Download Chapter 1 Line and Angle Relationships

Chapter 1
Line and Angle Relationships
1.1
Sets, Statements and Reasoning
Definition: A set is any collection of objects called elements. Braces { } are used to denote sets.
Examples:
Definition: A statement is a set of words and/or symbols that collectively make a claim that can
be classified as true or false.
Example: Determine which of the following sentences are statements. If a sentence is a statement,
give its truth value.
• Fullerton College opened in 1913
• 3(2) = 7
• Speak up
• A square has four 90o angles
• 5 + 7 − 21
Definition: The negation of a statement P is the claim opposite that of the original statement.
Indicated by “not P” or ∼P.
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Section 1.1
Example: Give the negation of each statement
P : x+5=9
Q : All students take math
R : George Washington was a president
Definition: A compound statement is formed by combining multiple statements.
Example:
Let P be the statement: “I eat cookies”
Let Q be the statement: “I get sick”
Let R be the statement: 7 + 2 = 9
Let S be the statement: “Elephants can fly”
From these simple statements we can build the following compound statements:
Statement Type
Key Word(s)
Example
Conjunction
Disjunction
Conditional (Implication)
First Statement
P
Second Statement
Q
Conjunction
P and Q
Disjunction
P or Q
True/False
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Section 1.1
Definition: In a conditional statement , such as ”If P , then Q” or P ⇒ Q.
P is the hypothesis and Q is the conclusion.
Definition: Reasoning is a process that leads to a conclusion. Three types of reasoning are:
• Intuition: Inspiration leading to a statement of a theory.
Ex: Geometry is fun.
• Induction: The use of specific observation or experiments to draw a general conclusion.
Ex: Consider the following sums. What can you conclude?
1+3
1+3+5
1+3+5+7
1+3+5+7+9
=
=
=
=
4
9
16
25
• Deduction: reasoning where knowledge and accepted facts guarantee the truth of a particular
conclusion. Also known as a proof.
Ex: Suppose statements (1) and (2) are true
(1) If it rains, then the street will be wet.
(2) It rained.
Conclusion:
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Section 1.1
Law of Detachment
Let P and Q be simple statements and assume 1 and 2 are true. Then a valid argument having
conclusion C has the form:
Example: Assume that premises (1) and (2) are true. If the argument is valid state the conclusion.
1.
Ex a):
If I go to the movies then I cry.
1.
2.
I am going to the movies.
2.
C.
Ex a):
If Ed goes to the beach, then he
will get sunburned.
Ed got sunburned.
C.
Definition: A counterexample is an example disproving the validity of an argument.
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Section 1.1
Examples: Assume that the given statements are true. Use deduction to state the conclusion, if
any. If no conclusion can be made, give a counter example.
Ex a): If the sum of two angles is 180o , then these angles are called “supplementary”. Angle 1
measures 125o and angle 2 measures 55o .
Ex b): If you put a cat in a tub of water, then the cat will not be happy. My cat is not happy.
Ex c): Every time Fred goes out with the boys, he gets in trouble with his wife. Fred is in trouble
with his wife.
Math 30 Geometry
1.2
Page 6
Section 1.2
Informal Geometry and Measurement
The following terms are described, but not formally defined:
• Point, Line, Plane, Space
• Straightness, flatness
• between A-X-B
• interior/exterior
Name
Figure
Symbol
line AB
←→
AB
line segment AB
AB
length AB
AB
ray AB
−→
AB
Notes
Math 30 Geometry
Name
Page 7
Figure
Section 1.2
Symbol
collinear and between
A-X-B
congruent
∼
=
angle ABC
∠ABC
∠CBA
∠B
measure of angle B
m∠B (book)
]B
]ABC
triangle ABC
4ABC
rectangle ABCD
ABCD
vertex/vertices
Notes
Math 30 Geometry
Name
arc AB
Page 8
Figure
Symbol
Section 1.2
Notes
>
AB
bisect
midpoint
right angle
straight angle
parallel
k
perpendicular
⊥
Note: Numbers can be equal, but line segments or angles are congruent!
