Lesson 1_Lines and Angle Relationship.pptx

Lines and Angles

1. Lines and Angle Relationship

2. Objectives Identify, name, and draw points, lines, segments, rays , planes and related concepts. Apply basic facts about points, lines, and planes and related concepts in solving problems. Find measures of pairs of angles. Construct midpoints, congruent segments and bisector.

3. Definitions and Postulates A MATHEMATICAL SYSTEM UNDEFINED TERMS DEFINED TERMS AXIOMS OR POSTULATES THEOREMS

4. UNDEFINED TERMS The most basic figures in geometry are undefined terms , which cannot be defined by using other figures. The undefined terms point , line , and plane are the building blocks of geometry.

5. Points that lie on the same line are collinear . K , L , and M are collinear. K , L , and N are noncollinear . Points that lie on the same plane are coplanar . Otherwise they are noncoplanar . M K L N

6. Example 1: Naming Points, Lines, and Planes A. Name four coplanar points. B. Name three lines. A, B, C, D Possible answer: AE, BE, CE

7. CHARACTERISTICS OF A GOOD DEFINITION A GOOD DEFINITION WILL POSSESS THESE QUALITIES 1. It names the term being defined. 2. It places the term into a set or category. 3. It distinguishes the defined term from other terms without providing unnecessary facts. 4. It is reversible. DEFINED TERMS

8. A GOOD DEFINITION WILL POSSESS THESE QUALITIES 1. It names the term being defined. 2. It places the term into a set or category. 3. It distinguishes the defined term from other terms without providing unnecessary facts. 4. It is reversible. DEFINED TERMS

9. Example 2: Drawing Segments and Rays Draw and label each of the following. A. a segment with endpoints M and N. B. opposite rays with a common endpoint T . M N T

10. Important Definitions

11. AXIOMS OR POSTULATES A postulate , or axiom , is a statement that is accepted as true without proof. Postulates about points, lines, and planes help describe geometric properties.

12. Through two distinct points, there is exactly one line. In Figure at the left, how many distinct lines can be drawn through point A ? Answer: An infinite (countless) number b) both points A and B at the same time ? Answer: Exactly one c) all points A , B , and C at the same time ? Answer: No line contains all three points. POSTULATE 1 Figure 1

13. The measure of any line segment is a unique positive number. The Ruler Postulate implies the following: 1. There exists a number measure for each line segment. 2. Only one measure is permissible. POSTULATE 2

14. POSTULATE 3 (Segment-Addition Postulate)

15. Using the Segment Addition Postulate M is between N and O . Find NO . 10 = 2 x NM + MO = NO Seg. Add. Postulate 17 + (3 x – 5) = 5 x + 2 – 2 – 2 Substitute the given values Subtract 2 from both sides. Simplify. 3 x + 12 = 5 x + 2 3 x + 10 = 5 x –3 x –3 x 2 2 5 = x Simplify. Subtract 3x from both sides. Divide both sides by 2.

16. If two lines intersect, they intersect at a point. POSTULATE 4

17. Through three non-collinear points, there is exactly one plane. POSTULATE 5 Name a plane that contains three non-collinear points.

18. If two distinct planes intersect, then their intersection is a line. POSTULATE 6

19. Given two distinct points in a plane, the line containing these points also lies in the plane. POSTULATE 7

20. Try this! 1. Two opposite rays. 3. The intersection of plane N and plane T . 4. A plane containing E , D , and B . 2. A point on BC. CB and CD Possible answer: D Possible answer: BD Plane T

21. Try This! 5. a line intersecting a plane at one point 6. a ray with endpoint P that passes through Q Draw each of the following.

22. Geometric Construction You can make a sketch or measure and draw a segment. These may not be exact. A construction is a way of creating a figure that is more precise. One way to make a geometric construction is to use a compass and straightedge.

23. Angles and Their Relationships

24. The set of all points between the sides of the angle is the interior of an angle . The exterior of an angle is the set of all points outside the angle. Angle Name  R ,  SRT ,  TRS , or  1 You cannot name an angle just by its vertex if the point is the vertex of more than one angle. In this case, you must use all three points to name the angle, and the middle point is always the vertex.

