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Chapter 4

Chapter 4. Jen Black and Kylie Suitum. Lesson 1: Coordinates and Distance. Origin: starting point of the coordinate plane Coordinates: numbers assigned to points on a coordinate plane. Two are used to locate a point on a plane. ( x -coordinate, y -coordinate)

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Chapter 4

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  1. Chapter 4 Jen Black and Kylie Suitum

  2. Lesson 1: Coordinates and Distance • Origin: starting point of the coordinate plane • Coordinates: numbers assigned to points on a coordinate plane. Two are used to locate a point on a plane. (x-coordinate, y-coordinate) • One-dimensional coordinate system:

  3. Two-dimensional coordinate system: Axes: 2 perpendicular lines used to find the coordinates X-axis: horizontal axis Y-axis: vertical axis

  4. Quadrants: 4 regions in the plane separated by the axes

  5. The Distance Formula • The distance between point P (x1, y1) and point Q (x2, y2) is

  6. Using the Distance Formula • What is the distance between (-2, -3) and (-4, 4)

  7. Lesson 2: Polygons and Congruence • Polygon: a connected set of at least three line segments in the same plane such that each segment intersects exactly two others, one at each endpoint. • Congruence: two triangles are congruent iff there is a correspondence between their vertices such that all of their corresponding sides and angles are equal. **Corollary: two triangles congruent to a third triangle are congruent to each other.

  8. Polygons Polygons have 3 or more line segments that are connected and form a closed area. If there are rounded edges on a shape, it is NOT a polygon.

  9. Lesson 3: ASA and SAS Congruence • The ASA Postulate: • If two angles and the included side of one triangle are equal to two angles and the included side of another triangle, the triangles are congruent.

  10. **when the images are being marked, the same markings (like the same number of lines on a side) mean those marked as such are equal • The SAS Postulate: • If two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, the triangles are congruent.

  11. Lesson 4: Congruence Proofs • Corresponding parts of congruent triangles are equal. (CPCTE)

  12. Given: AC=XZ; <ACB=<XZY; CB=ZY Prove: <CBA=<ZYX

  13. Lab: Proving Triangles Congruent • Triangles are congruent iff all of the corresponding sides and angles are equal. • In this lab, we determined which combination of sides and angles can prove two triangles are congruent. • ASA, SSS, AAS, and SAS all work. • AAA and SSA do not prove congruence but prove similarity.

  14. Lesson 5: Isosceles and Equilateral Triangles A triangle is: • Scaleneiff it has no equal sides • Isoscelesiff it has at least two equal sides • Equilateraliff all of its sides are equal Scalene triangle Isosceles triangle Equilateral triangle

  15. A triangle is: • Obtuseiff it has an obtuse angle (greater than 90˚, less than 180˚) • Rightiff it has a right angle (90˚) • Acuteiff all of its angles are acute (less than 90˚, greater than 0˚) • Equiangulariff all of its angles are equal (60˚) Acute Right Obtuse Equiangular

  16. Theorem 9: If two sides of a triangle are equal, the angles opposite them are equal. Given: In ΔBCD, BD = CD Prove: <C = <B Statements: • In ΔBCD, BD = CD • <D = <D • CD = BD • ΔBCD ΔCBD • <C = <B Reasons: • Given • Reflexive • Given • SAS • CPCTE D D C B C B

  17. Theorem 10: If two angles of a triangle are equal, the sides opposite them are equal Corollaries to Theorems 9 and 10: An equilateral triangle is equiangular An equiangular triangle is equilateral

  18. Lesson 6 : SSS Congruence Theorem 11: The SSS Theorem If three sides of one triangle are equal to the three sides of another triangle, the triangles are congruent. A E D F C B ΔABC ΔDEF

  19. Lesson 7: Constructions Construction 3:To copy a line segment Let AB be the given line segment. Set the radius of the compass to the length of AB by putting the metal point on A and the pencil point on B. Draw a line (l ) and mark as point C on it. With C as center, draw an arc of radius AB that intersects the line. (D in the figure below) CD = AB l A B C D AB = CD

  20. Construction 4: To copy an angle Let <A be the angle to be copied. Draw a ray BC as one side of the angle to be drawn. Draw an arc centered at A that intersects both of its sides. Do the same for <B with the same radius as <A . Then, measure from E to D with a compass and use that same radius to measure from F where you draw another arc. Lastly, draw BG <A = <B E F A D B C G F B C

  21. B F Construction 5: to copy a triangle l A C D E • Copy AC onto line l • Measure from A to B and use that radius to draw an arc centered at D • Measure from C to B and use that radius to draw an arc centered at E • Lastly, draw DF and EF ΔABC ΔAFE by SSS

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