CCGPS Analytic Geometry Proving Triangles Congruent (SSS, SAS, ASA, AAS, HL) Unit 1

SSS If the sides of one triangle are congruent to the sides of a second triangle, then the triangles are congruent. A B C D E F Unit 1 : SSS, SAS, ASA

Side-Side-Side SSS If all 3 sides of 2 triangles are congruent, the triangles are congruent. If AB  ED, and AC  EF BC  DF, , then the 2 triangles are congruent  Make sure that you write the congruency statement so that the corresponding vertices (and thus the corresponding sides) are in the same position in the congruency statement. ∆ABC  ∆EDF not ∆ABC  ∆DEF

* * * Included Angles Included Angle: Included Angle In a triangle, the angle formed by two sides is the included angle for the two sides. Lesson 4-3: SSS, SAS, ASA

* * * Included Sides Included Side: Included Side: The side of a triangle that forms a side of two given angles. Included Side: * * * Lesson 4-3: SSS, SAS, ASA 5

ASA Angle Side Angle If two angles and the included side of one triangle are congruent to the two angles and the included side of another triangle, then the triangles are congruent. A A B C D E F S Sides AB = ED A 6 Lesson 4-3: SSS, SAS, ASA

SAS Side Angle Side If two sides and the included angle of one triangle are congruent to the two sides and the included angle of another triangle, then the triangles are congruent. S A S Lesson 4-3: SSS, SAS, ASA 7

Steps for Proving Triangles Congruent Mark the Given. Mark … Reflexive Sides / Vertical Angles Choose a Method. (SSS , SAS, ASA) List the Parts … in the order of the method. Fill in the Reasons … why you marked the parts. Is there more? Lesson 4-3: SSS, SAS, ASA

Problem 1 - SSS AB @ CD BC DA AC CA 1. 2. 3. A B D C Statements Step 1: Mark the Given Step 2: Mark reflexive sides SSS Step 3: Choose a Method (SSS /SAS/ASA ) Step 4: List the Parts in the order of the method Step 5: Fill in the reasons Statements Reasons Step 6: Is there more? 1. AB @ CD 2. BC DA 3. AC CA Given A B D C Given Reflexive Property SSS Postulate Lesson 4-3: SSS, SAS, ASA

Unit 1 CCGPS Analytic Geom. SSS, SAS, ASA Problem 2 Step 1: Mark the Given Step 2: Mark vertical angles congruent SAS Step 3: Choose a Method (SSS /SAS/ASA) Step 4: List the Parts in the order of the method Step 5: Fill in the reasons Statements Reasons Step 6: Is there more? Given Vertical Angles. Given SAS Postulate Unit 1 CCGPS Analytic Geom. SSS, SAS, ASA

Problem 3 ASA X W Y Z Statements Reasons Step 1: Mark the Given Step 2: Mark reflexive sides ASA Step 3: Choose a Method (SSS /SAS/ASA) Step 4: List the Parts in the order of the method Step 5: Fill in the reasons Statements Reasons Step 6: Is there more? Z W Y X Given Reflexive Postulate Given ASA Postulate Lesson 4-3: SSS, SAS, ASA

AAS Angle Angle Side (corresponding) If two angles and a non included side of one triangle are congruent to the corresponding two angles and side of a second triangle, then the two triangles are congruent. IS NOT CONGRUENT WITH EITHER OF THE OTHER 2 Congruent Angles and side DO NOT correspond..

Hypotenuse Leg HL If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and corresponding leg of another right triangle, then the triangles are congruent. A B C D E F

Problem 1  AAS Statements Reasons Step 1: Mark the Given Step 2: Mark vertical angles AAS Step 3: Choose a Method (SSS /SAS/ASA/AAS/ HL ) Step 4: List the Parts in the order of the method Step 5: Fill in the reasons Statements Reasons Step 6: Is there more? Given Vertical Angle Thm Given AAS Postulate

Problem 2  HL Statements Reasons Step 1: Mark the Given Step 2: Mark reflexive sides HL Step 3: Choose a Method (SSS /SAS/ASA/AAS/ HL ) Step 4: List the Parts in the order of the method Step 5: Fill in the reasons Statements Reasons Step 6: Is there more? Given Given Reflexive Property HL Postulate