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## Lesson 4.4 - 4.5 Proving Triangles Congruent

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**Triangle Congruency Short-Cuts**If you can prove one of the following short cuts, you have two congruent triangles • SSS (side-side-side) • SAS (side-angle-side) • ASA (angle-side-angle) • AAS (angle-angle-side) • HL (hypotenuse-leg) right triangles only!**Shared side**SSS Vertical angles SAS Parallel lines -> AIA Shared side SAS**SOME REASONS For Indirect Information**• Def of midpoint • Def of a bisector • Vert angles are congruent • Def of perpendicular bisector • Reflexive property (shared side) • Parallel lines ….. alt int angles • Property of Perpendicular Lines**Side-Side-Side (SSS)**E B F A D C • AB DE • BC EF • AC DF ABC DEF**Side-Angle-Side (SAS)**B E F A C D • AB DE • A D • AC DF ABC DEF included angle**Angle-Side-Angle (ASA)**B E F A C D • A D • AB DE • B E ABC DEF included side**Angle-Angle-Side (AAS)**B E F A C D • A D • B E • BC EF ABC DEF Non-included side**Warning: No AAA Postulate**There is no such thing as an AAA postulate! E B A C F D NOT CONGRUENT**Warning: No SSA Postulate**There is no such thing as an SSA postulate! E B F A C D NOT CONGRUENT**Name That Postulate**(when possible) SAS ASA SSA SSS**This is called a common side.**It is a side for both triangles. We’ll use the reflexive property.**HL( hypotenuse leg ) is used**only with right triangles, BUT, not all right triangles. ASA HL**Name That Postulate**(when possible) Vertical Angles Reflexive Property SAS SAS Reflexive Property Vertical Angles SSA SAS**Name That Postulate**(when possible)**Name That Postulate**(when possible)**Let’s Practice**ACFE Indicate the additional information needed to enable us to apply the specified congruence postulate. For ASA: B D For SAS: AF For AAS:**G**K I H J Determine if whether each pair of triangles is congruent by SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write not possible. Ex 4 ΔGIH ΔJIK by AAS**B**A C D E Determine if whether each pair of triangles is congruent by SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write not possible. Ex 5 ΔABC ΔEDC by ASA**Determine if whether each pair of triangles is congruent by**SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write not possible. Ex 6 E A C B D ΔACB ΔECD by SAS**Determine if whether each pair of triangles is congruent by**SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write not possible. Ex 7 J K L M ΔJMK ΔLKM by SAS or ASA**Determine if whether each pair of triangles is congruent by**SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write not possible. J T Ex 8 L K V U Not possible**SSS (Side-Side-Side) Congruence Postulate**• If three sides of one triangle are congruent to three sides of a second triangle, the two triangles are congruent. If Side Side Side Then ∆ABC ≅ ∆PQR**Example 1**Prove: ∆DEF ≅ ∆JKL From the diagram, SSS Congruence Postulate. ∆DEF ≅ ∆JKL**SAS (Side-Angle-Side) Congruence Postulate**• If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent.**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, the two triangles are congruent. ∠A ≅ ∠D If Angle Side Angle ∠C ≅ ∠F Then ∆ABC ≅ ∆DEF**Example 2**Prove: ∆SYT ≅ ∆WYX**Side-Side-Side Postulate**• SSS postulate: If two triangles have three congruent sides, the triangles are congruent.**Angle-Angle-Side Postulate**• If two angles and a non included side are congruent to the two angles and a non included side of another triangle then the two triangles are congruent.**Angle-Side-Angle Postulate**• If two angles and the side between them are congruent to the other triangle then the two angles are congruent.**Side-Angle-Side Postulate**• If two sides and the adjacent angle between them are congruent to the other triangle then those triangles are congruent.**Which Congruence Postulate to Use?**1. Decide whether enough information is given in the diagram to prove that triangle PQR is congruent to triangle PQS. If so give a two-column proof and state the congruence postulate.**ASA**• If 2 angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the 2 triangles are congruent. A Q S C R B**AAS**• If 2 angles and a nonincluded side of one triangle are congruent to 2 angles and the corresponding nonincluded side of a second triangle, then the 2 triangles are congruent. A Q S C R B**AAS Proof**• If 2 angles are congruent, so is the 3rd • Third Angle Theorem • Now QR is an included side, so ASA. A Q S C R B**Example**• Is it possible to prove these triangles are congruent?**Example**• Is it possible to prove these triangles are congruent? • Yes - vertical angles are congruent, so you have ASA**Example**• Is it possible to prove these triangles are congruent?**Example**• Is it possible to prove these triangles are congruent? • No. You can prove an additional side is congruent, but that only gives you SS**Example**• Is it possible to prove these triangles are congruent? 2 1 3 4**Example**• Is it possible to prove these triangles are congruent? • Yes. The 2 pairs of parallel sides can be used to show Angle 1 =~ Angle 3 and Angle 2 =~ Angle 4. Because the included side is congruent to itself, you have ASA. 2 1 3 4**Included Angle**The angle between two sides H G I**E**Y S Included Angle Name the included angle: YE and ES ES and YS YS and YE E S Y**Included Side**The side between two angles GI GH HI**E**Y S Included Side Name the included side: Y and E E and S S and Y YE ES SY**Side-Side-Side Congruence Postulate**SSS Post. - If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent. If then,**1) Given**2) SSS Using SSS Congruence Post. Prove: • 1) • 2)