8.1 – Lines and Angles

1. 8.1 – Lines and Angles Defn: Space: The region that extends in all direction indefinitely. Plane: A flat surface without thickness that extends indefinitely in two directions..

2. 8.1 – Lines and Angles Defn: Line: A set of points extending indefinitely in opposite directions. Line AB Line l l AB AB AB  A  B Line Segment: A piece of a line that has two end points.  A  line segment AB B Ray: A part of a line with one end point extending indefinitely in one direction.  A  ray AB B

3. 8.1 – Lines and Angles Defn: Angle: A two dimensional plane whose sides consist of two rays that share the same end point. The shared point is called the vertex. B  A x  Angle BAC  BAC Angle CAB  CAB  C Angle A  A Angle x  x

4. 8.1 – Lines and Angles Identify each of the following figures. YI CF VE  TBS  SBT  THV  VHT  B  H

5. 8.1 – Lines and Angles Use the given figure to answer the questions. 1 What are the name(s) of the given angle? What are the other names of angle x?  BAC  AEC  CAB  CEA  A  E  1

6. 8.1 – Lines and Angles Classifying Angles Degree: A unit for measuring angles. The symbol denoting degrees is a raised circle (45°). Straight angle: Any angle that measures 180° (a straight line). Right angle: Any angle that measures 90°. Acute angle: Any angle whose measurement is between 0° and 90°. Obtuse angle: Any angle whose measurement is between 90° and 180°.

7. 8.1 – Lines and Angles Classifying Angles straight acute obtuse obtuse right

8. 8.1 – Lines and Angles Special Pairs of Angles Complimentary angles: Two angles whose sum is 90°. They are compliments of each other. Supplementary angles: Two angles whose sum is 180°. They are supplements of each other. What is the compliment of 15°? What is the supplement of 80°? 100° 180 – 80 = 90 – 15 = 75° What is the compliment of 29°? What is the supplement of 95°? 90 – 29 = 61° 85° 180 – 95 = What is the supplement of 29°? 151° 180 – 29 =

9. 8.1 – Lines and Angles Calculating the Measure of an Angle Given the figure below, what is the measure of CPD? Given the figure below, what is the measure of 1? 68° 176° 23° 119° 176 – 119 = 57° 68 – 23 = 45° What type of angle is 45°? What type of angle is 57°? acute acute

10. 8.1 – Lines and Angles Lines in a Plane Intersecting Lines: Lines in a plane that cross at a common point. If two lines intersect, they create four angles. 2 1 3 4 Vertical angles: Of the four angles formed from two intersecting lines, these are the two pairs of opposite angles. The measures of the opposite angles are the same. 2 = 4 1 = 3

11. 8.1 – Lines and Angles Lines in a Plane Parallel lines: Two or more lines in a plane that do not intersect. l1 line l line 2 l2 l1 l2 Perpendicular lines: Two lines in a plane that intersect at a 90° angle. l3 line 3  line 4 l3 l4 l4

12. 8.1 – Lines and Angles Lines in a Plane Adjacent angles: Angle that share a common side. DBC is adjacent to CBA as they share the common side BC. Adjacent angles formed from two intersecting lines are supplementary. 3 + 4 = 180° 1 + 2 = 180° 2 1 3 4 4 + 1 = 180° 2 + 3 = 180°

13. 8.1 – Lines and Angles Lines in a Plane Transversal lines: Any line that intersects two or more lines at different points. l3 l1 b a The position of the angles created by the transversal line have specific names. c l2 d f e g h Corresponding angles: a & e, b & f, d & h, and c & g Alternate interior angles: d & f and c & e

14. 8.1 – Lines and Angles Lines in a Plane If two parallel lines are cut by a transversal, then: (a) The corresponding angles are equal, (b) The alternate interior angles are equal. l1 l2 Corresponding angles: l3 l1 a = e, b = f, b a c l2 d d = h, and c = g f e Alternate interior angles: g h d = f and c = e

15. 8.1 – Lines and Angles Given the measure of one angle, calculate the measures of the other angles. 2 1 3 4 m4 = 127° m1 = 30° 53° 150° m1 = m2 = 180 – 30 = 180 – 127 = m2 = 127° m3 = 30° m3 = 53° m4 = 150°

16. 8.1 – Lines and Angles Given the measure of one angle, calculate the measures of the other angles. l1 l2 l3 l1 b a c l2 d f e g h md = 82° me = 98° 82° mf = 180 – 98 = ma = 98° mg = 98° mb = 82° mh = 82° mc = 98°

17. 8.2 – Perimeter Defn. Perimeter: The perimeter of any polygon is the total distance around it. The sum of the lengths of all sides of the polygon is the perimeter. Calculate the perimeter of each of the following: The length of a rectangle is 32 centimeters and its width is 15 centimeters. cm P = 32 15 + = 32 + + 15 94 cm P = 2(15) 2(32) + = 94

18. 8.2 – Perimeter Calculate the perimeter of each of the following: 18 meters The figure is a rectangle with the given measurements. 10 meters m + 20 = 56 36 P = 2(10) 2(18) + = 15 feet 9 feet 7 feet 14 feet ft + 9 + 14 = 45 P = 15 7 +

19. 8.2 – Perimeter Calculate the perimeter of each of the following: 17 inches All angles in the figure are 90°. 12 inches 22 inches 29 inches 29 – 12 = 17 inches 17 + 22 = 39 inches 39 in + = 17 + 136 + 12 + 22 P = 17 29 +

20. 8.2 – Perimeter A rectangular lot measures 60 feet by 120 feet. Calculate the cost of a fence to be installed around the perimeter of the lot if the fence costs $1.90 per foot. Calculate the perimeter. 240 ft P = 2(120) 120 + = 2(60) + = 360 Calculate the cost of the fence. $ 684.00 · C = 1.90 360 =

21. 8.2 – Perimeter Circumference – the length around the edge of a circle. Blue line = Diameter r Red line = Radius d d= 2r C =  d or C = 2  r

22. 8.2 – Perimeter C =  d or C = 2  r Find the exact circumference of a circle whose diameter is 20 yards. Find the approximate value of the circumference of a circle whose diameter is 7 meters. ( Use 3.14 as an approximation of .) C =  d C =  20 C =  d C = 3.14 (20) yds C = 20 3.14 x 20 6280 2 decimal places 62.80 yds