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Transcript
Lesson 5.1 and 5.2 Triangle Congruence: SSS, SAS, ASA, AAS, HL EQ: What information about two triangles allows you to conclude the triangles are congruent? Did you know… Triangle congruency is used in blueprints by construction engineers and managers? Two triangles are congruent if they have… exactly the same three sides and exactly the same three angles all corresponding sides are congruent all corresponding angles ae congruent We do not have to know that all three sides and all three angles are exactly the same to conclude that the triangles are congruent. Sometimes knowing that THREE out of six corresponding parts are congruent is enough to conclude that two triangles are congruent. There are five ways to prove or show that two triangles are congruent. 1) Side-Side-Side Congruence (SSS) – Two triangles with ALL THREE CORRESPONDING SIDES congruent. 2) Side-Angle-Side (SAS) – Two triangles with TWO CORRESPONDING SIDES and the INCLUDED ANGLE congruent. 3) Angle-Side-Angle (ASA) – Two triangles with TWO CORRESPONDING ANGLES and the INCLUDED SIDE congruent. 4) Angle-Angle-Side (AAS) – Two triangles with TWO CORRESPONDING ANGLES and the NONINCLUDED SIDE congruent. 5) Hypotenuse-Leg (HL)* – Two RIGHT triangles whose HYPOTENUSE and ONE LEG congruent. *Applies only to right triangles Understanding the difference between an included angle, an included side, and a non-included side. An included angle is an angle that is made up of the intersection of two sides. It is between the two sides. Summary: An included side is a side that is between two angles. A non-included side is a side that is not between two angles. Practice: Use the diagram to answer the questions that follow. 1. Given: ̅̅̅̅ 𝐶𝐴 and ̅̅̅̅ 𝐴𝑇, what angle is an included angle? ________ 2. Given: ∠𝐶 and ∠𝑇, what side is an included side? ________ 3. Given: ∠𝐴 and ∠𝑇, what side is a non-included side? _______ 4. Given: ∠𝐶 and ∠𝑇, what side is the included side? _______ 5. Given: ̅̅̅̅ 𝑇𝐶 and ̅̅̅̅ 𝐴𝑇, what angle is the included angle? _______ 6. Given: ∠𝑇 and ∠𝐴, what side is the non-included side? _______ Side-Side-Side Congruence Postulate Side-Angle-Side Congruence Postulate Angle-Side-Angle Congruence Postulate If two angles and the included side of one triangle is congruent to two corresponding angles and the included side of another triangle, then the triangles are congruent. Abbreviation: ASA ∆𝐴𝐵𝐶 ≅ ∆𝐷𝐸𝐹 Angle-Angle-Side Congruence Postulate Hypotenuse-Leg Congruence Postulate If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse of another right triangle, then the triangles are congruent. Abbreviation: HL ∆𝐴𝐵𝐶 ≅ ∆𝑋𝑌𝑍 Guided Practice Use SSS to explain why the triangles in each pair are congruent. 1. JKM > LKM 2.ABC > CDA _________________________ _________________________ _________________________ _________________________ _________________________ _________________________ _________________________ _________________________ 3.Use SAS to explain why WXY > WZY. ____________________________________________ ____________________________________________ ____________________________________________ Show that the triangles are congruent for the given value of the variable. 4. BCD ≅ FGH, x 6 5.PQR ≅ VWX, n 3 _______________________________ ______________________________ _______________________________ ______________________________ _______________________________ ______________________________ _______________________________ ______________________________ Determine whether you can use ASA to prove the triangles congruent. Explain. 1. KLM and NPQ 2.EFG and XYZ __________________________ _________________________ __________________________ _________________________ 3. KLM and PNM, given that M is the midpoint of NL 4.STW and UTV ______________________________ _______________________ ______________________________ _______________________