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4.4 Proving Triangles are Congruent: ASA and AASPowerPoint Presentation

4.4 Proving Triangles are Congruent: ASA and AAS

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## PowerPoint Slideshow about ' 4.4 Proving Triangles are Congruent: ASA and AAS' - derek-burris

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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 PostulateIf 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 TheoremGiven: congruent to two angles and the corresponding non-included side of a second triangle, then the triangles are congruent.A D, C F, BC EF

Prove: ∆ABC ∆DEF

Theorem 4.5: Angle-Angle-Side (AAS) Congruence TheoremYou are given that two angles of congruent to two angles and the corresponding non-included side of a second triangle, then the triangles are congruent.∆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 TheoremIs it possible to prove the triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning.

Ex. 1 Developing ProofA. In addition to the angles and segments that are marked, state the postulate or theorem you would use. Explain your reasoning.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 ProofIs it possible to prove the triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning.

Ex. 1 Developing ProofB. 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 ProofIs 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 Proof1

2

3

4

The two pairs of parallel sides can be used to show state the postulate or theorem you would use. Explain your reasoning.1 3 and 2 4. Because the included side WZ is congruent to itself, ∆WUZ ∆ZXW by the ASA Congruence Postulate.

Ex. 1 Developing Proof1

2

3

4

Given: AD state the postulate or theorem you would use. Explain your reasoning.║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 CongruentStatements: state the postulate or theorem you would use. Explain your reasoning.

BD BC

AD ║ EC

D C

ABD EBC

∆ABD ∆EBC

Reasons:

1.

Proof:Statements: state the postulate or theorem you would use. Explain your reasoning.

BD BC

AD ║ EC

D C

ABD EBC

∆ABD ∆EBC

Reasons:

1. Given

Proof:Statements: state the postulate or theorem you would use. Explain your reasoning.

BD BC

AD ║ EC

D C

ABD EBC

∆ABD ∆EBC

Reasons:

Given

Given

Proof:Statements: state the postulate or theorem you would use. Explain your reasoning.

BD BC

AD ║ EC

D C

ABD EBC

∆ABD ∆EBC

Reasons:

Given

Given

Alternate Interior Angles

Proof:Statements: state the postulate or theorem you would use. Explain your reasoning.

BD BC

AD ║ EC

D C

ABD EBC

∆ABD ∆EBC

Reasons:

Given

Given

Alternate Interior Angles

Vertical Angles Theorem

Proof:Statements: state the postulate or theorem you would use. Explain your reasoning.

BD BC

AD ║ EC

D C

ABD EBC

∆ABD ∆EBC

Reasons:

Given

Given

Alternate Interior Angles

Vertical Angles Theorem

ASA Congruence Theorem

Proof:Note: state the postulate or theorem you would use. Explain your reasoning.

- 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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