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26 Sept 2016 8:00 - 9:30 AM Geometry Proposed Agenda 1) Bulletin - (remember you can always go to Clarkmagnet.net to read the night before school) 2) G-CO Symmetries of a Quadrilateral 3) Rules of Transformations 4) SAS Postulate 5) ASA Postulate 6) Homework G-CO Symmetries of a Quadrilateral I Suppose ABCD is a quadrilateral for which there is exactly one rotation, through an angle larger than 0 degrees and less than 360 degrees, which maps it to itself. Further, no reflections map ABCD to itself. What shape is ABCD? Compare what you have to your neighbor. Be ready to share your results - with reasons. Quadrilaterals: Angles angles Quadrilaterals: 4 sides of equal length 2 pairs parallel sides Opposite angles equal, but not 90 degrees 2 reflection symmetries 1 rotation - 180 degrees G-CO Symmetries of a Quadrilateral I ABCD is a quadrilateral one rotation maps to itself No reflections G-CO Symmetries of a Quadrilateral I ABCD is a quadrilateral one rotation maps to itself (180degree) No reflections - no other symmetry for shape G-CO Symmetries of a Quadrilateral I ABCD is a parallelogram which is neither a rectangle nor a rhombus. G-CO Symmetries of a Quadrilateral II There is exactly one reflection and no rotation that sends the convex quadrilateral ABCD onto itself. What shape(s) could quadrilateral ABCD be? Explain. Compare what you have to your neighbor. Be ready to share your results - with reasons. Convex quadrilateral? Regular trapezoid Line of symmetry kite Line of symmetry Rules of Transformations In previous sections we have been working with three major transformations: Rules of Transformations In previous sections we have been working with three major transformations: translations, reflections and rotations. Rules of Transformations These three transformations have one major thing in common. These three transformations have one major thing in common. They will ALWAYS produce congruent figures. These are often called rigid motion transformations. Rigid Motion Refers to any transformation that ALWAYS maintains congruent shapes. Translations, Reflections and Rotations will ALWAYS produce congruent figures. So, these three are considered rigid motions. To understand this a little better, let’s look at the two triangles below. For this, we will assume that there is a translation that exists the will map ABC onto A’B’C’. A B C A’ B’ C’ What can we say about these 2 congruent triangles? A B C A’ B’ C’ Postulates A postulate is something that is assumed to be true based on reasoning or observations. For example a fundamental geometry postulate is written below: “Between two points, there is exactly one line.” “Between two points, there is exactly one line.” We don’t need to get expert advice to prove this is true. We can all agree that there is only one straight line that exists between two points. Because it is something we can all agree on, it is known as a postulate. SAS Postulate To increase our confidence in postulates we will use in the future and to save time then, we are going to use rigid motions here to see how the SAS postulate works. SAS Postulate If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent. Let’s start with the given information of the postulate. AB = DE BC = EF and ∠ABC = ∠DEF. Mark that on the figures. AB = DE BC = EF and ∠ABC = ∠DEF. D A F C B E 1. Translate triangle ABc so that point B maps to point E. Draw below. Mark the map of A as A’ and the map of C as C’. C D A’ A C’ B F E 1. Translate triangle ABc so that point B maps to point E. Draw below. Mark the map of A as A’ and the map of C as C’. D A’ C’ F E 2. Rotate triangle A’EC’ with center E so that A’ maps onto D. Draw that below. Mark the map of C’ as C”. D A’ C’ F E 2. Rotate triangle A’EC’ with center E so that A’ maps onto D. Draw that below. Mark the map of C’ as C”. C’’ D F C’ E 2. Rotate triangle A’EC’ with center E so that A’ maps onto D. Draw that below. Mark the map of C’ as C”. C’’ D F E 3. For the transformation in 2, how can you be certain that a rotation with center E would map A’ exactly onto D? C’’ D F E 3. For the transformation in 2, how can you be certain that a rotation with center E would map A’ exactly onto D? Line segment AB is the same length as line segment ED (congruent). C’’ We have only used D rigid motions so we have not changed any F lengths. They will fit on top of each other. E On the previous page, you should have ended up with a figure that looks like the one below. Mark the corresponding parts of the two triangles that are congruent based on our given information and the fact that the two transformations we did C’’ D will maintain congruence. F E In order to finish our “proof” investigation of the SAS postulate, we will do a reflection through line segment ED. If the two triangles are in fact congruent the C” should be able to map directly onto F because of this reflection. C’’ D F E How do you know that a reflection through DE exists that maps C” onto F? Explain below. D F E How do you know that a reflection through DE exists that maps C” onto F? Explain below. We are reflecting across DE, A”B” will not move, it will still be on DE, so that part of the triangle stays in position. D F E Also angle ABC is the same as (congruent to) angle DEF so the two edges of the angles will map onto each other, so then B”C” will be on top of EF which will also mean that C” had D mapped onto F. All three parts of triangle F ABC have mapped onto Triangle DEF. E As we were told, the triangles are indeed congruent. SAS 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 triangles are congruent. ASA Postulate ASA If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent. To do now: Optional: Complete diagram and explain how to show the two triangles are congruent. (Complete page 1C-12). Complete the homework problems 1- 6 on pages 1C-13 and 14. Draw more diagrams in the space at the bottom of the page and mark in the congruent parts CAREFULLY.