4.5 prove triangles congruent by asa and aas - rjs … · 4.5 prove triangles congruent by asa and...

Your Notes

4.5 Prove Triangles Congruent by ASA and AASGoal p Use two more methods to prove congruences.

VOCABULARY

Flow proof

POSTULATE 21: ANGLE-SIDE-ANGLE (ASA) CONGRUENCE POSTULATE

If two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the two triangles are congruent.

If Angle ∠A > ,

A C

B

D F

E

Side } AC > , and

Angle ∠C > ,

then nABC > .

THEOREM 4.6: ANGLE-ANGLE-SIDE (AAS) CONGRUENCE THEOREM

If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of a second triangle, then the two triangles are congruent.

If Angle ∠A > ,

A C

B

D F

E

Angle ∠C > , and

Side } BC > ,

then nABC > .

104 Lesson 4.5 • Geometry Notetaking Guide Copyright © Holt McDougal. All rights reserved.

Your Notes

4.5 Prove Triangles Congruent by ASA and AASGoal p Use two more methods to prove congruences.

VOCABULARY

Flow proof A flow proof uses arrows to show the flow of a logical argument.

POSTULATE 21: ANGLE-SIDE-ANGLE (ASA) CONGRUENCE POSTULATE

If two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the two triangles are congruent.

If Angle ∠A > ∠D ,

A C

B

D F

E

Side } AC > } DF , and

Angle ∠C > ∠F ,

then nABC > nDEF .

THEOREM 4.6: ANGLE-ANGLE-SIDE (AAS) CONGRUENCE THEOREM

If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of a second triangle, then the two triangles are congruent.

If Angle ∠A > ∠D ,

A C

B

D F

E

Angle ∠C > ∠F , and

Side } BC > } EF ,

then nABC > nDEF .


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Your Notes

Copyright © Holt McDougal. All rights reserved. Lesson 4.5 • Geometry Notetaking Guide 105

1. 60° 60°

S

T

V

U

W

2. S T

W V

Checkpoint Can nSTW and nVWT be proven congruent with the information given in the diagram? If so, state the postulate or theorem you would use.

Can the triangles be proven congruent with the information given in the diagram? If so, state the postulate or theorem you would use.

a. b. c.

Solutiona. There is not enough information to prove the triangles

are congruent, because no are known to be congruent.

b. Two pairs of angles and a pair of sides are congruent. The triangles are congruent by the .

c. The vertical angles are congruent, so two pairs of angles and their are congruent. The triangles are congruent by the

.

Example 1 Identify congruent triangles

Your Notes


1. 60° 60°

S

T

V

U

W

Yes; AAS Congruence Theorem

2. S T

W V

Yes; ASA Congruence Postulate

Checkpoint Can nSTW and nVWT be proven congruent with the information given in the diagram? If so, state the postulate or theorem you would use.

Can the triangles be proven congruent with the information given in the diagram? If so, state the postulate or theorem you would use.

a. b. c.

Solutiona. There is not enough information to prove the triangles

are congruent, because no sides are known to be congruent.

b. Two pairs of angles and a non-included pair of sides are congruent. The triangles are congruent by the AAS Congruence Theorem .

c. The vertical angles are congruent, so two pairs of angles and their included sides are congruent. The triangles are congruent by the ASA Congruence Postulate .

Example 1 Identify congruent triangles

LAH_GE_11_FL_NTG_Ch04_089-122.in105 105LAH_GE_11_FL_NTG_Ch04_089-122.in105 105 1/15/09 7:01:45 PM1/15/09 7:01:45 PM


In the diagram, ∠1 > ∠4 and

A E

B D

C

F

12

43

}

CF bisects ∠ACE. Write a flow

proof to show nCBF > nCDF.

SolutionGiven ∠1 > ∠4, } CF bisects ∠ACE.

Prove nCBF > nCDF

Linear Pair Postulate

Congruent Def. of ∠ bisector Supps. Thm.

Example 2 Write a flow proof

∠1 > ∠4∠1 and ∠2 are .∠3 and ∠4 are .

} CF bisects.∠ACE.

∠2 > } CF > } CF ∠ACF >

∠CBF > ∠CDF

3. In Example 2, suppose it is given that } CF bisects ∠ACE and ∠BFD. Write a flow proof to show nCBF > nCDF.

Checkpoint Complete the following exercise.

Your Notes

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In the diagram, ∠1 > ∠4 and

A E

B D

C

F

12

43

}

CF bisects ∠ACE. Write a flow

proof to show nCBF > nCDF.

SolutionGiven ∠1 > ∠4, } CF bisects ∠ACE.

Prove nCBF > nCDF

Given Linear Pair Postulate Given

Congruent Reflexive Prop. Def. of ∠ bisector Supps. Thm.

AAS Congruence Theorem

Example 2 Write a flow proof

∠1 > ∠4∠1 and ∠2 are supplements .∠3 and ∠4 are supplements .

} CF bisects.∠ACE.

∠2 > ∠3 } CF > } CF ∠ACF > ∠ECF

∠CBF > ∠CDF

3. In Example 2, suppose it is given that } CF bisects ∠ACE and ∠BFD. Write a flow proof to show nCBF > nCDF.

Given Given

Def. of ∠ Reflexive Prop. Def. of ∠ bisector bisector

ASA Congruence Postulate


}

CF bisects ∠ACE. }

CF bisects ∠BFD.

∠ACF > ∠ECF }

CF > }

CF ∠BFC > ∠DFC

∠CBF > ∠CDF

Your Notes



4. Theater Two actors are standing apart from each other on the edge of a stage. Spotlights are located and pointed as shown in the diagram. Can one of the actors move to another location on the stage without changing any of the angles of the triangle, without changing the distance to the other actor, and without requiring a spotlight to move?

20 ft40°


Games You and a friend are trying to find a flag hidden in the woods. Your friend is standing 75 feet away from you. When facing each other, the angle from you to the flag is 728 and the angle from your friend to the flag is 538. Is there enough information to locate the flag?

SolutionThe locations of you, your friend, and 72°

53°75 ftthe flag form a triangle. The measures

of and an of the triangle are known.

By the , all triangles with these measures are congruent. So, the triangle formed is unique and the flag location is given by the

.

Example 3 Choose a postulate or theorem

Homework

Your Notes



4. Theater Two actors are standing apart from each other on the edge of a stage. Spotlights are located and pointed as shown in the diagram. Can one of the actors move to another location on the stage without changing any of the angles of the triangle, without changing the distance to the other actor, and without requiring a spotlight to move?

20 ft40°

No. The measures of two angles and a nonincluded side of the triangle are known. By the AAS Congruence Theorem, all triangles with these measures are congruent. To create a different congruent triangle, one of the actors would have to move to a location that is off the stage, and the spotlight following that actor would have to move.


Games You and a friend are trying to find a flag hidden in the woods. Your friend is standing 75 feet away from you. When facing each other, the angle from you to the flag is 728 and the angle from your friend to the flag is 538. Is there enough information to locate the flag?

SolutionThe locations of you, your friend, and 72°

53°75 ftthe flag form a triangle. The measures

of two angles and an included side of the triangle are known.

By the ASA Congruence Postulate , all triangles with these measures are congruent. So, the triangle formed is unique and the flag location is given by the third vertex .

Example 3 Choose a postulate or theorem

Homework

Your Notes


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