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Lesson 15 NYS COMMON CORE MATHEMATICS CURRICULUM Name:__________________________________ M2 Period: ________ Date: __________ Lesson 15: Angle – Angle Similarity Learning Targets I can use the AA criteria to solve for missing angles or sides in triangle problems. I can prove two triangles to be similar by using Angle - Angle criteria Opening Exercise Recall and identify the property or theorem that applies to each diagram below: 1. 2. 3. 4. We will be using these theorems and more in creating two-column similarity proofs. Recall from yesterday: Angle-Angle Similarity – Notation: ____________ If ___________ angles of one triangle are ____________________ to _____________ angles of another triangle, then the two triangles are _____________________. Example: ̅̅̅̅ ⊥ 𝐵𝐶 ̅̅̅̅ , 𝐴𝐷 ̅̅̅̅ bisects ∠𝐵𝐴𝐶 Given: 𝐴𝐷 Prove: ∆𝐴𝐵𝐷~∆𝐴𝐶𝐷 Statement Reason ̅̅̅̅ 1. ̅̅̅̅ 𝐴𝐷 ⊥ 𝐵𝐶 Given 2. ̅̅̅̅ 𝐴𝐷 bisects ∠𝐵𝐴𝐶 Given 3. ∠𝐴𝐷𝐵, ∠𝐴𝐷𝐶 are right angles 4. ∠𝐴𝐷𝐵 ≅ ∠𝐴𝐷𝐶 5. Definition of angle bisector 6. ∆𝐴𝐵𝐷~∆𝐴𝐶𝐷 AA Similarity Lesson 15 NYS COMMON CORE MATHEMATICS CURRICULUM Name:__________________________________ Period: ________ Date: __________ Lesson 15: Angle – Angle Similarity Classwork (Form A) 1. Given: ∠𝑊 ≅ ∠𝑇 Prove: 𝑊𝑉𝑍 ~ 𝑇𝑉𝑆 Statement 1. ∠𝑊 ≅ ∠𝑇 Reason Given 2. ∠𝑊𝑉𝑍 ≅ ∠𝑇𝑉𝑆 3. AA Similarity ̅̅̅̅ ⊥ 𝐵𝐶 ̅̅̅̅ , 𝐴𝐵 ̅̅̅̅ ⊥ 𝐴𝐶 ̅̅̅̅ 2. Given: 𝐴𝐷 Prove: ∆𝐴𝐷𝐶 ~ ∆𝐵𝐴𝐶 Statement ̅̅̅̅ 1. ̅̅̅̅ 𝐴𝐷 ⊥ 𝐵𝐶 Reason Given 2. ̅̅̅̅ 𝐴𝐵 ⊥ ̅̅̅̅ 𝐴𝐶 3. ∠𝐵𝐴𝐶, ∠𝐴𝐷𝐶 are right angles 4. 5. ∠𝐶 ≅ ∠𝐶 6. AA Similarity M2 Lesson 15 NYS COMMON CORE MATHEMATICS CURRICULUM Name:__________________________________ Period: ________ Date: __________ ̅̅̅̅ ⊥ 𝑉𝑊 ̅̅̅̅̅ , 𝑊𝑌 ̅̅̅̅̅ ⊥ 𝑈𝑉 ̅̅̅̅ . 3. Given: 𝑈𝑋 Prove: ∆𝑈𝑋𝑉~∆𝑊𝑌𝑉. Statement ̅̅̅̅ ⊥ 𝑉𝑊 ̅̅̅̅̅ 1. 𝑈𝑋 Reason Given ̅̅̅̅̅ ⊥ 𝑈𝑉 ̅̅̅̅ 2. 𝑊𝑌 3. ∠𝑈𝑋𝑉, ∠𝑊𝑌𝑉 are right angles 4. 5. 