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Name: Unit 4: Triangles Period: Date: Notes Packet #2: Section 4.3/4.4/4.5: Triangle Congruence (PA) CRS CCSS Review: PPF 24-27 Use several angle properties to find an unknown angle measure G-CO. 8 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. EEI 20-23 Evaluate algebraic expressions by substituting integers for unknown Focus PPF: 28-32 Apply properties of 30°-60°-90°,45°-45°-90°, similar, and congruent triangles G-SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Extension: PPF: 28-32 Apply properties of30°-60°-90°,45°-45°-90°, similar, and congruent triangles PPF 33-36 Draw conclusions based on a set of conditions Solve multistep geometry problems that involve integrating concepts, planning, visualization, and/or making connections with other content areas Level 1 Objectives Name congruent triangles based on information provided (use CPCTC). Name and label corresponding congruent parts of congruent triangles. Set up and solve equations using congruent triangles. “Basic” Definition Congruent Triangles Definition of Congruent Triangles (CPCTC) Naming corresponding parts of triangles a) Name the congruent angles and sides for the following two triangles. Angles:________________ Sides:________________ ________________ ________________ ________________ ________________ Congruence Triangles: ▲ __________ ▲__________ 1 b) A ______ B ______ F ______ AC ______ DE ______ FE ______ c) ∆ABD ∆CBD. Name the congruent angles and sides Identify the congruent triangles in each figure and name the corresponding congruent parts: 1. 2. 2 3. If ▲ABC ▲DEC, name all corresponding sides and angles. Algebra Crossover: 1. ∆GHI ≅ ∆ RST, HI =14, GH = 12, GI = 10, and ST = 2x + 4. a. Draw and label a figure to show the congruent triangles. b. Find x 2. ∆ABC≅ ∆DEF, mA = 40°, mE = 60°, mF = (4x -20) a. Draw and label a figure to show the congruent triangles. b. Find x Homework Level 1 Page 195 #2-5,9-16,29-32 Congruence and Triangles Worksheet CPCTC Worksheet 3 Level 2 Objectives Recognize when triangles are congruent by Side-Side-Side and Side-Angle-Side postulates. Properties of Equality for Segments and Angles Segments Reflexive Property AB = AB Symmetric Property If AB = CD, then CD = AB Transitive Property If AB = CD and CD = EF, then AB = EF Angles Remember: A postulate is a statement that is accepted as true without proof Ex. Through any two points, there is exactly one line. A theorem is a statement that has been proven true. Ex. Linear Pair Theorem: Linear pairs are supplementary. Proving Triangles Congruent by SSS and SAS Directions: Determine which postulate, if any, can be used to prove that the two triangles are congruent. If the triangles cannot be shown to be congruent, write “cannot be determined.” 1. 2. 4. 5. 3. 6. 4 Directions: Complete each statement and tell which congruence shortcut you used to determine that the triangles are congruent. If the triangles cannot be shown to be congruent, write “cannot be determined.” 2) 3) ▲ABC ▲________ ▲EF G ▲________ Reason: _________ Reason: _________ ▲NPR ▲________ Reason: _________ Homework Level 2 SSS and SAS Congruence Worksheet 5 Level 3 Objectives Recognize when triangles are congruent by Angle-Side-Angle and Angle-Angle-Side postulates. Proving Triangles Congruent by ASA and AAS Directions: Determine which postulate, if any, can be used to prove that the two triangles are congruent. If the triangles cannot be shown to be congruent, write “cannot be determined.” 1) 2) 3) Directions: Complete each statement and tell which congruence shortcut you used to determine that the triangles are congruent. If the triangles cannot be shown to be congruent, write “cannot be determined.” 1) ▲STV ▲________ Reason: _________ 2) 3) ▲WXY ▲________ Reason: _________ ▲STU ▲________ Reason: _________ Homework Level 3 ASA and AAS Congruence Worksheet Triangle Congruence Postulate Worksheet 6 Level 4 Objectives Combine all congruence postulates including CPCTC to prove two triangles are congruent. Using the SSS, SAS, AAS, ASA and CPCTC 1) GIVEN: D is the midpoint of both GE and FH. PROVE: ▲FDE ▲HDG. What are the three congruent parts? Why are they congruent? ____________________________ because ____________________________________________ ____________________________ because ____________________________________________ ____________________________ because ____________________________________________ What is the reason ▲FDE ▲HDG? ________________ 2) GIVEN: BD bisects AC at D and AB CB. PROVE: ▲ABC ▲CBD What are the three congruent parts? Why are they congruent? ____________________________ because ____________________________________________ ____________________________ because ____________________________________________ ____________________________ because ____________________________________________ What is the reason ▲ABC ▲CBD? ________________ By proving that two triangles are congruent, you can deduce information about the other three parts. As you know a triangle has six basic parts and we have to get three parts of one equal to three parts of the other (the right three parts) in order for them to be congruent. This means we get the remaining three parts as a bonus because when figures are congruent all the parts of one are equal the corresponding parts to the other. The other three parts are equal for the reason: Corresponding Parts of Congruent Triangles are Congruent (CPCTC). 7 3) GIVEN: N is the midpoint of AB AX NY NX BY PROVE: X Y What are the three congruent parts? Why are they congruent? ____________________________ because ____________________________________________ ____________________________ because ____________________________________________ ____________________________ because ____________________________________________ ▲ _________ ▲ __________ because of __________ X Y because______________ 4) GIVEN: VB bisects EVO BV bisects EBO PROVE: BO What are the three congruent parts? Why are they congruent? ____________________________ because ____________________________________________ ____________________________ because ____________________________________________ ____________________________ because ____________________________________________ ▲ _________ ▲ __________ because of __________ BO because______________ Level 4 Homework Triangle Congruence Proof Worksheet 8