Warm Up Yes, ASA. Yes, AAS. Not enough info. Yes, SAS. Determine if the triangles are congruent. If so, state what theorem/postulate makes them congruent. Yes, ASA. Yes, AAS. Not enough info. Yes, SAS.

Isosceles Triangle Theorem

Review What theorems do we have to help us to prove triangles are congruent? SSS – Side, Side, Side SAS – Side, Included Angle, Side ASA – Angle, Included Side, Angle AAS – Angle, Angle, Non-included Side

Review Which relationships are we not allowed to use for proving triangles congruent? ASS or SSA AAA

Think and Discuss 1.) Discuss with your partner what it really means when ∆’s are congruent. Ways to start your conversation: I know that if triangles are congruent then… When triangles are congruent…. 2.) Does this mean that all corresponding parts of one triangle are congruent to all the parts of the other?

CPCTC Corresponding Parts of Congruent Triangles are Congruent. Once you’ve proven that triangles are congruent, you now can conclude that all their other corresponding parts are also congruent. EXAMPLE: If ∆𝑨𝑩𝑪≅∆𝑿𝒀𝒁 by ASA ≅, then we know that ∠𝑪≅∠𝒁, 𝑨𝑪 ≅ 𝒁𝑿 , 𝒂𝒏𝒅 𝑩𝑪 ≅ 𝒀𝒁 because Corresponding Parts of Congruent Triangles are Congruent.

Proof using CPCTC Statements Reasons 1.) 𝐷𝐴 ≅ 𝐷𝐸 ; 𝐴𝑇 ≅ 𝐸𝑇 1.) Given Given: See diagram Prove: ∠𝐴≅∠𝐸 Statements Reasons 1.) 𝐷𝐴 ≅ 𝐷𝐸 ; 𝐴𝑇 ≅ 𝐸𝑇 1.) Given 2.) 𝐷𝑇 ≅ 𝐷𝑇 2.) Reflexive POC (Shared Side) 3.) ∆𝐴𝐷𝑇≅∆𝐸𝐷𝑇 3.) SSS ≅ 4.) ∠𝐴≅∠𝐸 4.) Corresponding Parts of Congruent Triangles are Congruent (CPCTC)

YOU TRY! Statements Reasons 1.) 𝐴𝐶 ≅ 𝐷𝐶 ; ∠𝐴≅∠𝐷 1.) Given 2.)∠𝐴𝐶𝐵≅∠𝐷𝐶𝐸 Given: See diagram Prove: 𝐵𝐴 ≅ 𝐸𝐷 Statements Reasons 1.) 𝐴𝐶 ≅ 𝐷𝐶 ; ∠𝐴≅∠𝐷 1.) Given 2.)∠𝐴𝐶𝐵≅∠𝐷𝐶𝐸 2.) Vertical Angles are ≅. 3.) ∆𝐴𝐶𝐵≅∆𝐷𝐶𝐸 3.) ASA ≅ 4.) 𝐵𝐴 ≅ 𝐸𝐷 4.) CPCTC

BREAK! 5 minutes

Classifying Triangles by Sides Scalene – No congruent Sides Isosceles – At least 2 congruent sides Equilateral – All 3 congruent sides.

Components of an Isosceles Triangle B C VERTEX ANGLE LEG LEG BASE ANGLES BASE

Isosceles Triangle Theorem If two sides of a triangle are congruent, then the angles opposite them are congruent. A B C

Converse of Isosceles Triangle Theorem If two angles of a triangle are congruent, then the sides opposite them are congruent. A B C

Example 1: Therefore, 𝒎∠𝒁 𝒂𝒏𝒅 𝒎∠𝒀=𝟔 𝟎 𝒐 . Find the 𝒎∠𝒁 and the 𝒎∠𝒀. Since this is an isosceles triangle, that means that the base angles must be congruent to each other so ∠𝒁≅ ∠𝒀. Since all angles in a triangle have a sum of 180 degrees, that means that 𝒎∠𝒁+𝒎∠𝒀=𝟏𝟐𝟎. Therefore, 𝒎∠𝒁 𝒂𝒏𝒅 𝒎∠𝒀=𝟔 𝟎 𝒐 .

PRACTICE! 𝒎∠𝑴= 𝟑𝟎 𝒐 𝑷𝑵=𝟏𝟏 𝒄𝒎

Corollary to Isosceles Triangle Theorem If a triangle is equilateral, then it is equiangular.

Converse to the Corollary ofIsosceles Triangle Theorem If a triangle is equiangular, then it is equilateral.

Example 2: Using Algebra Solve for the missing variables. Since the vertex angle and a base angle are the same measure, that means this triangle must be equiangular. 2𝑥=60 𝒙=𝟑𝟎 Since it’s equiangular, it’s equilateral. 4𝑦−5=3 4𝑦=8 𝒚=𝟐

Practice! 𝒚=𝟐 𝒙=𝟕 Find the value of the missing variables. Since the triangle is isosceles… 4𝑦−2=2𝑦+2 2𝑦−2=2 2𝑦=4 𝒚=𝟐 Since the vertex angle is 80 degrees, the base angles must each be 50 degrees 6𝑥+8=50 6𝑥=42 𝒙=𝟕

Proving the Isosceles Triangle Theorem B C D Given: 𝐴𝐵 ≅ 𝐵𝐶 Prove: ∠𝐴≅∠𝐶 PLAN: Draw an auxiliary line that bisects ∠𝐴𝐵𝐶. Then prove the triangles are congruent. Then show that the angles are congruent by CPCTC. Statements Reasons Lines Used 1.) 𝐴𝐵 ≅ 𝐵𝐶 1.) Given 2.) Draw Angle Bisector of ∠𝐴𝐵𝐶 to opposite side and label point D. 2.) Every angle can be bisected 3.) ∠𝐴𝐵𝐷≅∠𝐶𝐵𝐷 3.) Def of ∠ Bisector 2 4.) 𝐵𝐷 ≅ 𝐵𝐷 4.) Reflexive Property of Congruence. 5.) ∆𝐴𝐵𝐷≅∆𝐶𝐵𝐷 5.) 𝑆𝐴𝑆 ≅ 1,3,4 6.) ∠𝐴≅∠𝐶 6.) CPCTC 5