4.2 Congruence & Triangles Geometry Mrs. Spitz Fall 2005.
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4.2 Congruence & Triangles Geometry Mrs. Spitz Fall 2005
Transcript of 4.2 Congruence & Triangles Geometry Mrs. Spitz Fall 2005.
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 ≅ TRE
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 ≅ DCE3. 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 ≅ DCE3. 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 ≅ DCE3. 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 ≅ DCE3. 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 theorem3. Vertical angles theorem
Proof:Statements:
1. AB║DC, AB≅DC
2. EAB ≅ EDC, ABE ≅ DCE3. 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 theorem3. Vertical angles theorem4. Given
Proof:Statements:
1. AB║DC, AB≅DC
2. EAB ≅ EDC, ABE ≅ DCE3. 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 theorem3. Vertical angles theorem4. Given
5. Definition of a midpoint
Proof:Statements:
1. AB║DC, AB≅DC
2. EAB ≅ EDC, ABE ≅ DCE3. 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 theorem3. Vertical angles theorem4. Given
5. Definition of a midpoint6. 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.
