4.5 using congruent triangles
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4.5 Using Congruent Triangles. Geometry Mr. Davenport Fall 2009. Objectives:. Use congruent triangles to plan and write proofs. Use congruent triangles to prove constructions are valid. Planning a proof. - PowerPoint PPT PresentationTRANSCRIPT
4.5 Using Congruent Triangles
GeometryMr. Davenport
Fall 2009
Objectives:
Use congruent triangles to plan and write proofs.
Use congruent triangles to prove constructions are valid.
Planning a proof
Knowing that all pairs of corresponding parts of congruent triangles are congruent can help you reach conclusions about congruent figures.
Planning a proof
For example, suppose you want to prove that PQS ≅ RQS in the diagram shown at the right. One way to do this is to show that ∆PQS ≅ ∆RQS by the SSS Congruence Postulate. Then you can use the fact that corresponding parts of congruent triangles are congruent to conclude that PQS ≅ RQS.
Q
S
P R
Ex. 1: Planning & Writing a Proof
Given: AB ║ CD, BC ║ DA
Prove: AB≅CDPlan for proof: Show that ∆ABD ≅ ∆CDB. Then use the fact that corresponding parts of congruent triangles are congruent.
B
A
C
D
Ex. 1: Planning & Writing a Proof
Solution: First copy the diagram and mark it with the given information. Then mark any additional information you can deduce. Because AB and CD are parallel segments intersected by a transversal, and BC and DA are parallel segments intersected by a transversal, you can deduce that two pairs of alternate interior angles are congruent.
B
A
C
D
Ex. 1: Paragraph Proof
Because AD ║CD, it follows from the Alternate Interior Angles Theorem that ABD ≅CDB. For the same reason, ADB ≅CBD because BC║DA. By the Reflexive property of Congruence, BD ≅ BD. You can use the ASA Congruence Postulate to conclude that ∆ABD ≅ ∆CDB. Finally because corresponding parts of congruent triangles are congruent, it follows that AB ≅ CD.
B
A
C
D
Ex. 2: Planning & Writing a Proof
Given: A is the midpoint of MT, A is the midpoint of SR.Prove: MS ║TR.Plan for proof: Prove that ∆MAS ≅ ∆TAR. Then use the fact that corresponding parts of congruent triangles are congruent to show that M ≅ T. Because these angles are formed by two segments intersected by a transversal, you can conclude that MS ║ TR.
A
M
T
R
S
Given: A is the midpoint of MT, A is themidpoint of SR.Prove: MS ║TR.
Statements:1. A is the midpoint of MT,
A is the midpoint of SR.2. MA ≅ TA, SA ≅ RA3. MAS ≅ TAR4. ∆MAS ≅ ∆TAR5. M ≅ T6. MS ║ TR
Reasons:1. Given
A
M
T
R
S
Given: A is the midpoint of MT, A is themidpoint of SR.Prove: MS ║TR.
Statements:1. A is the midpoint of MT,
A is the midpoint of SR.2. MA ≅ TA, SA ≅ RA3. MAS ≅ TAR4. ∆MAS ≅ ∆TAR5. M ≅ T6. MS ║ TR
Reasons:1. Given
2. Definition of a midpoint
A
M
T
R
S
Given: A is the midpoint of MT, A is themidpoint of SR.Prove: MS ║TR.
Statements:1. A is the midpoint of MT,
A is the midpoint of SR.2. MA ≅ TA, SA ≅ RA3. MAS ≅ TAR4. ∆MAS ≅ ∆TAR5. M ≅ T6. MS ║ TR
Reasons:1. Given
2. Definition of a midpoint3. Vertical Angles
Theorem
A
M
T
R
S
Given: A is the midpoint of MT, A is themidpoint of SR.Prove: MS ║TR.
Statements:1. A is the midpoint of MT,
A is the midpoint of SR.2. MA ≅ TA, SA ≅ RA3. MAS ≅ TAR4. ∆MAS ≅ ∆TAR5. M ≅ T6. MS ║ TR
Reasons:1. Given
2. Definition of a midpoint3. Vertical Angles
Theorem4. SAS Congruence
Postulate
A
M
T
R
S
Given: A is the midpoint of MT, A is themidpoint of SR.Prove: MS ║TR.