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Page 9
Section 1.2
Example: If M is the midpoint of F G, then
1. Find F M , if F G = 17
2. Find F G if M G = 2.7
3. Find an expression for F G if F M = 3x − 1 and M G = 2x + 4
4. Find x and F G if F M = x2 and M G = 24 − 5x
Example: Suppose that point B lies on AC between A and C. If AC = 13 and AB is 3 units longer
than BC, find x = AB and y = BC
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Section 1.2
Constructions are the art of making geometrics drawings with specific properties using only a compass, straightedge and a pencil. These drawings meet the desired properties exactly by logical rules
rather than by measurement (which would use a ruler and protractor). This is the pure form of
geometric drawing that does not involve numbers.
Example: Given AB
Construct: CD on line m so that AB ∼
= CD
Definition: A circle is a set of all points in a plane that are at a given distance from a particular
point (known as the “center” of the circle).
Definition: A radius (plural: radii) is any line segment joining the center to a point on the circle.
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Section 1.2
There are only three “operations” involved in making a construction drawing:
1. Draw a circle or arc with specified center and radius
C#1(A, AB)
C#2(P, r)
C#3(Q, AB)
has center A and radius AB
has center P and radius r
has center Q and radius AB
2. Draw a straight line through two specified points
←→
AP
draws a straight-line through points A and P
3. Draw a point, most often the intersection between two objects from 1 and 2
←→
C#1 ∩ BC @ Q
C#1 ∩ C#2 @ P and T
←→ ←→
AB ∩ CD @ R
circle #1 and line BC intersect at point Q
circle #1 and circle #2 intersect at point P and T
line AB and line CD intersect at point R
For any construction one can write a construction protocol, which is a list of the steps involved.
You may have to label (give names to) points in the drawing as they appear.
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Section 1.2
Exercise #1: Write a construction protocol for the following construction of a segment of a specified
length
Construct: CD on line m so that AB ∼
= CD
Steps:
•
•
•
Exercise #2: Follow the given construction protocol.
• C#1(A, r) with any r > 12 AB
• C#2(B, r)
• C#1 ∩ C#2 @ C and D
←→
• CD
←→
• CD ∩ AB @ M
A
B
Math 30 Geometry
1.3
Page 13
Section 1.3
Early Definitions and Postulates
Geometry is a mathematical system that is developed from undefined terms, defined terms, axioms (or postulates) and theorems.
Definition: An axiom or postulate is an assumed property of a mathematical system.
Definition: A theorem is a statement that can be proved.
Postulate 1:
Through two distinct points there is exactly one line.
Example: How many distinct lines can be drawn through
1. A given point A
2. Two given points A and B
3. Three given points A, B and C
Postulate 2: (Ruler Postulate)
The measure of any line segment is a unique positive number.
Definition: The distance between two points A and B is the length of the line segment AB that
joins the two points.
Postulate 3: (Segment-Addition Postulate)
If X is a point of AB and A-X-B, then AX + XB = AB
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Section 1.3
Example: Consider AB as shown.
1. Find AB if AX = 23.72 and XB = 9.14
2. Find AX if AB = 26 and XB = 31 (AX) + 2
Definition: Congruent Segments (∼
=) are two line segments that have the same length.
Example: If AB = 3.25 ft and CD = 39 in , is AB ∼
= CD?
Definition: The midpoint of a line segment is the point that separates a line segment into two
congruent parts.
Example: If M is the midpoint of AB with AM = 3x + 9 and M B = 8x − 6, find x and AB.
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Section 1.3
−→
←→
Definition: Ray AB, denoted as AB, is the union of AB and all the points X on AB such that B
is between A and X, i.e. A-B-X.
Example:
Definition: Opposite rays are two rays that share an endpoint and together make a line.
Definition: The intersection of two geometric figures is the set of points that the two figures have
in common.
Postulate 4:
If two lines intersect, they intersect at a point.
(Sidenote: When two lines share two (or more) points,
the lines coincide, and we say there is only one line.)
Definition: Parallel lines are lines that lie in the same plane but do not intersect.
Definition: A plane has infinite length and width, but no thickness.
Points that lie in the same plane are called coplanar.
Postulate 5: (Tripod Postulate)
Through three noncollinear points, there is exactly one plane.
Postulate 6:
If two distinct planes intersect, then their intersection is a line.
Postulate 7:
Given two distinct points in a plane, the line segment containing these points also lies in the plane.
Theorem 1.3.1: The midpoint of a line segment is unique.
Math 30 Geometry
1.4
Page 16
Section 1.4
Angles and their Relationships
Definition: An angle is the union of two rays that share a common endpoint.
