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Geometry

9. Geometry. Ancient and Modern Mathematics Embrace. Lines, Angles, and Circles. 9.1. Understand the basic properties of geometric objects such as points, lines, and planes. Work with the fundamental properties of angles.

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Geometry

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  1. 9 Geometry Ancient and Modern Mathematics Embrace

  2. Lines, Angles, and Circles 9.1 • Understand the basic properties of geometric objects such as points, lines, and planes. • Work with the fundamental properties of angles. • Solve problems involving the relationship among angles, arcs, and circles.

  3. Points, Lines, and Planes Euclid: a pointis “that which has no part,” a linehas “length but no breadth,” and a planehas “length and breadth only.” A point on a line divides the line into three parts—the point and two half lines. A rayis a half line with its endpoint included. A piece of a line joining two points and including the points is called a line segment.

  4. Points, Lines, and Planes Parallel lineslie on the same plane and have no points in common. Intersecting lines lie on the same plane and have a single point in common.

  5. Two lines that intersect forming right angles are called perpendicular lines.

  6. Angles Two rays having a common endpoint form an angle. We measure angles in units called degrees. The symbol ° represents the word degrees.

  7. Angles An angle whose measure is between 0° and 90° is called an acute angle. A right anglehas a measure of 90°.

  8. Angles An obtuse anglehas a measure between 90° and 180°. A straight anglehas a measure of 180°.

  9. Find the measure of <F

  10. Vertical Angles are Congruent Two intersecting lines form two pairs of angles called vertical angles.

  11. Find the measure of h

  12. Angles A pair of angles is complementaryif the sum of their measures is 90°. Two angles having an angle sum of 180° are supplementaryangles. (Complementary Angles make a Corner) (Supplementary Angles make a Straight Line)

  13. Angles If we intersect a pair of parallel lines with a third line, called a transversal, we form eight angles.

  14. Angles

  15. If m1 = 100, what are the measures of 2 through 8? m2 = m3 = m4 = m5 = m6 = m7 = m8 = Congruent or Supplementary?

  16. Angles • Example: If lines l and mare parallel, find the measure of angle 9.

  17. Angles • Example: If lines l and mare parallel, find the measure of angle 9. • Solution: • (corresponding angles) • (straight angle)

  18. Angles • Example: If lines land mare parallel, find the measure of angle 2.

  19. Angles • Example: If lines land mare parallel, find the measure of angle 2. • Solution: • (same side interior angles)

  20. Circles

  21. Circles An angle that has its vertex at the center of a circle is called a central angle.

  22. Circles • Example: A circle has a circumference of 12 meters. If central angle ACB has measure of 120°, then what is the length of the arc from A to B? (continued on next slide)

  23. Circles • Solution:

  24. Circles • Example: Use geometry to estimate the circumference of Earth. • Solution: Assume that lines l and m are parallel and cut by the transversal t. The point C is the center of the circle. Therefore, angles αandβ are equal. (continued on next slide)

  25. Circles To measure the circumference of Earth, place a vertical pole in the ground and wait until noon when the rays of the Sun and the pole form an angle of 0°. Suppose at that very moment, a friend 1,000 miles away also has a similar vertical pole, and the Sun’s rays make an angle of 15° with his pole. (continued on next slide)

  26. Circles

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