6. AA Similarity ̅̅̅̅ ∥ ̅̅̅̅ 4. Given: Trapezoid 𝐴𝐵𝐷𝐸, and 𝐴𝐵 𝐸𝐷 Prove: ∆𝐴𝐹𝐵 ~∆𝐷𝐹𝐸. Statement 1. Trapezoid 𝐴𝐵𝐷𝐸 Reason Given 2. ̅̅̅̅ 𝐴𝐵 ∥ ̅̅̅̅ 𝐸𝐷 3. ∠𝐸𝐷𝐹 ≅ ∠𝐵𝐴𝐹 4. Vertical Angles are congruent 5. AA Similarity M2 Lesson 15 NYS COMMON CORE MATHEMATICS CURRICULUM Name:__________________________________ Period: ________ Date: __________ Lesson 15: Angle – Angle Similarity Classwork (Form B) 1. Given: ∠𝑊 ≅ ∠𝑇 Prove: 𝑊𝑉𝑍 ~ 𝑇𝑉𝑆 Statement 1. Reason Given 2. 3. ̅̅̅̅ ⊥ 𝐵𝐶 ̅̅̅̅ , 𝐴𝐵 ̅̅̅̅ ⊥ 𝐴𝐶 ̅̅̅̅ 2. Given: 𝐴𝐷 Prove: ∆𝐴𝐷𝐶 ~ ∆𝐵𝐴𝐶 Statement 1. 2. 3. 4. 5. 6. Reason Given M2 Lesson 15 NYS COMMON CORE MATHEMATICS CURRICULUM Name:__________________________________ Period: ________ Date: __________ ̅̅̅̅ ⊥ 𝑉𝑊 ̅̅̅̅̅ , 𝑊𝑌 ̅̅̅̅̅ ⊥ 𝑈𝑉 ̅̅̅̅ . 3. Given: 𝑈𝑋 Prove: ∆𝑈𝑋𝑉~∆𝑊𝑌𝑉. Statement Reason 1. Given 2. Given 3. 4. 5. 6. ̅̅̅̅ ∥ ̅̅̅̅ 4. Given: Trapezoid 𝐴𝐵𝐷𝐸, and 𝐴𝐵 𝐸𝐷 Prove: ∆𝐴𝐹𝐵 ~∆𝐷𝐹𝐸. Statement Reason 1. Given 2. Given 3. 4. 5. M2 Lesson 15 NYS COMMON CORE MATHEMATICS CURRICULUM Name:__________________________________ M2 Period: ________ Date: __________ Lesson 15: Angle – Angle Similarity Homework (Form A) In the diagram at right, ̅̅̅̅ 𝐷𝐸 , ̅̅̅̅ 𝐸𝐹 , and ̅̅̅̅ 𝐹𝐷 are midsegments of ∆𝐴𝐵𝐶. Prove that ∆𝐴𝐵𝐶 ~∆𝐹𝐸𝐷. Statement Reason ̅̅̅̅ , and 𝐹𝐷 ̅̅̅̅ are midsegments of ∆𝐴𝐵𝐶. Given 1. ̅̅̅̅ 𝐷𝐸 , 𝐸𝐹 ̅̅̅̅ , _____ ∥ ̅̅̅̅ 2. ̅̅̅̅ 𝐷𝐸 ∥ 𝐵𝐶 𝐴𝐵 , _____ ∥ _____ Midsegments are ∥ to opposite side 3. _________ ≅ ∠𝐴𝐸𝐷, ∠𝐷𝐹𝐸 ≅ _________ Alternate Interior Angles are _________________ 4. ∠𝐴𝐸𝐷 ≅ ∠𝐶, ∠𝐶𝐸𝐹 ≅ _________ ___________________ angles are congruent 5. ∠𝐹𝐷𝐸 ≅ ___________, ∠𝐷𝐹𝐸 ≅ ___________ Transitive Property 6. Lesson 15 NYS COMMON CORE MATHEMATICS CURRICULUM Name:__________________________________ M2 Period: ________ Date: __________ Lesson 15: Angle – Angle Similarity Homework (Form B) In the diagram at right, ̅̅̅̅ 𝐷𝐸 , ̅̅̅̅ 𝐸𝐹 , and ̅̅̅̅ 𝐹𝐷 are midsegments of ∆𝐴𝐵𝐶. Prove that ∆𝐴𝐵𝐶 ~∆𝐹𝐸𝐷. Statement Reason ̅̅̅̅ , and 𝐹𝐷 ̅̅̅̅ are midsegments of ∆𝐴𝐵𝐶. Given 1. ̅̅̅̅ 𝐷𝐸 , 𝐸𝐹 2. ̅̅̅̅ 𝐷𝐸 ∥ ̅̅̅̅ 𝐴𝐶 , _____ ∥ ̅̅̅̅ 𝐴𝐵 , _____ ∥ _____ Midsegments are ∥ to opposite side 3. _________ ≅ ∠𝐴𝐸𝐷, ∠𝐷𝐹𝐸 ≅ _________ Alternate Interior Angles are _________________ 4. ∠𝐴𝐸𝐷 ≅ _________, ∠𝐶𝐸𝐹 ≅ _________ ___________________ are congruent 5. ∠𝐹𝐷𝐸 ≅ ___________, ∠𝐷𝐹𝐸 ≅ ___________ Transitive Property 6.
