4.2 Congruence and Triangles Geometry Ms. Reser. Standards/Objectives: Standard 2: Students will...

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4.2 Congruence and 4.2 Congruence and Triangles Triangles Geometry Geometry Ms. Reser

Transcript of 4.2 Congruence and Triangles Geometry Ms. Reser. Standards/Objectives: Standard 2: Students will...

Page 1: 4.2 Congruence and Triangles Geometry Ms. Reser. Standards/Objectives: Standard 2: Students will learn and apply geometric concepts Objectives:  Identify.

4.2 Congruence and 4.2 Congruence and TrianglesTrianglesGeometryGeometry

Ms. Reser

Page 2: 4.2 Congruence and Triangles Geometry Ms. Reser. Standards/Objectives: Standard 2: Students will learn and apply geometric concepts Objectives:  Identify.

Standards/Objectives:Standards/Objectives:

Standard 2: Students will learn and apply Standard 2: Students will learn and apply geometric conceptsgeometric concepts

Objectives:Objectives: Identify congruent figures and Identify congruent figures and

corresponding partscorresponding parts Prove that two triangles are congruentProve that two triangles are congruent

Page 3: 4.2 Congruence and Triangles Geometry Ms. Reser. Standards/Objectives: Standard 2: Students will learn and apply geometric concepts Objectives:  Identify.

AssignmentAssignment 4.2 Worksheet A and B4.2 Worksheet A and B Quiz 4.2 on page 210 to review for quiz Quiz 4.2 on page 210 to review for quiz

next time we meet.next time we meet.

Page 4: 4.2 Congruence and Triangles Geometry Ms. Reser. Standards/Objectives: Standard 2: Students will learn and apply geometric concepts Objectives:  Identify.

Identifying congruent Identifying congruent figuresfigures

Two geometric figures are congruent if Two geometric figures are congruent if they have exactly the same size and they have exactly the same size and shape. shape.

CONGRUENT

NOT CONGRUENT

Page 5: 4.2 Congruence and Triangles Geometry Ms. Reser. Standards/Objectives: Standard 2: Students will learn and apply geometric concepts Objectives:  Identify.

TrianglesTriangles

Corresponding anglesCorresponding angles

A A ≅ ≅ PP

B B ≅ ≅ QQ

C C ≅ ≅ RR

Corresponding SidesCorresponding Sides

AB AB ≅ PQ≅ PQ

BC ≅ QRBC ≅ QR

CA ≅ RPCA ≅ RP

A

B

C

Q

P R

Page 6: 4.2 Congruence and Triangles Geometry Ms. Reser. Standards/Objectives: Standard 2: Students will learn and apply geometric concepts Objectives:  Identify.

ZZ

If If ΔΔ ABC is ABC is to to ΔΔ XYZ, which angle is XYZ, which angle is to to C?C?

Page 7: 4.2 Congruence and Triangles Geometry Ms. Reser. Standards/Objectives: Standard 2: Students will learn and apply geometric concepts Objectives:  Identify.

Thm 4.3Thm 4.333rdrd angles thm angles thm

If 2 If 2 s of one s of one ΔΔ are are to 2 to 2 s of another s of another ΔΔ, then the , then the 3rd 3rd s are also s are also ..

Page 8: 4.2 Congruence and Triangles Geometry Ms. Reser. Standards/Objectives: Standard 2: Students will learn and apply geometric concepts Objectives:  Identify.

Ex: find xEx: find x

)

))

2222oo

8787oo

)

)) (4x+15)(4x+15)oo

Page 9: 4.2 Congruence and Triangles Geometry Ms. Reser. Standards/Objectives: Standard 2: Students will learn and apply geometric concepts Objectives:  Identify.

Ex: continuedEx: continued

22+87+4x+15=18022+87+4x+15=180

4x+15=714x+15=71

4x=564x=56

x=14x=14

Page 10: 4.2 Congruence and Triangles Geometry Ms. Reser. Standards/Objectives: Standard 2: Students will learn and apply geometric concepts Objectives:  Identify.

