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Transcript
Points Points do not have actual size. How to Sketch: A Using dots B C A How to label: Use capital letters Never name two points with the same letter (in the same sketch). Lesson 1-1 Point, Line, Plane 1 Lines Lines extend indefinitely and have no thickness or width. How to sketch : using arrows at both ends. n A B C How to name: 2 ways (1) small script letter – line n (2) any two points on the line - AB , BC, AC , BA, CA, CB Never name a line using three points - ABC Lesson 1-1 Point, Line, Plane 2 Collinear Points Collinear points are points that lie on the same line. (The line does not have to be visible.) A point lies on the line if the coordinates of the point satisfy the equation of the line. Ex: To find if A (1, 0) is collinear with A B C the points on the line y = -3x + 3. Substitute x = 1 and y = 0 in the equation. Collinear 0 = -3 (1) + 3 C 0 = -3 + 3 A 0=0 B The point A satisfies the equation, therefore the point is collinear Non collinear with the points on the line. Lesson 1-1 Point, Line, Plane 3 Postulate 1-5 Ruler Postulate The distance between any two points is the absolute value of the difference of the corresponding numbers (on a number line or ruler) Congruent Segments: two segments with the same length Midpoint: a point that divides the segment into two equal parts A B B is the midpoint, so AB = BC C Postulate 1-6 Segment Addition Postulate If three points A, B, and C are collinear and B is between A and C, then AB + BC = AC C B A Using the Segment Addition Postulate If DT = 60, find the value of x. Then find DS and ST. 2x - 8 D 3x - 12 S T Using the Segment Addition Postulate If EG = 100, find the value of x. Then find EF and FG. 4x - 20 E 2x + 30 F G Finding Lengths C is the midpoint of AB. Find AC, CB, and AB. 3x – 4 2x + 1 A C B Intersection of Figures The intersection of two figures is the set of points that are common in both figures. The intersection of two lines is a point. m Line m and line n intersect at point P. P n Lesson 1-1 Point, Line, Plane Continued……. 10 Planes A plane is a flat surface that extends indefinitely in all directions. How to sketch: Use a parallelogram (four sided figure) How to name: 2 ways (1) Capital script letter – Plane M (2) Any 3 non collinear points in the plane - Plane: ABC/ ACB / BAC / BCA / CAB / CBA A B M C Horizontal Plane Vertical Plane Lesson 1-1 Point, Line, Plane Other 11 3 Possibilities of Intersection of a Line and a Plane (1) Line passes through plane – intersection is a point. (2) Line lies on the plane - intersection is a line. (3) Line is parallel to the plane - no common points. Lesson 1-1 Point, Line, Plane 12 Throughout New York City there are free movie nights in the park. The movie screen in the middle of the park is an example of a plane (2D) in space (3D). Lesson 1-1 Point, Line, Plane 13 Intersection of Two Planes is a Line. B P A R Plane P and Plane R intersect at the line AB Lesson 1-1 Point, Line, Plane 14 Different planes in a figure: A D B C E H Plane EFGH F G Plane ABCD Plane BCGF Plane ADHE Plane ABFE Plane CDHG Etc. Lesson 1-1 Point, Line, Plane 15 Other planes in the same figure: Any three non collinear points determine a plane! A D B Plane AFGD Plane ACGE C Plane ACH E H F G Plane AGF Plane BDG Etc. Lesson 1-1 Point, Line, Plane 16 Coplanar Objects Coplanar objects (points, lines, etc.) are objects that lie on the same plane. The plane does not have to be visible. A D B C E H F G Are the following points coplanar? A, B, C ? A, B, C, F ? H, G, F, E ? E, H, C, B ? A, G, F ? C, B, F, H ? Lesson 1-1 Point, Line, Plane Yes No Yes Yes Yes No 17 Regents Questions on Planes Go to jmap.org -> Resources by Topics>Geometry->Planes Groups assigned G.G.1 through G.G.9 to complete and present whole class Lesson 1-1 Point, Line, Plane 18 Angle Relationships Adjacent angles are “side by side” and share a common ray. 45º 15º These are examples of adjacent angles. 