Postulate 8: (Protractor Postulate)
The measure of an angle is a unique positive number.
Types of Angles
acute angle
right angle
obtuse angle
straight angle
reflex angle
Postulate 9: (Angle-Addition Postulate)
If a point D lies in the interior of an angle ABC, then
]ABD + ]DBC = ]ABC
Example: Use the given figure to find ]T SV
a) ]T SW = 19o and ]W SV = 24o
T
W
S
V
b) ]T SW = 2xo and ]W SV = (7x − 8)◦
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Section 1.4
Definition: Two angles are adjacent if they have a common vertex and a common side between
them.
Definition: Congruent Angles (∼
=) are two angles with the same measure.
Example: Given: ∠1 ∼
= ∠2, ]1 = 3x + 7, ]2 = 4x − 12
Find: x, ]1, ]2
Definition: The bisector of an angle is the ray that separates the given angle into two congruent
angles.
Definition: Two angles are complementary if the sum of their measures is 90◦
Each angle in the pair is known as the complement of the other angle.
Definition: Two angles are supplementary if the sum of their measures is 180◦ .
Each angle in the pair is known as the supplement of the other angle.
Example: Given that ∠P and ∠Q are supplementary and ]P = 3(]Q) − 20o . Find ]P and ]Q.
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Definition: The pair of nonadjacent angles formed when
two lines intersect are called vertical angles
Constructions with angles.
Example (Construct congruent angle): Given ∠ABC
Construct: ∠HJK so that ∠ABC ∼
= ∠HJK
Section 1.4
Math 30 Geometry
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Example (Bisect Angle): Given ∠DEF
−−→
Construct: EG so that ∠DEG ∼
= ∠GEF
Theorem 1.4.1: There is one and only one angle bisector for any given angle.
Section 1.4
Math 30 Geometry
1.5
Page 20
Section 1.5
Introduction to Geometric Proof
Basic Tools you can use for a proof.
• Properties of Equality (let a, b, and c, be real numbers)
– Addition Property of Equality:
– Subtraction Property of Equality:
– Multiplication Property of Equality:
– Division Property of Equality:
• Other Algebraic Properties: (let a, b, and c, be real numbers (a, b, c, ∈ R))
– Distributive Property:
– Substitution Property:
– Transitive Property:
Example: Given: 3(x − 5) + 7 = 19
Prove: x = 9
Statement
1. 3(x − 5) + 7 = 19
Reasoning
1. Given
2.
2.
3.
3.
4.
4.
5.
5.
Math 30 Geometry
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Section 1.5
Recall the Segment-Addition Postulate:
If A-X-B, then AX + XB=AB
Example: Given: A-X-B on AB
Prove: XB=AB − AX
Statement
Reasoning
Properties of Inequality (let a, b, c ∈ R)
• Addition Property of Inequality:
• Subtraction Property of Inequality:
Example: Given: AB < CD with A-B-C-D
Prove: AC < BD
Statement
Reasoning
Math 30 Geometry
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Section 1.5
Recall the Angle-Addition Postulate:
If a point D lies in the interior of an angle ABC, then
]ABD + ]DBC = ]ABC
−→
−−→
Example: Given: ∠BAE with AC and AD (as shown)
Prove: ]BAE = ]BAC + ]CAD + ]DAE
Statement
Reasoning
Math 30 Geometry
1.6
Page 23
Section 1.6
Relationships: Perpendicular Lines
Definition: Perpendicular Lines are two lines that meet to form congruent adjacent angles.
Theorem 1.6.1: If two lines are perpendicular, then they meet to form right angles.
←→ ←→
Proof: Given: AB ⊥ CD intersecting at E
Prove: ∠AEC is a right angle.
Statement
←→ ←→
1. AB ⊥ CD intersecting at E
Reasoning
1. Given
Math 30 Geometry
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Section 1.6
Relations
A relation ”connects” two elements of a set of objects.
Relation
Objects related
Example
Possible properties of relations. Let R refer to a relation and a,b, and c be associated
1. Reflexive Property:
2. Symmetric Property:
3. Transitive Property:
Any relation that has reflexive, symmetric and transitive properties is called
an equivalence relation.
Math 30 Geometry
Page 25
Section 1.6
Example: Consider the relation “is greater than”. Which of the following properties apply?
1. Reflexive Property:
2. Symmetric Property:
3. Transitive Property:
Example: Consider the relation of congruence of angles. Which of the following properties apply?
