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Warm-up 1/12/12

Warm-up 1/12/12. Given the m A = 24, m B = 17, find m C. 2. The measure of a base angle of an isosceles triangle is 30. What is the measure of the vertex angle?. 180 – 24 – 17 = 139. 120. 180 – 30(2) = 120. 30. 30. Points of concurrency #2.

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Warm-up 1/12/12

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  1. Warm-up 1/12/12 • Given the m A = 24, m B = 17, find m C. • 2. The measure of a base angle of an isosceles triangle is 30. What is the measure of the vertex angle? 180 – 24 – 17 = 139 120 180 – 30(2) = 120 30 30

  2. Points of concurrency #2 • (1). Centroid or orthocenter or incenter or circumcenter • (2). Orthocenter • (3). Centroid • (4). Circumcenter • (5). Incenter • Orthocenter • Circumcenter • Orthocenter

  3. Essential Questions (1). How do we use proofs to show that triangles are congruent? (2). What is the difference in a theorem and a postulate? Congruence of Triangles

  4. Proof : a logical argument that shows a statement is true Two-column proof : a proof that has number statements and corresponding reasons that show an argument in a logical order Theorem: a statement that can be proven Postulate: a rule that is accepted without proof

  5. Writing a proof is part of logic and reasoning skills. To write a two-column proof, you may need definitions, properties, postulates, or other theorems that have already been proven.

  6. Algebraic Properties of Equality: Examples 1. Addition Property If x = 7 then x + 4 = 11 • Subtraction Property If x = 10 then x - 4 = 6 3. Multiplication Property If x = 2 then 3x = 6 4. Division Property If 2x =10 then x = 5 • Substitution Property If x=3 and x+y=7 then3+y=7

  7. Algebraic Properties of Equality: Examples 6. Reflexive Property 5 = 5 7. Symmetry Property If 4 = 3 + 1, then 3 + 1 = 4 8. Transitive Property If x=2 and 2 = 3 – 1, thenx = 3 – 1.

  8. Below is a two column algebraic proof, justify each step: Given: Prove: 3 = x Example 1: given multiplication property substitution/mult. Inverse subtraction property division property symmetric property

  9. Example 2: Given: 2(A + C) = 4A Prove: C = A Given Distributive Property Subtraction Prop. of Equality Division Prop. of Equality

  10. Congruent Figures are figures that have the same size and same shape Patty Paper Activity: Trace ∆ABC on your patty paper labeling each vertex. Now, slide the paper on top of ∆DEF so that ∡A matches with ∡D. Do the triangles match perfectly? YES Since ∆ABC and ∆DEF are exactly the same size and the same shape, we say that they are congruent and we write ∆ABC ≅ ∆DEF Note: the order in which the vertices are named is IMPORTANT

  11. The edges and angles that correspond when this translation occurs are called the corresponding parts of two triangles. Corresponding Angles: Corresponding Edges: D DE E EF F DF

  12. Note: Corresponding Parts of Congruent Triangles are Congruent!!! We abbreviate this In the diagram above, if m A = 50°, which angle must also measure 50°? In the diagram above, if BC = 8 in., which edge must also measure 8 in.? CPCTC Angle D EF

  13. If a diagram is not given, it is easy to determine the corresponding parts of congruent triangles. Note the order in which the vertices are named in each triangle. Try these: (3). If ∆RST ≅ ∆GIY, then (4). If ∆XYZ ≅∆PQB, then (5). If ∆ABH ≅∆LKC, then

  14. The pairs of triangles pictured below are congruent. Name the congruent triangles: (Use patty paper if needed) ABC HKG (6). ∆ ________  ∆ _________ A H B C G K

  15. NEF RXT (7). ∆ ________  ∆ _________ E F R N T X (8). ∆ ________  ∆ _________ WVK YQI W I Q V Y K

  16. Often two pairs of triangle will have a common edge (a segment that is common to both triangles). Identify the common edge in each picture below: (9). _______ (10). _______ (11). _______ ST NC AR A A T N P S Q C W M D R Note: In example #10, the two triangles are ______ congruent not

  17. In a triangle, the angle formed by two edges is called the included angle for the two edges. Identify the included angle for each pair of edges. (12). AB and BC (13). CD and DE (14). RS and ST Included angle: ____ Included angle: ____ Included angle: ___ B D S D A S R C E C B T

  18. To show that two triangles are congruent, you do not need to show all corresponding parts are congruent. Here are two postulates that can be used to prove that triangles are congruent: Side-Side-Side Postulate (SSS): If three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent. Side-Angle-Side Postulate (SAS): If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.

  19. Based only on the markings, determine why the pairs of triangles are congruent: (15). _________ (16). ________ SSS SAS

  20. Based only on the markings, determine why the pairs of triangles are congruent: SAS (17). _________ (18). ________ SAS SSS SAS (19). _________ (20). ________

  21. B (21) Given: ; Prove:  ABM CBM A C M Given Given Reflexive Property SSS  ABM CBM Flow Chart for Problem #21: ABM CBM MB  MB SSS Reflexive Given

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