Ex: ABCD is Ex: ABCD is to HGFE, find x to HGFE, find x and y.and y.

4x-3=9 5y-12=113

4x=12 5y=125

x=3 y=25

A B

D C

91°

86° 113°

9 cm

AB

DC

(5y-12)°

HG

F E

4x – 3 cm

Page 11: 4.2 Congruence and Triangles Geometry Ms. Reser. Standards/Objectives: Standard 2: Students will learn and apply geometric concepts Objectives:  Identify.

Thm 4.4Thm 4.4Props. of Props. of ΔΔss

Reflexive prop of Reflexive prop of ΔΔ - - Every Every ΔΔ is is to itself to itself ((ΔΔABC ABC ΔΔABC).ABC).

Symmetric prop of Symmetric prop of ΔΔ - - If If ΔΔABC ABC ΔΔPQR, then PQR, then ΔΔPQR PQR ΔΔABC.ABC.

Transitive prop of Transitive prop of ΔΔ - If - If ΔΔABC ABC ΔΔPQR & PQR & ΔΔPQR PQR ΔΔXYZ, then XYZ, then ΔΔABC ABC ΔΔXYZ.XYZ.

A

BB

CPP

Q

R

XY

Z

Page 12: 4.2 Congruence and Triangles Geometry Ms. Reser. Standards/Objectives: Standard 2: Students will learn and apply geometric concepts Objectives:  Identify.

Copy the congruent triangles shown then label the Copy the congruent triangles shown then label the vertices of your triangle so that vertices of your triangle so that ∆AMT ∆AMT ∆CDN. ∆CDN. Identify Identify all pairs of congruent corresponding angles and all pairs of congruent corresponding angles and corresponding sides.corresponding sides.

Start by labeling the Start by labeling the triangles. ∆AMT triangles. ∆AMT can be the triangle can be the triangle on the left and on the left and ∆CDN can be the ∆CDN can be the one on the right. one on the right. Do this now.Do this now.

∆∆AMT AMT ∆CDN ∆CDN

Page 13: 4.2 Congruence and Triangles Geometry Ms. Reser. Standards/Objectives: Standard 2: Students will learn and apply geometric concepts Objectives:  Identify.

Copy the congruent triangles shown then label the Copy the congruent triangles shown then label the vertices of your triangle so that vertices of your triangle so that ∆AMT ∆AMT ∆CDN. Identify ∆CDN. Identify all pairs of congruent corresponding angles and all pairs of congruent corresponding angles and corresponding sides. corresponding sides.

Here’s what it should Here’s what it should look like. Next look like. Next begin by labeling begin by labeling the drawing with the drawing with tick marks to keep tick marks to keep track of sides that track of sides that are congruent. are congruent. Write them down. Write them down. Hint: use ∆AMT Hint: use ∆AMT ∆CDN∆CDN

A

MT

C

DN

Page 14: 4.2 Congruence and Triangles Geometry Ms. Reser. Standards/Objectives: Standard 2: Students will learn and apply geometric concepts Objectives:  Identify.

Copy the congruent triangles shown then label the Copy the congruent triangles shown then label the vertices of your triangle so that vertices of your triangle so that ∆AMT ∆AMT ∆CDN. Identify ∆CDN. Identify all pairs of congruent corresponding angles and all pairs of congruent corresponding angles and corresponding sides. corresponding sides.

AM AM MT MT TA TA Hint: use Hint: use

∆∆AMT AMT ∆CDN ∆CDN

A

MT

C

DN

Page 15: 4.2 Congruence and Triangles Geometry Ms. Reser. Standards/Objectives: Standard 2: Students will learn and apply geometric concepts Objectives:  Identify.

Copy the congruent triangles shown then label the Copy the congruent triangles shown then label the vertices of your triangle so that vertices of your triangle so that ∆AMT ∆AMT ∆CDN. Identify ∆CDN. Identify all pairs of congruent corresponding angles and all pairs of congruent corresponding angles and corresponding sides. corresponding sides.

