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28 Oct 2015 9:50 - 11:20 Geometry Agenda 1) Properties of Parallelograms (p.25 your worksheets) 2) G-CO Is this a parallelogram? Group task. 3) Homework - revised 4) Don’t forget the Geometry Dictionary project - 4 pictures per week was the plan - catch up you have a long weekend Objectives Individual : Students more responsible for learning correcting - reflecting - teaching - editing Academic: Properties of parallelograms Properties of Parallelograms (p.25) A parallelogram is a quadrilateral that contains two pairs of parallel sides. In previous units we used parallel lines to prove the following two things about parallelograms: Theorem: Opposite angles of a parallelogram are congruent Theorem: Consecutive angles of a parallelogram are supplementary Do a sketch in space below box cont. In this unit, we are going to use what we know about the properties of congruent triangles to find some more properties of parallelograms. Let’s look at the parallelogram below. A D B C cont. Like we did before, draw a diagonal of the parallelogram connecting B and D, in other words draw line segment BD. A D B C Now use the space below to prove triangle ABD is congruent to triangle CDB Statement Reason 1. ABCD is a parallelogram 1. Given 2. Line segment AB is 2. Definition of a parallel to line segment DC; parallelogram line segment AD is parallel to line segment BC 3. There is a line BD 3. Through any 2 points there is exactly one line Now use the space below to prove triangle ABD is congruent to triangle CDB Statement 1. ABCD is a parallelogram Reason 1. Given 2. Line segment AB is parallel to line segment DC; line segment AD is parallel to line segment BC 2. Definition of a parallelogram 3. There is a line BD 3. Through any 2 points there is exactly one line 4. Line segment BD is congruent to line segment DB 4. Now use the space below to prove triangle ABD is congruent to triangle CDB Statement 1. ABCD is a parallelogram Reason 1. Given 2. Line segment AB is parallel to line segment DC; line segment AD is parallel to line segment BC 2. Definition of a parallelogram 3. There is a line BD 3. Through any 2 points there is exactly one line 4. Line segment BD is congruent to line segment DB 4. Reflexive property Now use the space below to prove triangle ABD is congruent to triangle CDB Statement 1. ABCD is a parallelogram Reason 1. Given 2. Line segment AB is parallel to line segment DC; line segment AD is parallel to line segment BC 2. Definition of a parallelogram 3. There is a line BD 3. Through any 2 points there is exactly one line 4. Line segment BD is congruent to line segment DB 4. Reflexive property 5. Angle ABD is congruent 5. For 2 parallel lines cut by to angle CDB; angle ADB is a transversal alternate congruent to CBD interior angles are congruent. Now use the space below to prove triangle ABD is congruent to triangle CDB Statement 1. Reason ABCD is a parallelogram 1. Given 2. Line segment AB is parallel to line segment DC; line segment AD is parallel to line segment BC 2. Definition of a parallelogram 3. There is a line BD 3. Through any 2 points there is exactly one line 4. Line segment BD is congruent to line segment DB 4. Reflexive property 5. Angle ABD is congruent to angle CDB; angle ABD is congruent to CDB 5. For 2 parallel lines cut by a transversal alternate interior angles are congruent. 6. Triangle ABC is congruent to triangle CDB \ ASA Two angles and the included side of one triangle are congruent to two angles and the included side of the other triangle Now use the space below to prove triangle ABD is congruent to triangle CDB Statement 1. ABCD is a parallelogram Reason 1. Given 2. Line segment AB is parallel to line segment DC; line segment AD is parallel to line segment BC 2. Definition of a parallelogram 3. There is a line BD 3. Through any 2 points there is exactly one line 4. Line segment BD is congruent to line segment DB 4. Reflexive property 5. Angle ABD is congruent to angle CDB; angle ABD is congruent to CDB 5. For 2 parallel lines cut by a transversal alternate interior angles are congruent. 6. Triangle ABD is congruent to triangle CDB 6. Angle-Side-Angle Now that you know that triangle ABD is congruent to triangle CDB, use what you know about corresponding parts of congruent triangles (that they are congruent) to write at least two things that would be true about every parallelogram. angle BAD is congruent to angle DCB line segment AD and CD are congruent; line segments DC and BA are congruent Theorem: Opposite angles of a parallelogram are congruent. Theorem: Opposite sides of a parallelogram are congruent. Page 2 Theorem: Diagonals of a parallelogram bisect each other. Going to prove this. Below is a parallelogram with two diagonals that intersect at point M. Use the space below to give a 2 column proof of how you know that triangle WMZ is congruent to triangle YMX. You may use the definition of a parallelogram and the theorems we did on the previous page. 6 mins - I think a 4 or 5 line proof is possible Statement 1. Quadrilateral WXYZ is a parallelogram Reason 1. Given 2. line segment WZ || line segment YX and line segement WX|| line segment ZY 2. Definition of a parallelogram 3. Angle ZWM is congruent to angle XYM 3. If 2 parallel lines are cut by a transversal, then alternate interior angles are congruent. 4. line segment WZ is congruent to line segment XY; line segment WX is congruent to line segment ZY 4. Definition of a parallelogram 5. Angle WMZ is congruent to angle YMX 5. Vertical angles are congruent Now that we know that triangle WMZ is congruent to triangle YMX, what can we say about line segment MX and line segment MZ? How do we know this? Now that we know that triangle WMZ is congruent to triangle YMX, what can we say about line segment MX and line segment MZ? They are congruent. How do we know this? Corresponding parts of congruent triangles are congruent. Since we have shown that MX = MZ, we know that M is the midpoint of line segment XZ. Is there any reason that the reasoning that we used above could not be used for triangle WMZ and triangle YMX (not YMZ). As long as we are looking at the diagram we have here where XMZ are colinear we can use this reasoning. Since we can say that M is the midpoint of XZ and WY, we have proved the following: Theorem: Diagonals of a parallelogram bisect each other. p. 334 textbook for further reference Homework - Due Monday Problems Adjusted from what I said on Monday. 4-4 Proving Congruence SSS, SAS Exercises starting on page 230 7, 8, 14 - 18, 22 and Extra Credit 23. Do a two column proof for question 14, please.