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 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 3. Calculate m 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 ______ ______, then ∆ABC ~ ∆RST . ______, then ∆ABC ~ ∆RST . Example 1 Determine which two of the three triangles are similar. Find the scale factor for the pair. State which theorem was used to support your answer. Similar Triangles ______ Scale factor ______ Theorem ______ Example 2 Are the triangles similar? If so, state the similarity and the postulate or theorem that justifies your answer. ______ Example 3 Find the values of x that makes ∆ABC ~ ∆DEF . x = ______ Example 4 A large tree has fallen against another tree and rests at an angle as shown in the figure. To estimate the length of the tree from the ground you make the measurements shown in the figure. What theorem or postulate can be used to show that the triangles in the figure are similar? ______ Explain how you can use similar triangles to estimate the length of the tree. Then estimate the length. ______ Example 5 Unit 8: Congruent and Similar Triangles Lesson 8.8 Use Proportionality Theorems Lesson 6.6 from textbook Objectives • Use proportionality theorems to calculate segments lengths and to determine parallel lines. • Apply proportions to solve problems involving missing lengths and angle measures in similar figures. Triangle Proportionality Triangle Proportionality Converse Theorem Theorem If a line parallel to one side of a triangle If a line divides two sides of a triangle intersects the other two sides, then it divides proportionally, then it is parallel to the third side the two sides proportionally RT RU If TU // QS , then ______If = , then ______TQ US Example 1 In the diagram, TU // QS , RS = 4, ST = 6, and QU = 9. What is the length of RQ ? RQ = ______ Example 2 Determine whether PS // QR . Explain. ______ Example 3 Use the figure to find the length of each segment. GF = ______FC = ______ ED = ______FE = ______ Example 4 Find the value of x. x = ______x = ______ Example 5 The figure is a diagram of a cross section of the attic of a house. A vent pipe comes through the floor 6 feet from the edge of the house. What is the distance x on the roof, from the edge of the roof to the vent pipe? ______ Example 6