ABC  ∆DEC

7. Def. of  ∆s

Example 4: Engineering Application For example, you can name polygon

The diagonal bars across a gate give it support. Since the angle measures and the lengths of the corresponding sides are the same, the triangles are congruent.

Given: PR and QT bisect each other.

PQS  RTS, QP  RT

Prove: ∆QPS∆TRS

1. For example, you can name polygon QP RT

3.PR and QT bisect each other.

4.QS TS, PS  RS

Example 4 Continued

1. Given

2.PQS  RTS

2. Given

3. Given

4. Def. of bisector

5.QSP  TSR

5. Vert. s Thm.

6.QPS  TRS

6. Third s Thm.

7.∆QPS  ∆TRS

7. Def. of  ∆s

Given: For example, you can name polygon MK bisects JL. JL bisects MK. JK ML.JK|| ML.

TEACH! Example 4

Use the diagram to prove the following.

Prove: ∆JKN∆LMN

2. For example, you can name polygon JK|| ML

1.JK ML

4.JL and MK bisect each other.

5.JN LN, MN  KN

TEACH! Example 4 Continued

1. Given

2. Given

3.JKN  NML

3. Alt int. s are .

4. Given

5. Def. of bisector

6.KNJ  MNL

6. Vert. s Thm.

7.KJN  MLN

7. Third s Thm.

8.∆JKN ∆LMN

8. Def. of  ∆s

RS For example, you can name polygon

Lesson Quiz

1. ∆ABC  ∆JKL and AB = 2x + 12. JK = 4x – 50. Find x and AB.

Given that polygon MNOP polygon QRST, identify the congruent corresponding part.

2. NO  ____ 3. T  ____

4. Given: C is the midpoint of BD and AE.

A  E, AB  ED

Prove: ∆ABC  ∆EDC

31, 74

P

Statements For example, you can name polygon

Reasons

1. A  E

1. Given

2. C is mdpt. of BD and AE

2. Given

3. AC EC; BC  DC

3. Def. of mdpt.

4. AB ED

4. Given

5. ACB  ECD

5. Vert. s Thm.

6. B  D

6. Third s Thm.

7. ABC  EDC

7. Def. of  ∆s

Lesson Quiz

4.