Unit 7 Congruency and Similarity Proving Triangles Congruent (SSS, SAS, ASA, AAS, and HL)

2 Congruent Figures Congruent figures are two figures that have the same size and shape. IF two figures are congruent THEN they have the same size and shape. IF two figures have the same size and shape THEN they are congruent. Two figures have the same size and shape IF they are congruent.

3 Methods of Proving Triangles Congruent SSS If three sides of one triangle are congruent to three sides of a second triangle, then the triangles are congruent. Included Angle:In a triangle, the angle formed by two sides is the included angle for the two sides, where the sides meet. Included Side:The side of a triangle that is shared by the two given angles. A B C D E F

4 Included Angles & Sides Included Angle: Included Side: * **

5 Methods for Proving Triangles Congruent ASA 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. SAS 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. A BC D E F A BC D E F

6 Methods of Proving Triangles Congruent AAS 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. 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 BC D E F A BC D E F

7 Steps for Proving Triangles Congruent 1.Mark the Given. ( What have I been told is congruent) 2.Mark anything else I know to be congruent … For ex. Common Sides / Vertical Angles 3.Choose a Method. (SSS, SAS, ASA, AAS, and HL) 4.Is there more than one possible answer ?? … 5.Fill in the Reasons … why you marked the parts. All statements must have a VALID reason.

8 Congruent Triangles - CPCTC CPCTC: Corresponding Parts of Congruent Triangles are Congruent Two triangles are congruent IF their corresponding parts (angles and sides) are congruent. B C A Q R P A ↔ P; B ↔ Q; C ↔ R Vertices of the 2 triangles correspond in the same order as the triangles are named. Corresponding sides and angles of the two congruent triangles: ≡ ≡ = = │ │

9 Problem 1  StatementsReasons Step 1: Mark the Given Step 2: Mark reflexive sides 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 Step 6: Is there more? AB D C SSS Given Reflexive Property SSS Postulate 1. AB  CD 2. BC  DA 3. AC  CA

10 Problem 2  Step 1: Mark the Given Step 2: Mark vertical angles 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 Step 6: Is there more? SAS Given Vertical Angles. SAS Postulate StatementsReasons

11 Problem 3 StatementsReasons Step 1: Mark the Given Step 2: Mark reflexive sides 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 Step 6: Is there more? ASA Given Reflexive Postulate ASA Postulate Z W Y X

Lesson 4-4: AAS & HL Postulate 12 Problem 4  StatementsReasons Step 1: Mark the Given Step 2: Mark vertical angles 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 Step 6: Is there more? AAS Given Vertical Angle Thm AAS Postulate

Lesson 4-4: AAS & HL Postulate 13 Problem 5  Step 1: Mark the Given Step 2: Mark reflexive sides 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 Step 6: Is there more? HL Given Reflexive Property HL Postulate StatementsReasons