Unit 8: Congruent and Similar Triangles Lesson 8.1 Apply Congruence and Triangles Lesson 4.2 from textbook Objectives • Identify congruent figures and corresponding parts of closed plane figures. • Prove that two triangles are congruent using definitions, properties, theorems, and postulates. Congruent Not Congruent *If two figures are congruent, then their corresponding parts are _______________________. In the diagram, ∆ABC ≅ ∆FED . Label the two triangles accordingly and mark all corresponding parts that are congruent. Congruence Statements: ________________________________________________________________ Example 1 Write a congruence statement for the triangles. Identify all pairs of corresponding congruent parts. Triangles _____________________________________ Corresponding Angles ____________________________________ Corresponding Sides _______________________________________ Example 2 In the diagram, DEFG ≅ SPQR . Find the value of x. _________________ Find the value of y. _________________ Example 3 In the diagram, a rectangular wall is divided into two sections. Are the sections congruent? Explain. _____________________________________________ _____________________________________________ Third Angles Theorem If two angles of one triangle are congruent to two angles of another triangles, then the third angles are _____________________________________________ Example 4 Find m<BDC. Example 5 Graph the triangle with vertices D(1, 2), E(7, 2), and F(5, 4). Then, graph a triangle congruent to ∆DEF . Example 6 Given: AD ≅ CB , DC ≅ BA , ∠ACD ≅ ∠CAB , ∠CAD ≅ ∠ACB Prove: ∆ACD ≅ ∆CAB Statements Reasons 1. 1. Given 2. 2. 3. 3. Given 4. 4. 5. ∆ACD ≅ ∆CAB 5. Properties of Congruent Triangles Reflexive Property For any triangle ABC, ∆ABC ≅ ________________. Symmetric Property If ∆ABC ≅ ∆DEF , then _______________________. Transitive Property If ∆ABC ≅ ∆DEF and ∆DEF ≅ ∆JKL , then _____________________. Unit 8: Congruent and Similar Triangles Lesson 8.2 Prove Triangles Congruent by SSS Lesson 4.3 from textbook Objectives • Use the Side-Side-Side (SSS) Congruence Postulate to prove that two triangles are congruent, along with other definitions, properties, theorems, and postulates. • Prove that two triangles are congruent in the coordinate plane using the Distance Formula and the SSS postulate. Side-Side-Side (SSS) Congruence Postulate If three sides of one triangle are congruent to three sides of a second triangle, then _________________________________________________ Example 1 Determine whether the congruence statement is true. Explain your reasoning. ∆DFG ≅ ∆HJK ∆ACB ≅ ∆CAD Example 2 Use the given coordinates to determine if ∆ABC ≅ ∆DEF . A(-3, -2), B(0, -2), C(-3, -8), D(10, 0), E(10, -3), F(4, 0) AB = __________ CA = __________ DE = __________ FD = __________ BC = __________ EF = __________ Example 3 Explanation: ____________________________________________________________________________________ ____________________________________________________________________________________ Example 4 Example 5 Statements Reasons 1. 1. 2. 2. 3. 3. Unit 8: Congruent and Similar Triangles Lesson 8.3 Prove Triangles Congruent by SAS and HL Lesson 4.4 from textbook Objectives • Use the Side-Angle-Side (SAS) and Hypotenuse-Leg (HL) Congruence Postulate to prove that two triangles are congruent, along with other definitions, properties, theorems, and postulates. • Use two-column proofs to justify statements about congruent triangles. Side-Angle-Side (SAS) Congruence Postulate If two sides and the included angle of one triangle are congruent to the corresponding to sides and corresponding and the corresponding included angle of a second triangle, then ___________________________________________________. Example 1 Decide whether enough information is given to prove that the triangles are congruent using the SAS Congruence Postulate. Hypotenuse-Leg Congruence Theorem If the leg and hypotenuse of a right triangle are congruent to the corresponding leg and hypotenuse of a second triangle, then ____________________________________________________. Example 2 State the third congruence that must be given to prove ∆ABC ≅ ∆DEF using indicated postulate. a) Given: AB ≅ DE ,CB ≅ FE , ______ ≅ ______ (SSS Congruence Postulate) b) Given: ∠A ≅ ∠D,CA ≅ FD , ______ ≅ _______ (SAS Congruence Postulate) c) Given: ∠B ≅ ∠E, AB ≅ DE , ______ ≅ _______ (SAS Congruence Postulate) Example 3 Statements Reasons 1. 1. 2. 2. Definition of perpendicular lines. 3. 3. Definition of a right triangle. 4. 4. 5. 5 6. 