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24 Oct 2016 8:00 - 9:30 AM Geometry Proposed Agenda 1) Bulletin 2) Student Survey - confidential - be honest - no names 3) Properties of Parallelograms - a quadrilateral with two pairs of parallel sides a) Opposite sides b) Opposite angles c) Diagonals - measuring 4) Homework Date Properties of Parallelograms p 55 A parallelogram is a quadrilateral that contains 2 pairs of parallel sides. In previous units we used parallel lines to prove the following 2 things about parallelograms: Theorem: Opposite angles of a parallelogram are congruent Theorem: Consecutive angles of a parallelogram are supplementary. More about the Properties of Parallelograms Given: Quadrilateral ABCD is a parallelogram. Prove: Triangle ABC is congruent to triangle CDB. Draw the diagonal connecting B and D. Draw in what we A B know. D C More about the Properties of Parallelograms Given: Quadrilateral ABCD is a parallelogram. Prove: Triangle ABC is congruent to triangle CDB. Draw the diagonal connecting B and D. Draw in what we A B * know. D * C More about the Properties of Parallelograms Given: Quadrilateral ABCD is a parallelogram. Prove: Triangle ABC is congruent to triangle CDB. Draw the diagonal connecting B and D. Draw in what we A B * know. D * C More about the Properties of Parallelograms Given: Quadrilateral ABCD is a parallelogram. Prove: Triangle ABC is congruent to triangle CDB. Draw the diagonal connecting B and D. Draw in what we A B * # know. # D * C Statement Reason 1. ABCD is a parallelogram. 1. Given 2. AB ||DC ; AD || BC 2. Definition of a parallelogram 3. There is a line BC 3. Through any 2 points there is exactly one line Statement Reason 4. ∠ABD =∠CDB: ∠ADB =∠CBD 4. If 2 parallel lines are cut by a transversal, then alternate interior angles are congruent. 5. BD = DB Statement Reason 4. ∠ABD =∠CDB: ∠ADB =∠CBD 4. If 2 parallel lines are cut by a transversal, then alternate interior angles are congruent. 5. DB = BD. 5. Reflexive property 6. △ABD = △CDB 6. ASA congruence postulate More about the Properties of Parallelograms Now we know △ABD = △CDB, using what we know about corresponding parts of congruent triangles write at least 2 things that would be true about every parallelogram. A B * # # D * C Theorem: Opposite sides of a parallelogram are congruent. Theorem: Opposite angles of a parallelogram are congruent. A B * # # D * C Name Date Parallelogram Diagonals: Data Collection For each parallelogram on the worksheet, measure the length of the diagonals and line segments defined in the table below: We will do the first one together Measurements B C Parallelogram AC 1 5.3 2 M 3 A D 4 5 6 7 8 9 10 BD AM (cm) MC DM MB Patterns? Measurements (cm) B C M A Parallelogram AC 1 5.3 2 3 D 4 5 6 7 8 9 10 BD AM MC DM MB Patterns? Measurements (cm) B C M A Parallelogram AC BD 1 5.3 4.2 2 3 D 4 5 6 7 8 9 10 AM MC DM MB Patterns? Measurements (cm) B C Parallelogram AC BD 1 5.3 4.2 2 M 3 A D 4 5 6 7 8 9 10 AM MC DM MB Patterns? Measurements (cm) B C Parallelogram AC 1 5.3 2 M 3 A D 4 5 6 7 8 9 10 BD AM 4.2 2.65 MC DM MB Patterns? Measurements (cm) B C Parallelogram AC 1 5.3 2 M 3 A D 4 5 6 7 8 9 10 BD AM 4.2 2.65 MC DM MB Patterns? Measurements (cm) B C AC 1 5.3 2 M A Parallelogram D BD AM MC DM MB 4.2 2.65 2.65 2.1 2.1 Patterns? For all three? 3 4 If you see 5 exactly on a6 cm value - 7 show the 8 zero eg. 9 10 3.0cm Complete then make a conjecture about the diagonals of parallelograms (a statement that appears to be true based on limited information) Not formally proven to be true. What pattern/patterns did we observe? 1, 2, 3 diagonals bisect each other (cut each other in half) 4, 5, 6 diagonals bisect each other 7, 8, 9 diagonals bisect each other 10, 11, 12 diagonals bisect each other What pattern/patterns did we observe? 1, 2, 3 diagonals bisect each other 4, 5, 6 diagonals bisect each other 7, 8, 9 diagonals bisect each other Diagonals congruent diagonals bisect each other Diagonals congruent 10, 11, 12 What was your conjecture? Share on white board Parallelogram Diagonals: Formal Proof Given: Quadrilateral ACDE is a parallelogram. Prove: Segment EC bisects segment AD. Segment AD bisects segment EC. Draw the given. C A D E Parallelogram Diagonals: Formal Proof Given: Quadrilateral ACDE is a parallelogram. Prove: Segment EC bisects segment AD. Segment AD bisects segment EC. Draw the given. C A D E Homework Complete the Parallelogram Diagonals: Formal Proof page (blue sheet).