Statements:1. A is the midpoint of MT,
A is the midpoint of SR.2. MA ≅ TA, SA ≅ RA3. MAS ≅ TAR4. ∆MAS ≅ ∆TAR5. M ≅ T6. MS ║ TR
Reasons:1. Given
2. Definition of a midpoint3. Vertical Angles
Theorem4. SAS Congruence
Postulate5. Corres. parts of ≅ ∆’s
are ≅
A
M
T
R
S
Given: A is the midpoint of MT, A is themidpoint of SR.Prove: MS ║TR.
Statements:1. A is the midpoint of MT,
A is the midpoint of SR.2. MA ≅ TA, SA ≅ RA3. MAS ≅ TAR4. ∆MAS ≅ ∆TAR5. M ≅ T6. MS ║ TR
Reasons:1. Given
2. Definition of a midpoint3. Vertical Angles Theorem4. SAS Congruence
Postulate5. Corres. parts of ≅ ∆’s
are ≅6. Alternate Interior Angles
Converse.
A
M
T
R
S
EC
D
B
A
Ex. 3: Using more than one pair of triangles.
Given: 1≅2, 3≅4.Prove ∆BCE≅∆DCEPlan for proof: The only information you have about ∆BCE and ∆DCE is that 1≅2 and that CE ≅CE. Notice, however, that sides BC and DC are also sides of ∆ABC and ∆ADC. If you can prove that ∆ABC≅∆ADC, you can use the fact that corresponding parts of congruent triangles are congruent to get a third piece of information about ∆BCE and ∆DCE.
21
43
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Given: 1≅2, 3≅4.Prove ∆BCE ∆DCE≅
Statements:1. 1≅2, 3≅42. AC ≅ AC3. ∆ABC ≅ ∆ADC4. BC ≅ DC5. CE ≅ CE6. ∆BCE≅∆DCE
Reasons:1. Given
EC
D
B
A43
21
Given: 1≅2, 3≅4.Prove ∆BCE ∆DCE≅
Statements:1. 1≅2, 3≅42. AC ≅ AC
3. ∆ABC ≅ ∆ADC4. BC ≅ DC5. CE ≅ CE6. ∆BCE≅∆DCE
Reasons:1. Given2. Reflexive property
of Congruence
EC
D
B
A43
21
Given: 1≅2, 3≅4.Prove ∆BCE ∆DCE≅
Statements:1. 1≅2, 3≅42. AC ≅ AC
3. ∆ABC ≅ ∆ADC
4. BC ≅ DC5. CE ≅ CE6. ∆BCE≅∆DCE
Reasons:1. Given2. Reflexive property
of Congruence3. ASA Congruence
Postulate
EC
D
B
A43
21
Given: 1≅2, 3≅4.Prove ∆BCE ∆DCE≅
Statements:1. 1≅2, 3≅42. AC ≅ AC
3. ∆ABC ≅ ∆ADC
4. BC ≅ DC5. CE ≅ CE6. ∆BCE≅∆DCE
Reasons:1. Given2. Reflexive property
of Congruence3. ASA Congruence
Postulate4. Corres. parts of ≅ ∆’s
are ≅
EC
D
B
A43
21
Given: 1≅2, 3≅4.Prove ∆BCE ∆DCE≅
Statements:1. 1≅2, 3≅42. AC ≅ AC
3. ∆ABC ≅ ∆ADC
4. BC ≅ DC5. CE ≅ CE6. ∆BCE≅∆DCE
Reasons:1. Given2. Reflexive property
of Congruence3. ASA Congruence
Postulate4. Corres. parts of ≅ ∆’s
are ≅5. Reflexive Property of
Congruence
EC
D
B
A43
21
Given: 1≅2, 3≅4.Prove ∆BCE ∆DCE≅
Statements:1. 1≅2, 3≅42. AC ≅ AC
3. ∆ABC ≅ ∆ADC
4. BC ≅ DC5. CE ≅ CE6. ∆BCE≅∆DCE
Reasons:1. Given2. Reflexive property of
Congruence3. ASA Congruence
Postulate4. Corres. parts of ≅ ∆’s are
≅5. Reflexive Property of
Congruence6. SAS Congruence Postulate
EC
D
B
A43
21
Given: QSRP, PT RT≅ Prove PS≅ RS
Statements:1. QS RP2. PT ≅ RT
Reasons:1. Given2. Given
43
21
TP
Q
R
S
Assignment:
Pgs. 233-234 #14, 15, 17, 18, 25, 26