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Warm Up 1. Name all sides and angles of ∆ FGH . 2. What is true about  K and  L ? Why? - PowerPoint PPT Presentation


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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.

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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.


Geometric figures are congruent if they are the same size and shape. 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.


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

Check It Out! Example 1

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

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


Example 2A: Using 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°

Add 16 to both sides.

2x = 106

x = 53

Divide both sides by 2.


Example 2B: Using 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

Check It Out! 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

Add 2 to both sides.

2x = 8

x = 4

Divide both sides by 2.


Check It Out! 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  ∆


Check It Out! For example, you can name polygon Example 3

Given:ADbisectsBE.

BEbisectsAD.

ABDE, A D

Prove:∆ABC  ∆DEC


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

5.ADbisectsBE,

6.BC EC,

AC DC

BE bisects AD

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.QSP  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.

Check It Out! 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

Check It Out! 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.


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