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W6-7, , Learning Area, Quarter, , Mathematics, Third, , I. LESSON TITLE, II. MOST ESSENTIAL LEARNING, COMPETENCIES (MELCs), , III. CONTENT/CORE CONTENT, IV. LEARNING, Suggested, Time Frame, PHASES, 15 minutes, A. Introduction, , B. Development, , 180 minutes, , Grade Level, Date, , Nine, , Similarity and Triangle Similarity Theorem, Lesson 1: Illustrates similarity of figures (M9GE-IIIg-1), Lesson 2: Proves the conditions for similarity of triangles., 1.1 SAS Similarity Theorem, 1.2 SSS Similarity Theorem, 1.3 AA Similarity Theorem, 1.4 Right Triangle Similarity Theorem, 1.5 Special Right Triangle Theorem (M9GE-IIIg-h-1), Similar Figures and the Proof of Triangle Similarity Theorems, Learning Activities, Can you identify which of the following pairs are similar?, a., , c., , b., , d., , How can you say that the pair of figures or image are similar?, Similarity, In this lesson, you will learn that there are triangles and other polygons that have, the same shape but do not necessarily have the same size. The illustrative example, below will give you an idea on how we can say that the given figures are similar., , If you will observe, trapezoid ABCD and trapezoid EFGH have the same shape., When you pair the corresponding vertices, the angles coincide. It shows that their, corresponding angles are congruent: ∠A ≅∠E ; ∠B ≅ ∠F ; ∠C ≅ ∠G ; and ∠D ≅ ∠H., Another thing is that the ratios of the measure of the lengths of their, corresponding sides are equal., Thus, in EFGH to ABCD,, , 𝐸𝐹, 𝐴𝐵, , 3, , 𝐹𝐺, , 6, , 𝐵𝐶, , = =, , =, , 5, , 10, , =, , 𝐺𝐻, 𝐶𝐷, , =, , 6, , 12, , =, , 𝐸𝐻, 𝐴𝐷, , 4, , 1, , = = . Here, the scale, 8, , 2, 𝐴𝐵 6 𝐵𝐶, 1, factor k is ⁄2. We could also turn it around as ABCD to EFGH where, = =, =, 𝐸𝐹 3 𝐹𝐺, 10 𝐶𝐷 12 𝐴𝐷 8 2, =, =, =, = = = 2. Now here, the scale factor k is 2., 5, 𝐺𝐻, 6, 𝐸𝐻 4 1, , Based on the illustrative example, two polygons are similar (the symbol is ∼) if, their vertices can be paired so that corresponding angles are congruent and the, lengths of their corresponding sides are proportional., To indicate that trapezoid ABCD is similar to trapezoid EFGH, you can write, ABCD ∼ EFGH. If you use this notation, write the corresponding vertices on the, same order.

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IV. LEARNING, PHASES, , Suggested, Time Frame, , Learning Activities, Example 2, a. Given: △CAR and △PET. State the proportions that must be true if △CAR, ∼△PET by SSS Similarity., b. Given the statement that shows the proportionality of the three, 𝐷𝑂 𝑂𝑁 𝐷𝑁, corresponding sides of the two triangles,, =, =, , name the two, 𝐾𝐸 𝐸𝑌, 𝐾𝑌, similar triangles., Solution, 𝐶𝐴 𝐴𝑅 𝐶𝑅, a., =, =, 𝑃𝐸 𝐸𝑇 𝑃𝑇, b. △DON and △KEY, 1.3 AA Similarity Theorem, If two angles of one triangle are congruent respectively to two angles of, another triangle, then the triangles are similar., Illustration, Given:∠A ≅∠O ; ∠C ≅∠D, , Prove:△CAT ∼△DOG, Proof:, Statements, ∠A ≅∠O ; ∠C ≅∠D, m∠A ≅ m∠O ; m∠C ≅ m∠D, m∠A + m∠C = m∠O ≅ m∠D, m∠A + m∠C + m∠T= 180, m∠O + m∠D + m∠G = 180, m∠A + m∠C + m∠T= m∠O + m∠D +, m∠G, m∠T = m∠G, ∠T = ∠G, ∴△CAT ∼ △DOG, , Reasons, Given, Definition of Congruent Angles, Addition Property, The sum of the measures of the, interior angles of a triangle is 180., Transitive Property, Addition Property, Definition of congruent angles, AAA Similarity Theorem, , Example1, Given: ̅̅̅̅, 𝑈𝑉 ‖ ̅̅̅̅, 𝐵𝐶, Prove : △ABC ∼ △AUV by AA, Similarity Theorem, Solution, , Proof:, Statements, ̅̅̅̅ ‖ 𝐵𝐶, ̅̅̅̅, 𝑈𝑉, ∠𝐴𝑈𝑉 ≅ ∠ABC, m∠BAC ≅ 𝑚∠UAV, ∠BAC ≅ ∠UAV, ∴ △ABC ∼ △AUV, , Reasons, Given, If two parallel lines are cut by a, transversal, corresponding angles are, congruent., Reflexive Property, Definition of congruent angles, AA Similarity Theorem, , Example 2, Given: ̅̅̅̅, 𝐴𝐵 ‖ 𝐷𝐶. Name at least two, pairs of corresponding angles that are, congruent to prove that △AOB ∼ △DOC by, AA Similarity Theorem.

