4 4 proving triangles are congruent asa and aas
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4.4 Proving Triangles are Congruent: ASA and AAS. Geometry Mrs. Spitz Fall 2004. Objectives:. Prove that triangles are congruent using the ASA Congruence Postulate and the AAS Congruence Theorem Use congruence postulates and theorems in real-life problems. Assignment.

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objectives
Objectives:
  • Prove that triangles are congruent using the ASA Congruence Postulate and the AAS Congruence Theorem
  • Use congruence postulates and theorems in real-life problems.
assignment
Assignment
  • 4.4 pp. 223-225 #1-22 all
  • Quiz after this section
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 triangles are congruent.Postulate 21: Angle-Side-Angle (ASA) Congruence Postulate
theorem 4 5 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 triangles are congruent.Theorem 4.5: Angle-Angle-Side (AAS) Congruence Theorem
theorem 4 5 angle angle side aas congruence theorem2
You are given that two angles of ∆ABC are congruent to two angles of ∆DEF. By the Third Angles Theorem, the third angles are also congruent. That is, B  E. Notice that BC is the side included between B and C, and EF is the side included between E and F. You can apply the ASA Congruence Postulate to conclude that ∆ABC  ∆DEF.Theorem 4.5: Angle-Angle-Side (AAS) Congruence Theorem
ex 1 developing proof
Is it possible to prove the triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning.Ex. 1 Developing Proof
ex 1 developing proof1
A. In addition to the angles and segments that are marked, EGF JGH by the Vertical Angles Theorem. Two pairs of corresponding angles and one pair of corresponding sides are congruent. You can use the AAS Congruence Theorem to prove that ∆EFG  ∆JHG.Ex. 1 Developing Proof
ex 1 developing proof2
Is it possible to prove the triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning.Ex. 1 Developing Proof
ex 1 developing proof3
B. In addition to the congruent segments that are marked, NP  NP. Two pairs of corresponding sides are congruent. This is not enough information to prove the triangles are congruent. Ex. 1 Developing Proof
ex 1 developing proof4
Is it possible to prove the triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning.

UZ ║WX AND UW

║WX.

Ex. 1 Developing Proof

1

2

3

4

ex 1 developing proof5
The two pairs of parallel sides can be used to show 1  3 and 2  4. Because the included side WZ is congruent to itself, ∆WUZ  ∆ZXW by the ASA Congruence Postulate.Ex. 1 Developing Proof

1

2

3

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ex 2 proving triangles are congruent
Given: AD ║EC, BD  BC

Prove: ∆ABD  ∆EBC

Plan for proof: Notice that ABD and EBC are congruent. You are given that BD  BC

. Use the fact that AD ║EC to identify a pair of congruent angles.

Ex. 2 Proving Triangles are Congruent
proof
Statements:

BD  BC

AD ║ EC

D  C

ABD  EBC

∆ABD  ∆EBC

Reasons:

1.

Proof:
proof1
Statements:

BD  BC

AD ║ EC

D  C

ABD  EBC

∆ABD  ∆EBC

Reasons:

1. Given

Proof:
proof2
Statements:

BD  BC

AD ║ EC

D  C

ABD  EBC

∆ABD  ∆EBC

Reasons:

Given

Given

Proof:
proof3
Statements:

BD  BC

AD ║ EC

D  C

ABD  EBC

∆ABD  ∆EBC

Reasons:

Given

Given

Alternate Interior Angles

Proof:
proof4
Statements:

BD  BC

AD ║ EC

D  C

ABD  EBC

∆ABD  ∆EBC

Reasons:

Given

Given

Alternate Interior Angles

Vertical Angles Theorem

Proof:
proof5
Statements:

BD  BC

AD ║ EC

D  C

ABD  EBC

∆ABD  ∆EBC

Reasons:

Given

Given

Alternate Interior Angles

Vertical Angles Theorem

ASA Congruence Theorem

Proof:
slide21
Note:
  • You can often use more than one method to prove a statement. In Example 2, you can use the parallel segments to show that D  C and A  E. Then you can use the AAS Congruence Theorem to prove that the triangles are congruent.
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