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

Section 2.1 Perpendicularity 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 y x origin

Section 2.1 Perpendicularity 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) y x (2,7) 7 2

Section 2.1 Perpendicularity

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

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

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

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

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

Section 2.2 Complementary & 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)

Section 2.2 Complementary & 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)

Answers 90-30 = 60 90-79 = 11 900-19030’ = 89060’-19030’ = 70030’ 90-x 90-(x+y)

Section 2.2 Complementary & 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.

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.

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

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

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

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

Section 2.3 Drawing 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!

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

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

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

Section 2.4 Congruent Supplements & Complements Sample Proof

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

Section 2.4 Congruent Supplements & Complements Sample Proof

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 = 180 6. 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.

Section 2.5 Addition 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.

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.

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.

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)

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.