1. Reflexive Property:
2. Symmetric Property:
3. Transitive Property:
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Section 1.6
Theorem 1.6.2: If two lines intersect, the vertical angles formed are congruent.
←→ ←→
Proof: Given: AC, BD intersecting at O
Prove: ∠2 ∼
= ∠4
Statement
←→ ←→
1. AC, BD intersecting at O
Reasoning
1. Given
Theorem 1.6.3: In a plane, there is exactly one line perpendicular to a given line at any point.
Theorem 1.6.4: The perpendicular bisector of a given line segment is unique.
Math 30 Geometry
1.7
Page 27
Section 1.7
The Formal Proof of a Theorem
The hypothesis of a statement describes the given condition (Given).
The conclusion of a statement describes what you need to establish (Prove).
Example: Give the hypothesis and conclusion of each statement.
(a) If two planes intersect, then they intersect at a line.
(b) All rectangles have diagonals that are congruent.
(c) Two lines are parallel when they are in the same plane and do not intersect.
Essential parts of a formal proof
1. Statement: States the theorem to be proved.
2. Drawing: Represents the hypothesis of the theorem.
3. Given: Describes the hypothesis using notation from the drawing.
4. Prove: Describes the conclusion using notation from the drawing.
5. Proof: List of claims/statements and justification/reasons that start with “given” and end
with “prove” and follow logically.
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Section 1.7
Definition: The converse of a statement interchanges hypothesis and conclusion.
Statement: If P , then Q
Converse:
Example: Statement: If I am hungry, I will eat.
Converse:
Recall the definition of perpendicular lines: Two lines that meet to form congruent adjacent angles.
Recall Theorem 1.6.1: If two lines are perpendicular, then they meet to form right angles.
P:
Q:
Theorem 1.7.1 is the converse of Theorem 1.6.1.
Theorem 1.7.1: If two lines meet to form a right angle, then these lines are perpendicular.
Proof:
Statement
1.
Reasoning
1.
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Page 29
Section 1.7
Theorem 1.7.2: If two angles are complementary to the same angle (or to congruent angles), then
these angles are congruent.
Theorem 1.7.3: If two angles are supplementary to the same angle (or to congruent angles), then
these angles are congruent.
Theorem 1.7.4: Any two right angles are congruent.
Theorem 1.7.5: If the exterior sides of two adjacent acute angles form perpendicular rays, then
these angles are complementary.
Theorem 1.7.6: If the exterior sides of two adjacent angles form a straight line, then these angles
are supplementary.
Proof:
Statement
1.
Reasoning
1.
Theorem 1.7.7: If two line segments are congruent, then their midpoints separate these segments
into four congruent segments.
Theorem 1.7.8: If two angles are congruent, then their bisectors separate these angles into four
congruent angles.
Chapter 2
Parallel Lines
2.1
The Parallel Postulate and Special Angles
Example: Given line l and point P not on l
Construct: P Q ⊥ l
Theorem 2.1.1: From a point not on a given line, there is exactly one line perpendicular
to the given line.
Definition: Parallel Lines are two lines in the same plane that never intersect.
General: The pairs of: two lines in a plane, a line and a plane, and two planes
are parallel if they do not intersect.
30
Math 30 Geometry
Page 31
Section 2.1
Postulate 10: (Parallel Postulate)
Through a point not on a line, exactly one line is parallel to the given line.
m
n
Terminology associated with parallel lines:
• A transversal is a line that intersects two or more lines
(all in the same plane) at distinct points.
The transversal is
t
1
3
2
4
5
• Two angles that lie in the same relative position are
corresponding angles
List all pairs of corresponding angles:
7
6
8
• Alternate interior angles are interior angles that have different vertices and lie on opposite
sides of the transversal.
List all pairs of alternate interior angles:
• Alternate exterior angles are exterior angles that have different vertices and lie on opposite
sides of the transversal.
List all pairs of alternate exterior angles:
Postulate 11:
If two parallel lines are cut by a transversal, the corresponding angles are congruent.
Math 30 Geometry
Page 32
Section 2.1
Example: Suppose m k n with transversal t and ]4 = 68◦
m
n
Find:
t
• ]1
1
• ]3
3
• ]5
2
4
5
• ]8
7
6
8
• ]7
Theorem 2.1.2: If two parallel lines are cut by a transversal, then alternate interior angles
are congruent.
