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Are There Congruence Shortcuts? Objectives: Explore shortcut methods for determining whether triangles are congruent Discover the SSS and SAS are valid congruence shortcuts but SSA is not Homework: 4.4 pg.224-225 # 4-9, 12-14, 23, 24 Triangle Inequalities Do now: Quiz 4.1- 4.3 A building contractor has just assembled two massive triangular trusses to support the roof of a recreation hall. Before the crane hoists them into place, the contractor needs to verify that the two triangular trusses are identical. Must the contractor measure and compare all six parts of both triangles? Wikipedia: In architecture a truss is a structure comprising one or more triangular units constructed with straight members whose ends are connected at joints referred to as nodes. http://www.youtube.com/watch?v=8hmPcxOBQA for SSA: http://www.mathopenref.com/congruentssa.ht ml Explain what the picture statement means. Create a picture statement to represent the SAS Triangle Congruence Conjecture. Explain what the picture statement means. In the third investigation you discovered that the SSA case is not a triangle congruence shortcut. Sketch a counterexample to show why. Which conjecture tells you that triangles are congruent? Y is a midpoint Closing the Lesson: The main points of this lesson are that SSS and SAS can be used to establish the congruence of triangles but SSA cannot. What is the reason why SSA fails? 4.4-4.5 Are There Congruence Shortcuts? Objectives: Explore shortcut methods for determining whether triangles are congruent Discover valid congruence shortcuts Homework: 4.5 pg.229 # 4, 6, 8, 10, 12, 14 Do Now: # 4-6,10(!) pg. 224 Do you need all six ? NO ! SSS SAS ASA AAS Side-Side-Side (SSS) 1. AB DE 2. BC EF 3. AC DF ABC DEF Side-Angle-Side (SAS) 1. AB DE 2. A D 3. AC DF ABC DEF included angle Included Angle The angle between two sides G I H Included Angle Name the included angle: E Y S YE and ES E ES and YS S YS and YE Y Angle-Side-Angle (ASA) 1. A D 2. AB DE ABC DEF 3. B E included side Included Side The side between two angles GI HI GH Included Side Name the included side: E Y S Y and E YE E and S ES S and Y SY Angle-Angle-Side (AAS) 1. A D 2. B E ABC DEF 3. BC EF Non-included side Warning: No SSA Conjecture There is no such thing as an SSA conjecture! E B F A C D NOT CONGRUENT Warning: No AAA Conjecture There is no such thing as an AAA conjecture! E B A C D NOT CONGRUENT F The Congruence Conjectures SSS correspondence ASA correspondence SAS correspondence AAS correspondence SSA correspondence AAA correspondence Name That Conjecture (when possible) SAS SSA ASA SSS Name That Conjecture (when possible) AAA ASA SAS SSA Name That Conjecture (when possible) Vertical Angle Reflexive Property SAS SAS Vertical Angles SAS Reflexive Property SSA Let’s Practice Indicate the additional information needed to enable us to apply the specified congruence postulate. For ASA: B D For SAS: AC FE For AAS: A F Name That Conjecture (when possible) 10. The perimeter of ABC is 180 m. Is ABC ADE? Which conjecture supports your conclusion? Name That Conjecture (when possible) Indicate the additional information needed to enable us to apply the specified congruence postulate. For ASA: For SAS: For AAS: Determine whether the triangles are congruent, and name the congruence shortcut. If the triangles cannot be shown to be congruent, write “cannot be determined.” Solve: pg.224-226 # 11-19 pg.229 # 6-9 Pg.230 # 13, 15, 18 Shortcut for showing triangle congruence allows us to avoid proving the congruence of all six pairs of corresponding parts. There are four shortcuts for proving triangle congruence. Students discovered SSS and SAS in the previous lesson. Now they have discovered ASA and SAA. SSA and AAA are not congruence shortcuts. Other arrangements of the letters, such as ASS (not a shortcut) and AAS (a shortcut), are included among these six. 4.6 Corresponding Parts of Congruent Triangles Objectives: Show that pairs of angles or pairs of sides are congruent by identifying related triangles Homework: lesson 4.6 pg.233 # 3, 4, 5, pg. 231 # 26 Do now: CPCTC-Corresponding Parts of Congruent Triangles are Congruent Paragraph proof Together: # 1, 2 pg 233 # 12 4.6 Corresponding Parts of Congruent Triangles Objectives: Show that pairs of angles or pairs of sides are congruent by identifying related triangles Homework: lesson 4.6 pg.233-234 # 6, 7, 9, 18 CPCTC-Corresponding Parts of Congruent Triangles are Congruent Do now: In Chapter 3, you used inductive reasoning to discover how to duplicate an angle using a compass and straightedge. Now you have the skills to explain why the construction works using deductive reasoning. The construction is shown at right.Write a paragraph proof explaining why it works. Practice: # 3 (from h/w) # 3 (from h/w) #5 (h/w) #6 Closure: To show that two segments or angles (the targets) are congruent, you will often find two congruent triangles in which these segments or angles are corresponding parts.