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

2. 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

3. 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

4. Section 2.1 Perpendicularity

5. Section 2.1 Perpendicularity
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 = = 10 (y: 8, -2)
Segment AD = = 10
The coordinate of D is (-4, -2)
Area = length x width
Area = 14 x 10 = 140 square units

7. Section 2.1 Perpendicularity
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 2.2 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 - x)
Memory Helper: In alphabetical and numerical order
C (90)…… S (180)

11. Section 2.2 Complementary & Supplementary Angles
Sample Problems
Find the measure of the complement of an angle whose measure is
30, 79, ’
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-(x+y)

13. 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.

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 2.3 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!

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

21. 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

22. 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

23. Section 2.4 Congruent Supplements & Complements
Sample Proof

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

25. Section 2.4 Congruent Supplements & Complements
Sample Proof

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

27. 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.

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.