6. Example 4 Statements Reasons 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. Unit 8: Congruent and Similar Triangles Lesson 8.4 Prove Triangles Congruent by ASA and AAS Lesson 4.5 from textbook Objectives • Use the Angle-Side-Angle (ASA) and Angle-Angle-Side (AAS) Congruence Postulates to prove that two triangles are congruent, along with other definitions, properties, theorems, and postulates. • Use two-column proofs to justify statements about congruent triangles. Angle-Side-Angle (ASA) Congruence Postulate Angle-Angle-Side (AAS) Congruence Theorem Example 1 Is it possible to prove that the two triangles are congruent? If so, state the postulate or theorem you would use. ________________________ _______________________ Example 2 State the third congruence that must be given to prove ∆ABC ≅ ∆DEF using indicated postulate. a) Given: AB ≅ DE ,∠A ≅ ∠D, ______ ≅ ______ (AAS Congruence Postulate) b) Given: ∠A ≅ ∠D,CA ≅ FD , ______ ≅ _______ (ASA Congruence Postulate) Example 3 Tell whether you can use the given information to determine whether ∆ABC ≅ ∆DEF . Explain your reasoning. ∠A ≅ ∠D,AB ≅ DE , AC ≅ DF __________________________ ∠B ≅ ∠E,∠C ≅ ∠F, AC ≅ DE __________________________ Example 4 Given: X is the midpoint of VY and WZ . Prove: ∆VWX ≅ ∆YZX Statements Reasons 1. 1. 2. 2. 3. 3. 4. 4. Unit 8: Congruent and Similar Triangles Lesson 8.5 Using Congruent Triangles Lesson 4.6 from textbook Objectives • Use congruent triangles to plan and write proofs about their corresponding parts. Corresponding Parts of Congruent Triangles are Congruent Theorem (CPCTC) If ___________________________ are congruent then the ____________________________ of the congruent triangles are also _____________________. Given congruent parts: ________________________ ∆ABC ≅ ∆DEF by the ________________ Other corresponding congruent parts: __________________________________________ Example 1 Tell which triangles you can show are congruent in order to prove the statement. What postulate or theorem would you use? ∠A ≅ ∠D GK ≅ HJ ____________________ ____________________ ____________________ ____________________ Example 2 _____________________ ________________________ Example 3 Given: ∠Q ≅ ∠S,∠RTQ ≅ ∠RTS Prove: QT ≅ ST *FIRST PROVE TRIANGLES ARE CONGRUENT* Statements Reasons 1. 1. 2. 2. 3. 3. 4. 4. Example 4 Given: NM ≅ KM Prove: ∠MLK ≅ ∠MPN Statements Reasons 1. 1. 2. 2. 3. 3. 4. 4. Example 5 Use the diagram to write a plan for a proof: PLAN: Prove: ∠A ≅ ∠C Unit 8: Congruent and Similar Triangles Lesson 8.6 Prove Triangles Similar by AA Lesson 6.4 from textbook Objectives • Identify similar triangles using the Angle-Angle (AA) Similarity Postulate. • Find measures of similar triangles using proportional reasoning. ACTIVITY: Question: What can you conclude about two triangles if you know two pairs of corresponding angles are congruent? 1. Draw ∆EFG so that m<E = 40 o 2. Draw ∆RST so that <R = 40 o and m<T = 50 o, and m<G = 50 o. and is not congruent to ∆EFG . 3. Calculate m<F and m<S using Triangle Sum Theorem. ___________________________________ 4. Measure and record the side lengths of both triangles. (to the nearest mm). ___________________________________________ 5. Are the triangles similar? Explain. ____________________________________________________ 6. If all we know is that two angles in two different triangles are congruent, can we conclude that the triangles are similar? ______________________________ Angle-Angle (AA) Similarity Postulate If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar. _______________________________________________ Example 1 Determine whether the triangles are similar. If they are, write a similarity statement. Explain your reasoning. ______________________________________ ___________________________________ Example 2 Use the diagram to complete the information. MN ON MO ∆MON ~ _______ __ = = ___________________ ? ? ? 16 ? 12 ? = _________ = _______________ 12 10 16 y x = ____________ y = ____________ Example 3 The A-frame building shown in the figure has a balcony that is 16 feet long, 16 feet high, and parallel to the ground. The building is 28 feet wide at its base. How tall is the A-frame building? Height = ________________ Unit 8: Congruent and Similar Triangles Lesson 8.7 Prove Triangles Similar by SSS and SAS Lesson 6.5 from textbook Objectives • Use the similarity theorems such as the Side-Side-Side (SSS) Similarity Theorem and the Side- Angle-Side (SAS) Similarity Theorem to determine whether two triangles are similar. • Find measures of similar triangles using proportional reasoning. Side-Side-Side (SSS) Side-Angle-Side (SAS) Similarity Theorem Similarity Theorem If the corresponding side lengths of two If two sides of one triangle are proportiona l to triangles are proportional , then the triangles two sides of another triangle and their included are similar angles are congruent, then the triangles are similar. If ___________________________________ If ___________________________________ ___________________,