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Section 2.1 Perpendicularity

b. a. Section 2.1 Perpendicularity. Perpendicular (  ) : Lines, rays, or segments that intersect at right angles. Oblique : 2 intersecting lines that are not perpendicular. y. x. Section 2.1 Perpendicularity. Coordinate Plane: formed by the intersection of the x-axis and the y-axis

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Section 2.1 Perpendicularity

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  1. b a Section 2.1Perpendicularity • Perpendicular () : Lines, rays, or segments that intersect at right angles. • Oblique : 2 intersecting lines that are not perpendicular.

  2. y x Section 2.1Perpendicularity • Coordinate Plane: formed by the intersection of the x-axis and the y-axis • x-axis - the horizontal number line • y-axis - the vertical number line • Origin: the point where the number lines intersect origin

  3. y x Section 2.1Perpendicularity (2,7) • Coordinates : (aka ordered pair) a set of numbers in the form (x,y) that represents a point on the coordinate plane x is the distance from the y-axis (right - left) y is the distance from the x-axis (up - down) 7 2

  4. Section 2.1Perpendicularity

  5. Section 2.1Perpendicularity • Find the area of the rectangle below A(-4,8) B(10,8) D C(10, -2)

  6. Answer • Segment AB = 4 + 10 = 14 (x: -4, 10) • Segment DC = 4 + 10 = 14 • Segment BC = 8 + 2 = 10 (y: 8, -2) • Segment AD = 8 + 2 = 10 • The coordinate of D is (-4, -2) • Area = length x width • Area = 14 x 10 = 140 square units

  7. Section 2.1Perpendicularity • Find the perimeter of the rectangle below A(-4,8) B(10,8) D C(10, -2)

  8. Answer • Perimeter is the sum of the sides • P = 2 (l + w) P = 2 (14 + 10) = 2 (24) = 48 units

  9. Discussion Diagram F 30 E A 45 H G 45 D B C

  10. Section 2.2Complementary & Supplementary Angles • Complementary angles: two angles whose sum is 90 degrees • Complement: that which an angle needs to have a measure of 90. (90 - x) • Supplementary angles: two angles whose sum is 180 degrees • Supplement: that which an angle needs to have a measure of 180. (180 - x) • Memory Helper: In alphabetical and numerical order C (90)…… S (180)

  11. Section 2.2Complementary & Supplementary Angles Sample Problems • Find the measure of the complement of an angle whose measure is • 30, 79, 19030’ • Express the measure of the complement of an angle whose measure is represented by • x, (3a), (r - 40), (x+y)

  12. Answers • 90-30 = 60 • 90-79 = 11 • 900-19030’ = 89060’-19030’ = 70030’ • 90-x • 90-3a • 90-(r-40) • 90-(x+y)

  13. Section 2.2Complementary & Supplementary Angles More Sample Problems • Two angles are complementary. The measure of the larger angle is five times the measure of the smaller angle. Find the measure of the larger angle. • The supplement of the complement of an acute angle is always(1) an acute angle (2) an obtuse angle(3) a straight angle (4) a right angle.

  14. Answers • Let x = measure of smaller angle 5x = measure of larger angle x + 5x = 90 6x = 90 x = 15, so 5x = 75 • If two angles are complementary (sum=90), they are each acute, thus the complement of an acute angle must be less than 90. The supplement (sum=180) of an acute angle must be greater than 90. Thus, the supplement of the complement of an acute angle is always obtuse.

