Name: Ms. D’Amato Date: Block: Proving Triangles are Congruent 18. Be able to identify: 1. an angle included between 2 sides (called the ) 2. a side included between 2 angles (called the ) A Q C B P R ̅̅̅̅ . B is said to be “included between” ̅̅̅̅ and ̅̅̅̅ is said to be “included between” A and C. Examples: 19. 1. ̅̅̅̅ and ̅̅̅̅ What angle is included between ? 2. ̅̅̅̅ ? What is the “included angle” between ̅̅̅̅ & 3. What side is included between P and Q? The definition of congruent s requires that all 6 corresponding angles and sides be congruent. There are 5 other ways to prove 2 s congruent that require only half as much information. A. - If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent. B. - 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. Example 1: Example 2: B A P D E C F ABC by X Q R PQR Y by Z In the following examples you must: a. mark vertical s and, if lines //, other s b. mark reflexive s or reflexive sides . Examples: Are the 2 s ? Write yes or no. If yes, which method proves the congruence, SSS or SAS? 1. 2. _______ _______ 4. 3. _______ _______ 5. _______ _______ 7. 6. _______ _______ 8. _______ _______ C. _______ _______ _______ _______ 9. _______ _______ _______ _______ - If two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the two triangles are congruent. Notice that the side must be “included between” the 2 angles. Examples: Are the s by the ASA Postulate? 1. 2. _______ D. Examples: 3. _______ _______ _______ - If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of a second triangle, then the two triangles are congruent. Are the s below by the AAS theorem? 1. 2. _______ E. 4. 3. _______ 4. _______ _______ - If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of a second right triangle, then the two triangles are congruent. WARNINGS: (1) AAA does not prove 2 s . Why are angles alone not sufficient? (2) SSA (two sides and a not-included angle) does not prove 2 s . Two different sized s might be formed. (See the diagrams below.) 5” 40 4" 5” 40 4” 20. We often prove 2 triangles congruent in order to show that one of the six parts of the definition is true. Review: By the definition of congruent s, if ABC SNO, then __________ __________ __________ __________ __________ __________ *We need 3 of these 6 statements in order to prove the triangles congruent. The other 3 statements can be stated true after proving the s .* T Example: Given: E is the midpoint of ̅̅̅̅ . ̅̅̅̅̅ ̅̅̅ Prove: MET JET M Statements E Reasons 1. 1. Given 2. 2. Definition of a midpoint 3. 3. Given 4. 4. Reflexive property 5. TME 5. 6. MET 6. CPCTC means Note: Def of s means the same as CPCTC J

Download
# Name: Date: Ms. D`Amato Block: Proving Triangles are Congruent