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Lesson 2

Lesson 2. Line Segments and Angles. Measuring Line Segments. The instrument used to measure a line segment is a scaled straightedge like a ruler or meter stick. Units used for the length of a line segment include inches (in), feet (ft), centimeters (cm), and meters (m).

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Lesson 2

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  1. Lesson 2 Line Segments and Angles

  2. Measuring Line Segments • The instrument used to measure a line segment is a scaled straightedge like a ruler or meter stick. • Units used for the length of a line segment include inches (in), feet (ft), centimeters (cm), and meters (m). • We usually place the “zero point” of the ruler at one endpoint and read off the measurement at the other endpoint.

  3. Rulers

  4. We denote the length of by • So, if the line segment below measures 5 inches, then we write • We never write B A

  5. Congruent Line Segments • In geometry, two figures are said to be congruent if one can be placed exactly on top of the other for a perfect match. The symbol for congruence is • Two line segments are congruent if and only if they have the same length. • So, • The two line segments below are congruent.

  6. Segment Addition • If three points A, B, and C all lie on the same line, we call the points collinear. • If A, B, and C are collinear and B is between A and C, we write A-B-C. • If A-B-C, then AB+BC=AC. This is known as segment addition and is illustrated in the figure below. A B C

  7. Example R • In the figure, suppose RS = 7 and RT = 10. What is ST? • We know that RS + ST = RT. • So, subtracting RS from both sides gives: S T

  8. Midpoints C • Consider on the right. • The midpoint of this segment is a point M such that CM = MD. • M is a good letter to use for a midpoint, but any letter can be used. M D

  9. Example A • In the figure, it is given that B is the midpoint of and D is the midpoint of • It is also given that AC = 13 and DE = 4.5. Find BD. • Note that BC is half of AC. So, BC = 0.5(13) = 6.5. • Note that CD equals DE. So, CD = 4.5. • Using segment addition, we find that BD = BC + CD = 6.5 + 4.5 = 11. B C D E

  10. Example P • In the figure T is the midpoint of • If PT = 2(x – 5) and TQ = 5x – 28, then find PQ. • We set PT and TQ equal and solve for x: T Q

  11. Example continued

  12. Measuring Angles • Angles are measured using a protractor, which looks like a half-circle with markings around its edges. • Angles are measured in units called degrees (sometimes minutes and seconds are used too). • 45 degrees, for example, is symbolized like this: • Every angle measures more than 0 degrees and less than or equal to 180 degrees.

  13. A Protractor

  14. The smaller the opening between the two sides of an angle, the smaller the angle measurement. • The largest angle measurement (180 degrees) occurs when the two sides of the angle are pointing in opposite directions. • To denote the measure of an angle we write an “m” in front of the symbol for the angle.

  15. Here are some common angles and their measurements. 1 2 4 3

  16. Congruent Angles • Remember: two geometric figures are congruent if one can be placed exactly on top of the other for a perfect match. • So, two angles are congruent if and only if they have the same measure. • So, • The angles below are congruent.

  17. Types of Angles • An acute angle is an angle that measures less than 90 degrees. • A right angle is an angle that measures exactly 90 degrees. • An obtuse angle is an angle that measures more than 90 degrees. right obtuse acute

  18. A straight angle is an angle that measures 180 degrees. (It is the same as a line.) • When drawing a right angle we often mark its opening as in the picture below. right angle straight angle

  19. 1 2 Adjacent Angles • Two angles are called adjacent angles if they share a vertex and a common side (but neither is inside the opening of the other). • Angles 1 and 2 are adjacent:

  20. Angle Addition • If are adjacent as in the figure below, then C A D B

  21. M A H T Example • In the figure, is three times and • Find • Let Then • By angle addition,

  22. A D C B Angle Bisectors • Consider below. • The angle bisector of this angle is the ray such that • In other words, it is the ray that divides the angle into two congruent angles.

  23. A C B Complementary Angles • Two angles are complementary if their measures add up to • If two angles are complementary, then each angle is called the complement of the other. • If two adjacent angles together form a right angle as below, then they are complementary. 1 2

  24. Example • Find the complement of • Call the complement x. • Then

  25. Example • Two angles are complementary. • The angle measures are in the ratio 7:8. • Find the measure of each angle. • The angle measures can be represented by 7x and 8x. Then

  26. Supplementary Angles • Two angles are supplementary if their measures add up to • If two angles are supplementary each angle is the supplement of the other. • If two adjacent angles together form a straight angle as below, then they are supplementary. 1 2

  27. Example • Find the supplement of • Call the supplement x. • Then

  28. Example • One angle is more than twice another angle. If the two angles are supplementary, find the measure of the smaller angle. • Let x represent the measure of the smaller angle. Then represents the measure of the larger angle. Then

  29. Perpendicular Lines • Two lines are perpendicular if they intersect to form a right angle. See the diagram. • Suppose angle 2 is the right angle. Then since angles 1 and 2 are supplementary, angle 1 is a right angle too. Similarly, angles 3 and 4 are right angles. • So, perpendicular lines intersect to form four right angles. 2 1 4 3

  30. The symbol for perpendicularity is • So, if lines m and n are perpendicular, then we write • The perpendicular bisector of a line segment is the line that is perpendicular to the segment and that passes through its midpoint. m m perpendicular bisector n A B

  31. Vertical Angles • Vertical angles are two angles that are formed from two intersecting lines. They share a vertex but they do not share a side. • Angles 1 and 2 below are vertical. • Angles 3 and 4 below are vertical. 3 2 1 4

  32. 2 1 3 • The key fact about vertical angles is that they are congruent. • For example, let’s explain why angles 1 and 3 below are congruent. Since angles 1 and 2 form a straight angle, they are supplementary. So, • Likewise, angles 2 and 3 are supplementary. So, So, angles 1 and 3 have the same measure and they’re congruent.

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