25. Naming Angles A surveyor recorded the angles formed by a transit (point A ) and three distant points, B , C , and D . Name three of the angles. Possible answer:  BAC  CAD  BAD

26. You can use the Protractor Postulate to help you classify angles by their measure. The measure of an angle is the absolute value of the difference of the real numbers that the rays correspond with on a protractor. If OC corresponds with c and OD corresponds with d , m  DOC = | d – c | or | c – d |. The measure of an angle is a unique positive number. POSTULATE 7

27. TYPES OF ANGLES A reflex angle is one whose measure is between 180° and 360°.

28. POSTULATE 9 (Angle-Addition Postulate)

29. m  DEG = 115°, and m  DEF = 48°. Find m  FEG Using the Angle Addition Postulate m  DEG = m  DEF + m  FEG 115  = 48  + m  FEG 67  = m  FEG  Add. Post. Substitute the given values. Subtract 48 from both sides. Simplify. –48° –48°

30. Example: Finding the Measure of an Angle KM bisects  JKL , m  JKM = (4 x + 6)°, and m  MKL = (7 x – 12)°. Find m  JKM .

31. Example Continued Step 1 Find x. m  JKM = m MKL (4 x + 6) ° = (7 x – 12) ° 4 x -7x = -12 - 6 –3 x = -18 18 = 3 x 6 = x Def. of  bisector Substitute the given values. Simplify. Divide both sides by 3. Simplify. Step 2 Find m  JKM . m  JKM = 4 x + 6 = 4 (6) + 6 = 30 Substitute 6 for x. Simplify.

32. CLASSIFYING PAIRS OF ANGLES Many angle relationships involve exactly two angles (a pair)—never more than two angles and never less than two angles! Based from the definition, name 1 pair of adjacent angle.

33. A Linear Pair of angles is a pair of adjacent whose non common sides are opposite rays. CLASSIFYING PAIRS OF ANGLES

34. Tell whether the angles are only adjacent, adjacent and form a linear pair, or not adjacent. Identifying Angle Pairs  AEB and  BED  AEB and  BED have a common vertex, E , a common side, EB , and no common interior points. Their noncommon sides, EA and ED , are opposite rays. Therefore,  AEB and  BED are adjacent angles and form a linear pair.

35. Tell whether the angles are only adjacent, adjacent and form a linear pair, or not adjacent. Identifying Angle Pairs  AEB and  BEC  AEB and  BEC have a common vertex, E , a common side, EB , and no common interior points. Therefore,  AEB and  BEC are only adjacent angles.

36.  DEC and  AEB  DEC and  AEB share E but do not have a common side, so  DEC and  AEB are not adjacent angles. Tell whether the angles are only adjacent, adjacent and form a linear pair, or not adjacent. Identifying Angle Pairs

37. CLASSIFYING PAIRS OF ANGLES

38. CLASSIFYING PAIRS OF ANGLES

39. Find the measure of each of the following. Finding the Measures of Complements and Supplements A. complement of  F B. supplement of  G 90  – 59  = 31  (180 – x )  180 – (7 x +10)  = 180 – 7 x – 10 = (170 – 7 x )  (90 – x ) 

40. x = 108 , m<P = 54 , m<Q = 36 Example

41. An angle is 10° more than 3 times the measure of its complement. Find the measure of the complement. Using Complements and Supplements to Solve Problems x = 3(90 – x ) + 10 x = 270 – 3 x + 10 x = 280 – 3 x 4 x = 280 x = 70 The measure of the complement,  B , is (90 – 70 ) = 20 . Substitute x for m A and 90 – x for mB. Distrib. Prop. Divide both sides by 4. Combine like terms. Simplify. Step 1 Let m  A = x °. Then  B , its complement measures (90 – x )°. Step 2 Write and solve an equation.

42. An angle’s measure is 12° more than the measure of its supplement. Find the measure of the angle. x = (½)(180 – x ) + 12 x = 90 – (1/2) x + 12 x = 102 – (½) x (3/2) x = 102 x = 68 The measure of the angle is 68 . Substitute x for m A and 180 - x for mB. Distrib. Prop. Divide both sides by 3/2. Combine like terms. Simplify.

43. Another angle pair relationship exists between two angles whose sides form two pairs of opposite rays. Vertical angles are two nonadjacent angles formed by two intersecting lines.  1 and  3 are vertical angles, as are  2 and  4 . CLASSIFYING PAIRS OF ANGLES

44. Check It Out! Name a pair of vertical angles. Do they appear to have the same measure? Check by measuring with a protractor.  EDG and  FDH are vertical angles and appear to have the same measure. Check m  EDG ≈ m  FDH ≈ 45°