Proof: Given: a k b with transversal t
Prove: ∠3 ∼
= ∠6
Statement
1.
Reasoning
1.
Math 30 Geometry
Page 33
Section 2.1
Theorem 2.1.3: If two parallel lines are cut by
a transversal, then alternate exterior angles are
congruent.
Theorem 2.1.4: If two parallel lines are cut by
a transversal, then interior angles on the same
side of the transversal are supplementary.
Theorem 2.1.5: If two parallel lines are cut by
a transversal, then exterior angles on the same
side of the transversal are supplementary.
Theorem 2.1.5: If two parallel lines are cut by a transversal, then exterior angles on the
same side of the transversal are supplementary.
Proof: Given: a k b with transversal t
Prove: ∠1 and ∠8 are supplementary
Statement
1.
Reasoning
1.
Math 30 Geometry
Page 34
Section 2.1
Example: Given m k n with transversal t with
m
]1 = (x + 2)(x − 7) and
]8 = (x − 2)(x − 5) − 2.
Find x
n
t
1
3
2
4
5
7
6
8
Math 30 Geometry
2.2
Page 35
Indirect Proof
Notation: P → Q means
∼ P means
Name
Symbols
Sentence
Conditional
Converse
Inverse
Contrapositive
Note:
Conditional and contrapositive statements are
Also, converse and inverse statements are either both true or both false.
Example:
• Conditional: If I play the guitar, then I am a musician.
P:
Q:
• Converse:
• Inverse:
• Contrapositive:
Section 2.2
Math 30 Geometry
Page 36
Section 2.2
Example: Assume that each of the following premises are true.
(1) If I go on a roller coaster, then I will get sick.
(2) I didn’t get sick.
Conclusion:
Law of Contraposition
Law of Detachment
1. P −→ Q
2. P
3. ∴ Q
The law of contraposition is a form of reasoning used for indirect proof. An indirect proof can
often be used to prove a a conditional statement P → Q where Q denies some claim.
For example:
(Ex 1) If alternate interior angles are not congruent, then the lines are not parallel.
(Ex 2) If the interior angles of a polygon sum to 360◦ , then the polygon is not a triangle.
Method of Indirect Proof
To prove P → Q:
Step 1:
Step 2:
Step 3:
Math 30 Geometry
Page 37
Section 2.2
Example: Prove the following statement:
If alternate interior angles are not congruent when two lines are cut by a transversal,
then the lines are not parallel.
m
P :
n
Q:
t
1
3
2
4
5
7
Example: Prove the following statement:
If ∠EAG in the figure is obtuse, then ∠1 and ∠2 are not supplementary.
P :
Q:
6
8
Math 30 Geometry
Page 38
Example: (proving a positive statement) Prove the following statement:
If plane T intersects parallel planes P and Q in lines l and m respectively,
as shown in the figure, then l k m
P :
Q:
Example: (Proving Uniqueness) Prove the following statement:
The angle bisector of an angle is unique.
Section 2.2
Math 30 Geometry
2.3
Page 39
Section 2.3
Proving Lines Parallel
Recall: If two parallel lines are cut by a transversal, then:
(a) corresponding angles are congruent (Postulate 11)
(b) alternate interior angles are congruent (Theorem 2.1.2)
(c) alternate exterior angles are congruent (Theorem 2.1.3)
(d) interior, same side angles are supplementary (Theorem 2.1.4)
(e) exterior, same side angles are supplementary (Theorem 2.1.5)
Last time: If lines are parallel, what do we know about angles?
Next: If we know something about the angles, can we know about the lines being parallel?
Theorem 2.3.1: If two lines are cut by a transversal so that corresponding angles are
congruent, then these lines are parallel.
Proof:
Math 30 Geometry
Page 40
Section 2.3
Theorem 2.3.2: If two lines are cut by a transversal so that alternate interior angles are
congruent, then these lines are parallel.
Theorem 2.3.3: If two lines are cut by a transversal so that alternate exterior angles are
congruent, then these lines are parallel.
Proof: Given:
Prove:
Example: Name the lines that must be parallel (if any), if
a. ∠1 ∼
= ∠3
b. ∠1 ∼
= ∠6
c. ∠4 ∼
= ∠6
d. If ]6 = 108◦ , then find ]2 so that r k s
Theorem 2.3.4: If two lines are cut by a transversal so that interior angles on the same
side of the transversal are supplementary, then these lines are parallel.