  15. Solving Equation Word Problems Steps: • Read the problem a few times • Line by line, write the definitions of terms in the problem • Line by line, write the givens • Line by line, write algebraic expressions • Set up the equation. Look for key terms, such as exceeds, less than, difference, etc. • Solve • Check your work

  16. Practice Problem • The measure of a supplement of an angle exceeds three times the measure of the complement of the angle by 10. Find the measure of the angle. • Steps: 1. Read the problem a few times 2. Line by line, write the definitions of terms in the problem: supplement, complement

  17. 3. Line by line, write the givens: The measure of a supplement of an angle exceeds three times the measure of the complement of the angle by 10. 4. Line by line, write algebraic expressions: Let x = unknown angle Let 180-x = measure of the supplement of an angle Let 90-x = measure of the complement of the angle

  18. 5. Set up the equation Read the given (180-x) + 10 = 3(90-x) 6. Solve (180-x) + 10 = 270-3x 3x - x = 270 – 180 – 10 2x = 80 x = 40 7. Check (180-40) + 10 = 3(90-40) 150 = 150

  19. Section 2.3Drawing Conclusions • Methods, Suggestions • Must memorize definitions, theorems, etc. • Symbols give away information. Be familiar with them. • Draw as much information from each given as possible. • Decide what information will make your case. • Draw a valid conclusion!

  20. Section 2.3Drawing Conclusions Sample Proof Given Definition of bisector Given Definition of bisector Substitution

  21. Section 2.4Congruent Supplements & Complements • Theorem • If angles are supplementary to the same angle, then they are congruent. • Assumes only one angle • Theorem • If angles are supplementary to congruent angles, then they are congruent. • Assumes more than one angle

  22. Section 2.4Congruent Supplements & Complements • Theorem • If angles are complementary to the same angle, then they are congruent. • Assumes only one angle • Theorem • If angles are complementary to congruent angles, then they are congruent. • Assumes more than one angle

  23. Section 2.4Congruent Supplements & Complements Sample Proof

  24. Sample Answer Statements Reasons 1. PB  AD 1. Given 2.  PBC is a right angle 2. Definition of perpendicular 3. 2 and 1 are complementary 3. If two angles form a right angle, they are complementary. 4. QC  AD 4. Given 5. QCA is a right angle 5. Definition of perpendicular 6. 3 and 4 are complementary 6. Same as 3 7. m1 = m3 7. Given 8. 1 and 3 are congruent 8. If two angles have the same measure, then they are congruent. 9. 2 is congruent to 4 9. If angles are complementary to congruent angles, then they are congruent. 10. m2 = m4 10. Definition of congruent

  25. Section 2.4Congruent Supplements & Complements Sample Proof

  26. Sample Answer Statements Reasons 1. EJ  EK 1. Given 2. JEK is a right angle 2. Definition of perpendicular 3. mJEK = 90º 3. Definition of a right angle 4. CED is a straight angle 4. Assumption 5. mJEK + m1 + m2 = 180 5. Definition of a straight angle 6. 90º + m1 + m2 = 1806. Substitution 7. m1 and m2 = 90 7. Subtraction 8. 1 and 2 are complementary 8. If the sum of two angles is 90º, then they are complementary.

  27. Section 2.5Addition and Subtraction Properties More Theorems! • If a segment is added to congruent segments, the sums are congruent. • If congruent segments are added to congruent segments, the sums are congruent. • If an angle is added to congruent angles, the sums are congruent. • If congruent segments are added to congruent segments, the sums are congruent.

  28. Section 2.5 Addition and Subtraction Properties • Subtraction Theorems • If a segment (or angle) is subtracted from congruent segments or (angles), the differences are congruent. • If congruent segments (or angles) are subtracted from congruent segments (or angles), the differences are congruent.

  29. Section 2.6 Multiplication and Division Properties • Theorems • If segments (or angles) are congruent, their like multiples are congruent. • If segments (or angles) are congruent, their like divisions are congruent.

  30. Section 2.7 Transitive and Substitution Properties • Transitive Theorems • If angles (or segments) are congruent to the same angle (or segment), they are congruent to each other. • If angles (or segments) are congruent to congruent angles (or segments), they are congruent to each other. Note: The relation “is perpendicular to” is never transitive. • Substitution (Same as in algebra)

  31. Section 2.8 Vertical Angles • Definitions • opposite rays: 2 collinear rays with a common endpoint that extend in opposite directions • vertical angles: angles formed when two opposite rays (lines) intersect • Theorem • Vertical Angles are congruent.

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