80º 45º 35º 55º 85º 20º 130 º 50º These angles are NOT adjacent. 100 º 50º 35º 35º 55º 45º Complementary Angles sum to 90° 50 40° ° Complementary angles add up to 90º. 30º 40º 60º Adjacent and Complementary Angles 50º Complementary Angles but not Adjacent Supplementary Angles sum to 180° 30° 150 ° Supplementary angles add up to 180º. 40º 120 º 60º Adjacent and Supplementary Angles 140 º Supplementary Angles but not Adjacent Vertical Angles are opposite one another. Vertical angles are congruent. 100 ° 100 ° Vertical Angles are opposite one another. Vertical angles are congruent. 80 ° 80° Lines l and m are parallel. l||m Note the 4 angles that measure 120°. 120 ° 120 ° Line n is a transversal. 120 ° 120 ° n l m Lines l and m are parallel. l||m Note the 4 angles that measure 60°. 60 ° 60 ° 60 ° l m 60 ° Line n is a transversal. n Lines l and m are parallel. l||m There are many pairs of angles that are supplementary. Line n is a transversal. There are 4 pairs of angles that are vertical. 60 ° 120 ° 60 ° 120 ° 120 ° 60 ° 120 ° 60 ° n l m If two lines are intersected by a transversal and any of the angle pairs shown below are congruent, then the lines are parallel. This fact is used in the construction of parallel lines. Let’s Practice 1201 2 60 ° ° 60 3 4 120 ° 120 5 660° °7 8 ° 120 60 ° ° m<1=120° Find all the remaining angle measures. Practice Time! 1) Find the missing angle. ?° 36° 1) Find the missing angle. ?° 36° 2) Find the missing angle. ?° 64° 2) Find the missing angle. ?° 64° 3) Solve for x. 2x 3x° ° 3) Solve for x. 2x 3x° ° 4) Solve for x. 2x + 5 4) Solve for x. 2x + 5 5) Find the missing angle. ?° 168° 5) Find the missing angle. ?° 168° 6) Find the missing angle. 58° ?° 6) Find the missing angle. 58° ?° 7) Solve for x. 4x 5x 7) Solve for x. 4x 5x 8) Solve for x. 2x + 10 3x + 20 8) Solve for x. 2x + 10 3x + 20 9) Lines l and m are parallel. l||m Find the missing angles. 42 ° c° a ° b° d° e° g° f° l m 10) Lines l and m are parallel. l||m Find the missing angles. 81 ° c° a ° b° d° e° g° f° l m In the figure a || b. 13. Name the angles congruent to 3. 14. Name all the angles supplementary to 6. 15. If m1 = 105° what is m3? 16. If m5 = 120° what is m2? Find the missing angles. 70 ° Hint: The 3 angles in a triangle sum to 180°. d ° 70 b° ° 65 ° Find the missing angles. 70 ° Hint: The 3 angles in a triangle sum to 180°. 75 ° 40 ° 70 ° 65 ° Find the missing angles. 45 ° Hint: The 3 angles in a triangle sum to 180°. d ° 50 b° ° 75 ° Find the missing angles. 45 ° Hint: The 3 angles in a triangle sum to 180°. 20 ° 85 ° 50 ° 75 ° Another practice problem 40 ° 120 ° Find all the missing angle measures, and name the postulate or theorem that gives us permission to make our statements. Ex. : Interior Angles of a Quadrilateral P 80° Find the measure of each angle in the quadrilateral? 70° x° Q 2x° R S Find the measure of the missing angle in the figure below 135 100 70 quadrilateral 135 + 100 + 70 + x = 360 305 + x = 360 -305 -305 x x = 55 The End