Lesson 23: Base Angles of Isosceles Triangles Day 1

Transcription

1 Lesson 23: Base Angles of Isosceles Triangles Day 1 Learning Targets I can examine two different proof techniques via a familiar theorem. I can complete proofs involving properties of an isosceles triangle. Opening Exercise: Complete the missing statements or reasons below to prove the triangles are congruent using the SAS triangle congruence criteria. Example 1) Given: AB = DC, ABC = DCB Prove: ABC DCB 1. AB = DC ABC DCB BC = CB 3. Reflexive Property 4. ABC BCD A D 5. Corresponding angles ABC DCB, name one other pair of sides that must be congruent: Name two other pairs of angles that must be congruent: Example 2) Given: AB RS, AB RS, CB TS Prove: ABC RST. Statement 1.) AB RS Reason 1.) 2.) AB RS 3.) CB TS 4.) B S 2.) 3.) 4.) 5.) ABC RST 5.) If ABC RST, name one other pair of sides that must be congruent: If ABC RST, name two other pairs of angles that must be congruent:

2 C P C T C When you already know that two triangles are congruent, you are then allowed to say that all non mentioned corresponding parts of the triangles must also be congruent. Write it as an If, then Statement:. Discussion Isosceles triangle ABC is given. What we already know about an isosceles triangle? 1. at least two congruent sides 2. at least e pair of congruent base angles Prove Base Angles of an Isosceles are Congruent: Using SAS Auxillary Segment: Draw the angle bisector AD of A, where D is the intersection of the bisector and BC. We are going to use this auxiliary line towards our SAS criteria. Given: Isosceles ABC, with AB = AC, AD bisects BAC Prove: B C. 1.) 2.) 3.) 4.) 5.) 6.) 1.) Given 2.) Given 3.) < bisector creates 2 < s (Definition of angle bisector 4.) Reflexive Property 5.) SAS 6.) CPCTC If you recognize isosceles triangles in a proof, then you may use the following reasons: 1) If two sides of a triangle are congruent, then 2) If two angles of a triangle are congruent, then CPCTC is an abbreviation for the phrase: Corresponding Parts of Congruent Triangles are Congruent." It can be used as a justification in a proof after you have proven two triangles congruent.

3 Classwork Form B Examples 1 : Proving Corresponding Parts Congruent Example 2 Example 3) Given: JK = JL; JR bisects KL. Prove: JR KL.

4 Classwork Form A Examples 1 : Proving Corresponding Parts Congruent Example 2 1.) AB DC 1.) 2.) 2.) Given 3.) BC BC 3.) 4.) 4.) SAS 5.) <A < D 5.) 1.) 1.) 2.) 2.) 3.) 3.) 4.) 4.) SSS 5.) 5.) CPCTC 6.) 6.) Angle bisector divides an angle into 2 angles (Definition of angle bisector) Example 3) Given: JK = JL; JR bisects KL. Prove: JR KL.

5 Lesson 23: Base Angles of Isosceles Triangles Day 1 Exit Ticket/ Work Time/HW Lesson 23 M1 1. Based upon the diagram below, if the given congruency of sides exists, name the isosceles triangle and identify, by appropriate name, the pair of base angles that are congruent. Name of Isosceles Triangle Pair of Congruent Angles 2. Given: ABC ACB, AX bisects BAC Prove: ABX ACX

6 Examples 1 : Proving Corresponding Parts Congruent 1. AB = DC 1. Given 2. ABC DCB 2. Given 3. BC = CB 3. Reflexive Property 4. ABC BCD 4. SAS 5. A D 5. Corresponding angles of congruent triangles are congruent Modified Example 2 1. AC = AD 1. Given 2. CB = DB 2. Given 3. AB = BA 3. Reflexive Property 4. ACB BDA 4. SSS 5. CAB BAD 5. Corresponding angles of congruent triangles are congruent Modified Example 3) Given: JK = JL; JR bisects KL. Prove: JR KL. 1. JK = JL 1. Given 2. K L 2. Base angles of an isosceles triangle are congruent 3. KR = RL 3.. Definition of a segment bisector 4. JRK JRL 4. SAS 5. JRK JRL 5. Corresponding angles of congruent triangles are congruent 6. m JRK + m JRL = Linear pairs form supplementary angles 7. 2( JRK) = Substitution Property of Equality 8. JRK = Division Property of Equality 9. JR KL. 9. Definition of perpendicular