Math 30 Geometry
Page 41
Section 2.3
Theorem 2.3.5: If two lines are cut by a transversal so that exterior angles on the same
side of the transversal are supplementary, then these lines are parallel.
Proof: Given:
Prove:
Example: Suppose quadrilateral ABCD is a parallelogram.
Which angles are supplementary?
Theorem 2.3.6: (Transitivity of parallel for lines) If two lines are each parallel to a third
line, then these lines are parallel to each other.
Proof:
Math 30 Geometry
Page 42
Section 2.3
Theorem 2.3.7: If two coplanar lines are each perpendicular to a third line, then these
lines are parallel to each other.
Proof:
Example: Suppose ]1 = (3x − 12)◦ and ]2 = (2x + 17)◦ .
What values of x would guarantee m k n?
Math 30 Geometry
Example: Given line l and point P not on l
Construct: m through P so that m k l
Page 43
Section 2.3
Math 30 Geometry
2.4
Page 44
Section 2.4
The Angles of a triangle
Definition: A triangle is the union of three line segments that are determined by three noncollinear
points.
Notation for a triangle:
4ABC, 4BCA, 4CAB,
4ACB, 4BAC, 4CBA
A
C
Type of triangle
Scalene
Isosceles
Equilateral
Figure
B
Distinguishing feature by sides
Math 30 Geometry
Type of triangle
Page 45
Figure
Section 2.4
Distinguishing feature by angles
Acute
Obtuse
Right
Equiangular
Example: Given: 4ABC and line m through vertex B
so that m k AC
Prove: ]1 + ]2 + ]3 = 180◦
Theorem 2.4.1: In a triangle, the sum of the measures of the interior angles is 180◦ .
Math 30 Geometry
Page 46
Section 2.4
Example: In 4ABC if ]2 = 67◦ and ]4 = 129◦ , find ]1
A
1
2
4
3
B
C
D
Definition: A corollary is a theorem that follows easily (very short proof) from another theorem.
Corollary 2.4.2: Each angle of an equilateral triangle measures 60◦ .
Corollary 2.4.3: The acute angles of a right triangle are complementary.
Example: Suppose ∠C is a right angle in 4ABC. If ]A is six more than one-half the ]B, find ]A
and ]B
Corollary 2.4.4: If two angles of one triangle are congruent to
two angles of another triangle, then the third angles are also
congruent.
Definition: An exterior angle of a triangle is an angle formed by a side of the triangle and an
extension of an adjacent side.
Corollary 2.4.5: The measure of an exterior angle of a triangle
equals the sum of the measures of the two nonadjacent angles.
Math 30 Geometry
2.5
Page 47
Section 2.5
Convex Polygons
Definition: A polygon is a closed plane figure whose sides are line segments that intersect only
at the endpoints.
In a convex polygon the measure of every interior angle is between 0◦ and 180◦ .
Convex Polygons
Concave Polygons
Not Polygons
Polygons are named according how many side they have.
Name of Polygon
number of sides
Name of Polygon
number of sides
3
8
4
9
5
10
6
17
7
n
Definition: A diagonal is a line segment that joins two non-consecutive vertices of a polygon.
Math 30 Geometry
Page 48
Section 2.5
Figure
# of sides
# of diagonals
Theorem 2.5.1: The total number of diagonals D in a polygon of n sides
is given by the formula
D=
n(n − 3)
2
Example: How many diagonals are in an octagon?
Theorem 2.5.2: The sum S of the measures of the interior angles of a polygon with n sides is
given by S = (n − 2) · 180◦ . Note that n > 2 for any polygon.
Example: Find the sum of the measures of the interior angles of decagon.
Example: Find the number of sides in a polygon whose sum of its interior angles is 2880◦ .
Math 30 Geometry
Page 49
Section 2.5
Definition: An equilateral polygon
Definition: An equiangular polygon
Definition: An regular polygon is a polygon that is both equilateral and equiangular.
Corollary 2.5.3: The measure I of each interior angle of a regular polygon or equiangular
polygon of n sides is:
I=
(n − 2) · 180◦
n
Example: Find the measure of each interior angle in a regular nonagon.
Example: If an interior angle of a regular polygon is 162◦ , how many sides does the polygon have?
Example: What is the largest possible degree measure of an interior angle of a regular polygon?
