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

4.2 Congruence and 4.2 Congruence and TrianglesTrianglesGeometryGeometry

Ms. Reser

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

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.

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

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

ZZ

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

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 ..

Ex: find xEx: find x

)

))

2222oo

8787oo

)

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

Ex: continuedEx: continued

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

4x+15=714x+15=71

4x=564x=56

x=14x=14

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

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

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

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


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

∆∆AMT AMT ∆CDN ∆CDN

A

MT

C

DN


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



Hint: use Hint: use


A

MT

C

DN

A A M M T T



Hint: use Hint: use


A

MT

C

DN

A A C C M M D D T T NN

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

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

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

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

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