Chapter 3; Lines and Angles

1. Chapter 3; Lines and Angles By Annie Balunas and Kasey French

2. Lesson 1: Number Operations and Equality • Reflexive: a = a (Any number is equal to itself) • Substitution: If a = b, a can be substituted for b in any expression • Addition: If a = b, then a + c = b + c • Subtraction: If a = b, then a – c = b – c • Multiplication: If a = b, then ac = bc • Division: If a = b, and c ≠ 0, then a/c = b/c

3. Example Problem In triangle ABC, <A = 90° and <B is twice the size of <C. How large is <C? 1.) <A + <B + <C = 180° ( The sum of the angles of a triangle are 180 °) 2.) If <A = 90 °, then 90 ° + <B + <C + = 180 ° (Substitution) 3.) If <B + <C + 90 ° = 180 ° , then <B + <C = 90 ° (Subtraction) 4.) If <B + <C = 90 ° and <B = 2<C, then 2<C + <C = 90 ° (Substitution) 5.) If 3<C = 90 ° , then <C = 30 ° (Division)

4. Lesson 2: The Ruler and Distance Important Terms and Definitions: • A straight-edge is used to draw a line determined by 2 points. • A ruler is used to measure the distance between the points. • Every point that is on a line has a corresponding real number called a coordinate. • For every pair of points on a line, there corresponds a real number called distance.

5. Straight Edge tool

6. Lesson 2: Theorems, Definitions, and Postulates • Theorem 1: Betweeness of Points (BoP) • If A-B-C, then AB + BC = AC • Definition: Betweeness of Points • A-B-C iff a < b < c (or c > b > a) • Postulate 3: The Ruler Postulate • The points on a line can be numbered so that positive number differences measure distance

7. Betweeness of Points Theorem 4 8 14 AB = 8 – 4, AB = 4 BC = 14 – 8, BC= 6 AC= AB + BC, AC= 4 + 6, AC= 10

8. Lesson 3: The Protractor and Angle Measure Terms to Know: • A unit of measure is a degree. • Rays that correspond to a protractor are a half-rotation. • Rays that correspond to a circular protractor are a rotation of rays. • To every ray, there corresponds a real number called a coordinate. • To every pair of rays, there corresponds a real number called the measure of the angle.

9. degree Half-Rotation Protractor

10. Lesson 3: Terms (continued) Terms to Describe Angles: • Acute: angle measure of less than 90 ° • Right: angle measure of exactly 90 ° • Obtuse: angle measure of more than 90 °, but less than 180 ° • Straight: angle measure of exactly 180 °

11. Lesson 3: Theorems, Definitions, and Postulates • Theorem 2: Betweeness of Rays (BoRT) • If OA-OB-OC, then <AOB + <BOC = <AOC • Definitions: Betweeness of Rays • OA-OB-OC iff a < b < c (or c > b > a) • Postulate 4: The Protractor Postulate • The rays in a half-rotation can be numbered from 0 to 180 so that positive number differences measure angles.

12. Lesson 4: Bisection Terms to Know: • A midpoint is the point that divides a line segment into two equal segments • An angle bisector is a line that divides an angle into two equal angles • Something is congruentif it coincides exactly • A corollary is a theorem that can easily be proven as a consequence of a postulate or another theorem

13. Two pairs of congruent segments Angle Bisector

14. Lesson 4: Corollaries Ruler Postulate Corollary A line segment has exactly one midpoint. Protractor Postulate Corollary An angle has exactly one ray that bisects it.

15. Lesson 5: Complementary and Supplementary Angles Important Definitions • Two angles are complementaryiff their sum is 90° • Each angle is a complement of the other, iffthey equal 90° • Two angles are supplementaryiff their sum is 180° • Each angle is the supplement of the other, iffthey equal 180°

16. Lesson 5: (continued) Theorems • Theorem 3: Complements of the same angle are equal • Theorem 4: Supplements of the same angle are equal

17. Lesson 6: Linear Pairs and Vertical Angles Important Definitions • Two rays that point in the opposite direction are called opposite rays • Two angles are a linear pairiff they have a common side and their other sides are opposite rays •Two angles are vertical raysiff the sides of one angle are opposite rays to the sides of the other

18. Lesson 6: (continued) Theorems • Theorem 5: The angles in a linear pair are supplementary • Theorem 6: Vertical angles are equal

19. Lesson 7: Perpendicular and Parallel Lines Important Definitions • Two lines are perpendiculariff they form a right angle • Two lines are paralleliff they lie in the same plane and do not intersect

20. Lesson 7: (continued) Theorems and Corollaries • Theorem 7: Perpendicular lines form four right angles • Corollary to the Definition of a Right Angle: All right angles are equal • Theorem 8: If the angles in a linear pair are equal, then their sides are perpendicular

21. Lab #3: Proof 1 Given: <1 and <2 are supplementary; m<1 = m<2 Prove: <1 and <2 are right <‘s

22. Lab #3 Proof 1. <1 and <2 are supplementary (Given) 2. <1 + <2 = 180° (Definition of supp. <‘s) 3. <1 = <2 (Given) 4. <1 + <1 = 180 ° (Substitution 2,3) 5. 2<1 = 180 ° (Simplify) 6. <1 = 90 ° (Division) 7. 90 ° + <2 = 180 ° (Substitution 2,6) 8. <2 = 90 ° (Subtraction) 9. <1 and <2 are right angles (Right angles= 90 °)

23. Chapter 3 Summary • Chapter 3 taught us the basics of lines and angles, including specific types of both. This chapter also defined certain angles by their degree measure (complementary, right, obtuse, etc…) and lines by their length and relation to other lines(line segment, ray, line and parallel/perpendicular) • It taught us how to find lines and their distances using the Ruler Postulate and Betweeness of Points theorem/definition. • It also taught us how to create angles and their measurement through the Protractor Postulate and Betweeness of Rays theorem/definition • We learned how to find the relationship between lines and angles such as the equality of vertical pairs.