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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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
• 4.4 pp. 223-225 #1-22 all
• Quiz after this section
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
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
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
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
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
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
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
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

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

4

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
Statements:

BD  BC

D  C

ABD  EBC

∆ABD  ∆EBC

Reasons:

1.

Proof:
Statements:

BD  BC

D  C

ABD  EBC

∆ABD  ∆EBC

Reasons:

1. Given

Proof:
Statements:

BD  BC

D  C

ABD  EBC

∆ABD  ∆EBC

Reasons:

Given

Given

Proof:
Statements:

BD  BC

D  C

ABD  EBC

∆ABD  ∆EBC

Reasons:

Given

Given

Alternate Interior Angles

Proof:
Statements:

BD  BC

D  C

ABD  EBC

∆ABD  ∆EBC

Reasons:

Given

Given

Alternate Interior Angles

Vertical Angles Theorem

Proof:
Statements:

BD  BC

D  C

ABD  EBC

∆ABD  ∆EBC

Reasons:

Given

Given

Alternate Interior Angles

Vertical Angles Theorem

ASA Congruence Theorem

Proof:
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.