4.5 Proving 4.5 Proving ΔΔs are s are : : ASA and AASASA and AAS

& HL & HL

Objectives:Objectives:

• Use the ASA Postulate to prove triangles Use the ASA Postulate to prove triangles congruentcongruent

• Use the AAS Theorem to prove triangles Use the AAS Theorem to prove triangles congruentcongruent

Postulate 4.3 Postulate 4.3 ((ASAASA):):Angle-Side-Angle Congruence Angle-Side-Angle Congruence

PostulatePostulate

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

Theorem 4.5 (Theorem 4.5 (AASAAS): ): Angle-Angle-Side Congruence Angle-Angle-Side Congruence

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

Example 1:Example 1:

Is it possible to prove these triangles are Is it possible to prove these triangles are congruent? If so, state the postulate or theorem congruent? If so, state the postulate or theorem you would use. Explain your reasoning.you would use. Explain your reasoning.

Example 1:Example 1:

In addition to the In addition to the angles and segments angles and segments that are marked, that are marked, EGF EGF JGH by the JGH by the Vertical Angles Vertical Angles Theorem. Two pairs Theorem. Two pairs of corresponding of corresponding angles and one pair of angles and one pair of corresponding sides corresponding sides are congruent. Thus, are congruent. Thus, you can use the you can use the AAS AAS Congruence Congruence TheoremTheorem to prove that to prove that ∆EFG ∆EFG ∆JHG. ∆JHG.

Example 2:Example 2:

Is it possible to prove Is it possible to prove these triangles are these triangles are congruent? If so, congruent? If so, state the postulate or state the postulate or theorem you would theorem you would use. Explain your use. Explain your reasoning.reasoning.

Example 2:Example 2:

In addition to the In addition to the congruent segments congruent segments that are marked, NP that are marked, NP NP. Two pairs of NP. Two pairs of corresponding sides corresponding sides are congruent. This are congruent. This is is not enough not enough informationinformation to prove to prove the triangles are the triangles are congruent. congruent.

Example 3:Example 3:

AD llAD ll CECE

Is it possible to prove Is it possible to prove these triangles are these triangles are congruent? If so, state congruent? If so, state the postulate or the postulate or theorem you would use. theorem you would use. Explain your Explain your reasoning.reasoning.

Answer: ASA Congruence PostulateAnswer: ASA Congruence Postulate

Hypotenuse- Leg (HL) Congruence Theorem:

If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and leg of a second right triangle, then the two triangles are congruent.

Example: because of HL.

A X

B C Y Z

Examples: Determine if the triangles are congruent. State the postulate or

theorem.

Triangles are congruent when you have…

SSS AAS

SAS ASA

HL

Triangles are not congruent when you have…

ASS or SSA AAA

AssignmentAssignment

Geometry:Geometry:WorksheetWorksheet