Using Congruent Triangles

GeometryMrs. SpitzFall 2004

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

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

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: AB DE, AC DF, BC EF ≅ ≅ ≅Prove CAB≅FDE

Statements:1. AB ≅ DE2. AC ≅ DF3. BC ≅ EF4. ∆CAB ≅ ∆FDE5. CAB ≅ FDE

Reasons:1. Given

43

21

E

A

C

B

F

D

Given: AB DE, AC DF, BC EF ≅ ≅ ≅Prove CAB≅FDE

Statements:1. AB ≅ DE2. AC ≅ DF3. BC ≅ EF4. ∆CAB ≅ ∆FDE5. CAB ≅ FDE

Reasons:1. Given2. Given

43

21

E

A

C

B

F

D

Given: AB DE, AC DF, BC EF ≅ ≅ ≅Prove CAB≅FDE

Statements:1. AB ≅ DE2. AC ≅ DF3. BC ≅ EF4. ∆CAB ≅ ∆FDE5. CAB ≅ FDE

Reasons:1. Given2. Given3. Given

43

21

E

A

C

B

F

D

Given: AB DE, AC DF, BC EF ≅ ≅ ≅Prove CAB≅FDE

Statements:1. AB ≅ DE2. AC ≅ DF3. BC ≅ EF4. ∆CAB ≅ ∆FDE5. CAB ≅ FDE

Reasons:1. Given2. Given3. Given4. SSS Congruence

Post

43

21

E

A

C

B

F

D

Given: AB DE, AC DF, BC EF ≅ ≅ ≅Prove CAB≅FDE

Statements:1. AB ≅ DE2. AC ≅ DF3. BC ≅ EF4. ∆CAB ≅ ∆FDE5. CAB ≅ FDE

Reasons:1. Given2. Given3. Given4. SSS Congruence

Post5. Corres. parts of ≅

∆’s are ≅.

43

21

E

A

C

B

F

D

Given: QSRP, PT RT≅ Prove PS≅ RS

Statements:1. QS RP2. PT ≅ RT

Reasons:1. Given2. Given

43

21

TP

Q

R

S