Transcript Section 1.1 Introduction to Geometry

Section 2.1
Perpendicularity
Geometry

Perpendicular () : Lines, rays, or
segments that intersect at right
angles.
b
ab
a
• Oblique : 2 intersecting lines that are not
perpendicular.
1
Geometry




Section 2.1
Perpendicularity
Coordinate Plane:
formed by the
intersection of the xaxis and the y-axis
x-axis - the horizontal
number line
y-axis - the vertical
number line
Origin: the point where
the number lines
y
x
origin
intersect
2
Geometry

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
(2,7)
7
x
2
y is the distance from the
x-axis (up - down)
3
Geometry
Given: a  b
Prove 1  2
Section 2.1
Perpendicularity
a
2
1
b
Statements
Reasons
1. a  b
2. 1 is a right angle
1. Given
3. 2 is a right angle
4. 1  2
2. If two lines are perpendicular, they
form a right angle.
3. If two lines are perpendicular, they
form a right angle.
4. If angles are right angles, they are
congruent.
4
Section 2.1
Perpendicularity
Geometry

Find the area of the rectangle below
A(-4,8)
D
B(10,8)
C(10, -2)
5
Answer
Geometry







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
6
Section 2.1
Perpendicularity
Geometry

Find the perimeter of the rectangle below
A(-4,8)
D
B(10,8)
C(10, -2)
7
Geometry


Answer
Perimeter is the sum of the sides
P = 2 (l + w)
P = 2 (14 + 10)
= 2 (24) = 48 units
8
Discussion Diagram
Geometry
F

A
30
E
45
H
G
45
B
C
D
9
Geometry



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)
10
Geometry
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)
11
Answers
Geometry







90-30 = 60
90-79 = 11
900-19030’ = 89060’-19030’ = 70030’
90-x
90-3a
90-(r-40)
90-(x+y)
12
Geometry
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.
13
Answers
Geometry


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.
14
Geometry
Solving Equation Word Problems
Steps:
1.
Read the problem a few times
2.
Line by line, write the definitions of terms in
the problem
3.
Line by line, write the givens
4.
Line by line, write algebraic expressions
5.
Set up the equation. Look for key terms, such
as exceeds, less than, difference, etc.
6.
Solve
7.
Check your work
15
Geometry
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

16
Geometry
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
17
Geometry
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
18
Geometry

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!
19
Section 2.3
Drawing Conclusions
Geometry
Sample Proof
Given:
X
XB bis AC
XC bis BD
Prove: AB  CD
A
Statements
XB bis AC
AB  BC
B
C
Reasons
Given
Definition of bisector
BC  CD
Given
Definition of bisector
AB  CD
Substitution
XC bis BD
D
20
Geometry

Section 2.4
Congruent 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
21
Geometry

Section 2.4
Congruent 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
22
Geometry
Section 2.4
Congruent Supplements & Complements
Sample Proof
Given: PB  AD
m1 = m3
QC  AD
P
2
Prove: m2 = m4
A
Statements
R
B
1
Q
3
4
C
D
Reasons
23
Sample Answer
Geometry
R
P
2
Statements
1. PB  AD
2.  PBC is a right angle
3. 2 and 1 are complementary
4. QC  AD
5. QCA is a right angle
6. 3 and 4 are complementary
7. m1 = m3
8. 1 and 3 are congruent
9. 2 is congruent to 4
10. m2 = m4
Reasons
A
1
B
1. Given
2. Definition of perpendicular
3. If two angles form a right
angle, they are complementary.
4. Given
5. Definition of perpendicular
6. Same as 3
7. Given
8. If two angles have the same
measure, then they are congruent.
9. If angles are complementary to
congruent angles, then they are
congruent.
10. Definition of congruent
Q
3
4
C
D
24
Geometry
Section 2.4
Congruent Supplements & Complements
Sample Proof
K
J
Given: EJ  EK
Prove: 1 and 2 are complementary.
Statements
1
C
2
E
D
Reasons
25
Sample Answer
Geometry
K
J
1
C
Statements
1. EJ  EK
2. JEK is a right angle
3. mJEK = 90º
4. CED is a straight angle
5. mJEK + m1 + m2 = 180
6. 90º + m1 + m2 = 180
7. m1 and m2 = 90
8. 1 and 2 are complementary
2
E
D
Reasons
1. Given
2. Definition of perpendicular
3. Definition of a right angle
4. Assumption
5. Definition of a straight angle
6. Substitution
7. Subtraction
8. If the sum of two angles is 90º,
then they are complementary.
26
Geometry
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.
 If congruent segments are added to
congruent segments, the sums are
congruent.
27
Geometry

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.
28
Geometry

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.
29
Geometry

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)
30
Section 2.8
Vertical Angles
Geometry

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.
31