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Bell Work for 11/2 Plot the Following Points on a Coordinate Plane: A(-4,6) B(-2,1) C(-3,-2) D(1,5) E(3,0) F(2,-3) Connect the points to form two triangles, ABC and DEF What are some similarities you notice between the two triangles? What are some differences? Looking at the points, how can we form triangle DEF using what we know about the points that form ABC? Congruent Triangles Geometry Unit 4 Objective for Today • Identify congruent polygons based on properties of correspondence. Congruency • What did we mean when we said two segments or angles were congruent? • What might we mean when we say polygons are congruent? Congruency • Congruent Figures have the same size and shape • If two figures can be manipulated in one of three ways to occupy the same exact space, then the figures are congruent • Three Ways to manipulate and keep congruent • Translation • Reflection • Rotation Translation Translation—Creating an image of a figure by plotting a duplicate of each point moved a certain number of points along the x and/or y axis Example: Translate the points A(5,6);B(-2,4); and C(3,-3): • Up Two • Left Three • Up Two and Left Three Reflection Reflection—Creating an image of a figure by plotting a duplicate of each point on the opposite side of a specific line, the same distance away from the line Example: Reflect the points A(5,6);B(-2,4); and C(3,-3): • Across the X-Axis • Across the Y-Axis Rotation Rotation—Creating an image of a figure by plotting a duplicate of each point rotated in a clockwise or counterclockwise direction a certain number of degrees around a certain point Example: Translate the points A(5,6);B(-2,4); and C(3,-3): • 90 degrees clockwise around the origin • 90 degrees counter clockwise around the origin • 180 degrees around the origin Correspondence • Each point created using these three methods is said to correspond with the point used to make it • Similarly, segments and angles can correspond too • If all corresponding parts of a figure are the same as the original, than the two are congruent Objective for Today • Identify congruent polygons based on properties of correspondence. Congruence Practice Practice Practice Proving Congruency Congruency Bell Work for 11/3 Are ABC and DEF congruent in this case? • We cannot know, we need 3 sides and 3 angles to prove congruency Proving Figures Congruent • What do we need in order to prove that two figures are congruent? Practice Bell Work for 11/5 Quiz Today Everything off of your desks except a pencil and scratch paper. SSS Postulate Bell Work for 11/9 • Would you use SSS or SAS to prove the following pairs of triangles congruent? If there is not enough information, write not enough information SSS Postulate Using SSS and SAS Using SSS and SAS Two New Triplets Using ASA Bell Work for 11/10 Recap of Triplets to Prove Triangles • SSS • SAS • ASA • AAS What is not enough to prove triangles congruent? • AAA • SSA Bell Work for 11/11 Bell Work for 11/13 CPCTC • Corresponding Parts of Congruent Triangles are Congruent • If we know that two triangles are congruent, we know all the pairs of corresponding parts are also congruent • CPCTC is simply the converse of the above statement • In order to use CPCTC, we must always prove triangles congruent first Bell Work for 11/16 Types of Triangles Isosceles Triangles • Isosceles Triangles are triangles that have two congruent sides • We call the congruent sides the legs and the other the base • Isosceles Triangles also have two congruent angles • The congruent angles are called the Base angles and the other is the Vertex angle Using Isosceles Triangles Using Isosceles Triangles Equilateral Triangles Bell Work for 11/17 Practice Practice • Given the Angle Measures, fill in the all the other remaining angle measures Find the value of each variable Bell Work for 11/18 Types of triangles • By Angles Right Triangles Right Triangles • Properties of Right Triangles Practice Bell Work for 11/19 Bell Work for 11/30 • Write Which Postulate or Theorem you would use to prove each pair of triangles congruent. If there is not enough information, state so. Steps for All Triangle Proofs 1. Identify the corresponding parts of the triangles 2. Find which ones you can prove are congruent using the ways we have learned • • • • Midpoints and Bisectors Parallel lines and transversals Vertical Angles Reflexive Property 3. Once we have congruent parts, figure out which triangle congruence theorem to use • SSS, SAS, ASA, AAS, HL (These are the ONLY ones you can use) 4. Optional: After step 3, use CPCTC to find other congruent parts Bell Work for 12/1 • Copy down the figure to the right. • How many triangles can you find in the figure? Overlapping Triangles • Overlapping triangles are triangles that share a common part • Either a side or an angle • Common parts can help us prove congruence because they provide a part that we know is congruent • What property tells us something is congruent to itself? Common Parts • To identify common parts, draw each pair of overlapping triangles as the two individual triangles and see what is the same in both of them • Whatever side or angle shares the EXACT same letters is the common part Finding Common Parts Finding Common Parts Finding Common Parts Finding Common Parts Bell Work for 12/2 Find the common parts for each pair of triangles Using Congruent Triangles • Because overlapping triangles contain multiple triangles, we can use the information we find about certain ones to prove things about other ones in the figure • We can use CPCTC and the triangle congruence Theorems to do this Bell Work for 12/3 • Name a pair of overlapping triangles in each figure, and state which postulate we can use to prove them congruent