What about the angle What about the angle measures?measures?

Hint: use Hint: use

∆∆AMT AMT ∆CDN. The ∆CDN. The letters of the letters of the congruency congruency statement line up. statement line up.

A

MT

C

DN

Page 16: 4.2 Congruence and Triangles Geometry Ms. Reser. Standards/Objectives: Standard 2: Students will learn and apply geometric concepts Objectives:  Identify.

Copy the congruent triangles shown then label the Copy the congruent triangles shown then label the vertices of your triangle so that vertices of your triangle so that ∆AMT ∆AMT ∆CDN. Identify ∆CDN. Identify all pairs of congruent corresponding angles and all pairs of congruent corresponding angles and corresponding sides. corresponding sides.

What about the angle What about the angle measures?measures?

Hint: use Hint: use

∆∆AMT AMT ∆CDN. The ∆CDN. The letters of the letters of the congruency congruency statement line up. statement line up.

A

MT

C

DN

A A M M T T

Page 17: 4.2 Congruence and Triangles Geometry Ms. Reser. Standards/Objectives: Standard 2: Students will learn and apply geometric concepts Objectives:  Identify.

Copy the congruent triangles shown then label the Copy the congruent triangles shown then label the vertices of your triangle so that vertices of your triangle so that ∆AMT ∆AMT ∆CDN. Identify ∆CDN. Identify all pairs of congruent corresponding angles and all pairs of congruent corresponding angles and corresponding sides. corresponding sides.

What about the angle What about the angle measures?measures?

Hint: use Hint: use

∆∆AMT AMT ∆CDN. The ∆CDN. The letters of the letters of the congruency congruency statement line up. statement line up.

A

MT

C

DN

A A C C M M D D T T NN

Page 18: 4.2 Congruence and Triangles Geometry Ms. Reser. Standards/Objectives: Standard 2: Students will learn and apply geometric concepts Objectives:  Identify.

Some useful informationSome useful information Often in proving any figures congruent, but Often in proving any figures congruent, but

especially triangles . . . the following theorems are especially triangles . . . the following theorems are especially helpful:especially helpful:

1.1. Reflexive Property – Segment Reflexive Property – Segment Segments i.e. Segments i.e. BD BD BD with shared sides of a triangle (the side BD with shared sides of a triangle (the side in the middle).in the middle).

2.2. Third Angles Theorem – when you know the other Third Angles Theorem – when you know the other two angles are there and congruent.two angles are there and congruent.

3.3. Vertical Angles Theorem – the X in the triangles Vertical Angles Theorem – the X in the triangles that look like bowties.that look like bowties.

4.4. Last step is usually “Definition of Congruence” Last step is usually “Definition of Congruence” because this is where the congruency statement because this is where the congruency statement is given . . . is given . . . ∆ABC ∆ABC ∆DEF ∆DEF

Page 19: 4.2 Congruence and Triangles Geometry Ms. Reser. Standards/Objectives: Standard 2: Students will learn and apply geometric concepts Objectives:  Identify.

GivenGiven: seg RP : seg RP seg MN, seg PQ seg MN, seg PQ seg NQ seg NQ , seg RQ , seg RQ seg MQ, m seg MQ, mP=92P=92oo and m and mN is N is 9292oo..ProveProve: : ΔΔRQP RQP ΔΔMQNMQN

R

P

Q

N

M

9292oo

9292oo

Page 20: 4.2 Congruence and Triangles Geometry Ms. Reser. Standards/Objectives: Standard 2: Students will learn and apply geometric concepts Objectives:  Identify.

Statements ReasonsStatements Reasons1. 1. given1. 1. given2. m2. mP=mP=mN 2. subst. prop =N 2. subst. prop =3. 3. P P N N 3. def of 3. def of s s 4. 4. RQP RQP MQN 4. vert MQN 4. vert s thms thm5. 5. R R M 5. 3M 5. 3rdrd s thms thm6. 6. ΔΔRQP RQP ΔΔ MQN 6. def of MQN 6. def of ΔΔss