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IV. LEARNING, PHASES, , Suggested, Time Frame, , Learning Activities, 1.5.2, , , The 30-60-90 Right Triangle Theorem, In a 30-60-90 triangle, the length of the hypotenuse is 2 times the, length of the shorter leg and the length of the longer leg is √3 times, 1, the length of the shorter leg.On the other hand, the shorter leg is, 2, , the hypotenuse or, Illustration, , √3, 3, , times the longer leg., , Given:△ABC is a 30-60-90 triangle., Prove:c = 2a and b = a√3, Proof:, , C. Engagement, , 60 minutes, , Draw △ADC so that △ABC ≅ △ADC. m∠BAC + m∠DAC = m∠BAD = 60. m∠B, = m∠D = m∠BAD = 60. This shows that △ABDis equiangular, and hence,, equilateral. It follows that c = 2a. Using Pythagorean Theorem, a 2 + b2 = (2a)2 = 4a2., When simplified, b2 = 3a2 or b = a√3., Learning Task, Directions: Answer each of the following., 1. How do you find the scale factor of similar polygons?, 2. Illustrate or draw: △ART ∼△PEN. Then, complete each statement: ∠A ≅___ ;, 𝐴𝑅, ∠R ≅ ___ ; ∠T ≅ ___ ;, = ___ = ___, 𝑃𝐸, 3. Complete the following statement., , a. If PQRS∼ TUVW, then, 𝑃𝑆, = ___ ; x = ___, 𝑇𝑊, b. The scale factor of PQRS ∼ TUVW is ___., , ∠R≅ ___ ; ∠Q≅ ___ ;, , 4. In the given figure, △ADE ∼ △ABC. Which triangle similarity theorem justifies, this similarity? Show proof toyour answer., , 5. a. Using the figure below, name the three similar triangles., b. Write the proportions that exist among corresponding parts of similar, triangles.

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IV. LEARNING, PHASES, D. Assimilation, , V. ASSESSMENT, , (Learning Activity Sheets for, Enrichment, Remediation, or Assessment to be given, on Weeks 3 and 6), , Suggested, Time Frame, 60 minutes, , 60 minutes, , Learning Activities, Directions: Answer each of the following accordingly., 1. How do you find similar polygons?, 2. Are all squares similar? Explain your answer., 3. Using the figure on the right, are the, two triangles similar?, If so, state the triangle similarity, theorem and justify, your answer., 4. a. Given: △CUP and △JAR. State the proportions that must be true if △ CUP ∼, △JAR by SSS Similarity., b. Given the statement that shows the proportionality of the three, 𝑇𝑅, 𝑅𝑌, 𝑇𝑌, corresponding sides of the two triangles,, =, =, , name the two similar, 𝑂𝑈, 𝑈𝑇, 𝑂𝑇, triangles., 5. How do you solve a 30-60-90 right triangle given only the hypotenuse?, Directions: Answer each of the following accordingly., 1. Complete each statement., a. If for two polygons corresponding angles are ___ and corresponding sides are, ___, then the polygons are similar., b. If the scale factor between two similar triangles is one, then the triangles, are___., c. To find the length of the hypotenuse of a 45-45-90 triangle, multiply the, length of one of the legs by ___., 2. Fill in the statements and reasons that are left blank in proving the proportionality, of the given triangles., ̅̅̅̅ . 𝐷𝐸, ̅̅̅̅̅⊥ 𝐵𝐶, ̅̅̅̅ , 𝐹𝐺, ̅̅̅̅ ⊥ 𝐵𝐶, ̅̅̅̅ ., Given: ∆ABC is isosceles with base 𝐵𝐶, 𝐷𝐸 𝐵𝐸, Prove:, =, 𝐹𝐺 𝐶𝐺, Proof:, Statements, 1. ∆ABC is isosceles, 2., ̅̅̅̅̅⊥ BC, ̅̅̅̅, FG, ̅̅̅̅ ⊥ BC, ̅̅̅̅, 3. DE, 4. ∠BED and ∠CGF are right angles., 5. ∆BED and ∆CGF are right, triangles., 6., 𝐷𝐸 𝐵𝐸, 7. ∴, =, 𝐹𝐺 𝐶𝐺, , Reasons, 1., 2. Base angles of an isosceles, triangle are congruent., 3., 4., 5., 6. Right Triangle Similarity Theorem, 7., , 3. The following pairs of triangles are similar. State a theorem that supports your, answer., a., , b., , 4. Given: p ∥ q, Which of the following is not necessarily true?, a. AB : AC = AE : AD, b. ∠ACD ≅ ∠ABE, c. AB : BC = AE : ED, d. AB : ED = AE = BC, e. m∠BED + m∠CDA = 180, , c.

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IV. LEARNING, PHASES, VI. REFLECTION, , Suggested, Time Frame, 20 minutes, , Learning Activities, , , , Prepared by:, , Edgar V. Tuico, , The learners communicate the explanation of their personal assessment, as indicated in the Learner’s Assessment Card., The learners will write their personal insights about the lesson in their, notebook using the prompts below:, I understand that ___________________., I realize that ________________________., I need to learn more about __________., Checked by: MA. FILIPINA M. DRIO