4.2 Congruence & Triangles

GeometryMrs. SpitzFall 2005

Objectives:

• Identify congruent figures and corresponding parts

• Prove that two triangles are congruent

4.2 Work

• 4.2 pgs. 205-207 #4-35 (Skip 22, 23, 34) Be prepared next time we meet to draw on the board for participation points.

• Quiz 4.2 on page 210 to review for quiz next time we meet.

Identifying congruent figures

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

CONGRUENT

NOT CONGRUENT

Congruency

• When two figures are congruent, there is a correspondence between their angles and sides such that corresponding angles are congruent and corresponding sides are congruent.

Triangles

Corresponding angles

A ≅ PB ≅ QC ≅ R

Corresponding Sides

AB ≅ PQBC ≅ QRCA ≅ RP

A

B

C

Q

P R

How do you write a congruence statement?

• There is more than one way to write a congruence statement, but it is important to list the corresponding angles in the same order. Normally you would write ∆ABC ≅ ∆PQR, but you can also write that ∆BCA ≅ ∆QRP

Ex. 1 Naming congruent parts

• The congruent triangles. Write a congruence statement. Identify all parts of congruent corresponding parts.

E

F

D

R

T

S

Ex. 1 Naming congruent parts

• The diagram indicates that ∆DEF ≅ ∆RST. The congruent angles and sides are as follows:

• Angles: D≅ R, E ≅ S, F ≅T

• Sides DE ≅ RS, EF ≅ ST, FD ≅ TR

E

F

D

R

T

S

Ex. 2 Using properties of congruent figures

• In the diagram NPLM ≅ EFGH

• A. Find the value of x.

• You know that LM ≅ GH. So, LM = GH.

8 = 2x – 3 11 = 2x11/2 = x

L

M

NP

E

F

G

H

8 m

110°

87°10 m

72°(7y+9)°

(2x - 3) m

Ex. 2 Using properties of congruent figures

• In the diagram NPLM ≅ EFGH

• B. Find the value of y

• You know that N ≅ E. So, mN = mE.

72°= (7y + 9)°

63 = 7y

9 = y

L

M

NP

E

F

G

H

8 m

110°

87°10 m

72°(7y+9)°

(2x - 3) m

Third Angles Theorem

• If any two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent.

• If A ≅ D and B ≅ E, then C ≅ F.

B

A CE

D F

Ex. 3 Using the Third Angles Theorem

• Find the value of x. • In the diagram, N ≅ R and L ≅ S. From the Third Angles Theorem, you know that M ≅ T. So mM = mT. From the Triangle Sum Theorem, mM=180° - 55° - 65° = 60°

• mM = mT 60° = (2x + 30)° 30 = 2x 15 = x

M

N L S

R T(2x + 30)°

55° 65°

Ex. 4 Proving Triangles are congruent

• Decide whether the triangles are congruent. Justify your reasoning.

• From the diagram, you are given that all three pairs of corresponding sides are congruent.

• RP ≅ MN, PQ ≅ NQ, QR ≅ QM. Because P and N have the same measure, P ≅ N. By vertical angles theorem, you know that PQR ≅ NQM. By the Third Angles Theorem, R ≅ M.

• So all three pairs of corresponding sides and all three pairs of corresponding angles are congruent. By the definition of congruent triangles, ∆PQR ≅ ∆NQM.

Q

R

M

N

P

92°

92°

Ex. 5 Proving two triangles are congruent

• The diagram represents triangular stamps. Prove that ∆AEB≅∆DEC.

• Given: AB║DC, AB DC≅ . E is the midpoint of BC and AD.

• Prove ∆AEB ≅∆DEC• Plan for proof: Use the

fact that AEB and DEC are vertical angles to show that those angles are congruent. Use the fact that BC intersects parallel segment AB and DC to identify other pairs of angles that are congruent.

E

A B

D C

Proof:Statements:

1. AB║DC, AB DC≅

2. EAB ≅ EDC, ABE ≅ DCE

3. AEB ≅ DEC4. E is the midpoint of

AD, E is the midpoint of BC.

5. AE ≅ DE, BE ≅ CE

6. ∆AEB ≅ ∆DEC

Reasons:E

A B

D C

Given: AB║DC, AB DC. E is the ≅midpoint of BC and AD.Prove ∆AEB ∆DEC≅

Proof:Statements:

1. AB║DC, AB≅DC

2. EAB ≅ EDC, ABE ≅ DCE

3. AEB ≅ DEC4. E is the midpoint of

AD, E is the midpoint of BC.

5. AE ≅ DE, BE ≅ CE

6. ∆AEB ≅ ∆DEC

Reasons:

1. Given

Proof:Statements:

1. AB║DC, AB≅DC

2. EAB ≅ EDC, ABE ≅ DCE

3. AEB ≅ DEC4. E is the midpoint of

AD, E is the midpoint of BC.

5. AE ≅ DE, BE ≅ CE

6. ∆AEB ≅ ∆DEC

Reasons:

1. Given

2. Alternate interior angles theorem

Proof:Statements:

1. AB║DC, AB≅DC

2. EAB ≅ EDC, ABE ≅ DCE

3. AEB ≅ DEC4. E is the midpoint of

AD, E is the midpoint of BC.

5. AE ≅ DE, BE ≅ CE

6. ∆AEB ≅ ∆DEC

Reasons:

1. Given

2. Alternate interior angles theorem

3. Vertical angles theorem

Proof:Statements:

1. AB║DC, AB≅DC

2. EAB ≅ EDC, ABE ≅ DCE

3. AEB ≅ DEC4. E is the midpoint of

AD, E is the midpoint of BC.

5. AE ≅ DE, BE ≅ CE

6. ∆AEB ≅ ∆DEC

Reasons:

1. Given

2. Alternate interior angles theorem

3. Vertical angles theorem

4. Given

Proof:Statements:

1. AB║DC, AB≅DC

2. EAB ≅ EDC, ABE ≅ DCE

3. AEB ≅ DEC4. E is the midpoint of

AD, E is the midpoint of BC.

5. AE ≅ DE, BE ≅ CE

6. ∆AEB ≅ ∆DEC

Reasons:

1. Given

2. Alternate interior angles theorem

3. Vertical angles theorem

4. Given

5. Definition of a midpoint

Proof:Statements:

1. AB║DC, AB≅DC

2. EAB ≅ EDC, ABE ≅ DCE

3. AEB ≅ DEC4. E is the midpoint of

AD, E is the midpoint of BC.

5. AE ≅ DE, BE ≅ CE

6. ∆AEB ≅ ∆DEC

Reasons:

1. Given

2. Alternate interior angles theorem

3. Vertical angles theorem

4. Given

5. Definition of a midpoint

6. Definition of congruent triangles

What should you have learned?

• To prove two triangles congruent by the definition of congruence—that is all pairs of corresponding angles and corresponding sides are congruent.

• In upcoming lessons you will learn more efficient ways of proving triangles are congruent. The properties on the next slide will be useful in such proofs.

Theorem 4.4 Properties of Congruent Triangles

• Reflexive property of congruent triangles: Every triangle is congruent to itself.

• Symmetric property of congruent triangles: If ∆ABC ≅ ∆DEF, then ∆DEF ≅ ∆ABC.

• Transitive property of congruent triangles: If ∆ABC ≅ ∆DEF and ∆DEF ≅ ∆JKL, then ∆ABC ≅ ∆JKL.