Corollary 2.5.4: The sum of the four angles of a quadrilateral is 360◦ .
Math 30 Geometry
Page 50
Section 2.5
Corollary 2.5.5: The sum of the measure of the exterior angles of a regular polygon, one
at each vertex is 360◦ .
Proof:
Corollary 2.5.6: The measure E of each exterior angle of a regular polygon or equiangular
polygon of n sides is
E=
360◦
n
Definition: An polygram is a star shaped figure that results when the sides of a convex polygon
with five or more sides are extended. When the polygon is regular the polygram is regular.
Chapter 3
Triangles
3.1
Congruent Triangles
We will consider two congruent triangles, if they fit perfectly over each other.
4ABC ∼
= 4DEF if the three angles and the three sides are congruent, i.e.
∠A ∼
= DE, BC ∼
= EF , AC ∼
= DF
= ∠D, ∠B ∼
= ∠E, ∠C ∼
= ∠F ; and AB ∼
Notation: If two vertices of two triangles are corresponding, we denote this with the symbol ↔
With A ↔ A1 , B ↔ B1 , C ↔ C1 , we have 4ABC ∼
= 4A1 B1 C1
The order of the vertices in naming the two triangles has to match up with corresponding vertices.
Definition: Two triangles are congruent if the six parts (three sides and three angles) of the first
triangle are congruent to the six corresponding parts of the second triangle.
51
Math 30 Geometry
Page 52
Section 3.1
Note that congruence of triangles is an equivalence relation:
1. Reflexive:
2. Symmetric:
3. Transitive:
Construction: Construct a triangle whose sides have length of the given line segments
Math 30 Geometry
Page 53
Section 3.1
Postulate 12: Side-Side-Side (SSS)
If three sides of one triangle are congruent to three sides of a second triangle, then the triangles are
congruent.
Example:
Given: AB and CD bisect each other at M and AC ∼
= BD
∼
Prove: 4ACM = 4BDM
Math 30 Geometry
Page 54
Section 3.1
Notation:
Two sides of a triangle that meet at an angle are said to include that angle.
Two angles or a triangle that have a common side are said to include that side.
Example:
Answer the questions about 4XY Z
X
• Which angle is included by ZX and Y Z?
• Which sides include ∠X?
Z
Y
• Which is the included side for ∠Z and ∠X?
• Which angles include XY ?
Postulate 13: Side-Angle-Side (SAS)
If two sides and the included angle of one triangle are congruent to two sides and the included angle
of a second triangle, then the triangles are congruent.
Definition: Identity (or reflexive property of congruence) is the reason we cite when verifying a
segment or angle is congruent to itself.
Postulate 14: Angle-Side-Angle (ASA)
If two angles and the included side of one triangle are congruent to two angles and the included
side of a second triangle, then the triangles are congruent.
Math 30 Geometry
Page 55
Section 3.1
Theorem 3.1.1: Angle-Angle-Side (AAS) If two angles and the nonincluded side of one
triangle are congruent to two angles and the nonincluded side of a second triangle, then
the triangles are congruent.
Example:
Given: P Q ⊥ M N and ∠1 ∼
= ∠2
Prove: 4M QP ∼
= 4N QP
Math 30 Geometry
Page 56
Section 3.1
Note: Side-Side-Angle (SSA): If two sides and a nonincluded angle of one triangle are congruent
to two sides and a nonincluded angle of a second triangle, then the triangles do not have to be
congruent. There may be two non-congruent triangles with congruent SSA.
Note: Angle-Angle-Angle (AAA): If three angles of one triangle are congruent to three angles
of a second triangle, then the triangles are not necessarily congruent. They are similar.
Math 30 Geometry
3.2
Page 57
Section 3.2
Corresponding Parts of Congruent Triangles
Definition: CPCTC stands for corresponding parts of congruent triangles are congruent.
Example:
Suppose 4ABC ∼
= 4DEF by SAS, then
by CPCTC we have:
Example:
Given: AD bisects ∠BAC and AB ∼
= AC
Prove: ∠B ∼
∠C
=
Math 30 Geometry
Page 58
Section 3.2
Example:
Given: KN ∼
= LM and KL ∼
= NM
Prove: KL k N M
Right Triangles
Construction: Construct a right triangle using the given line segments as a leg and the hypotenuse.
Theorem 3.2.1: Hypotenuse-Leg (HL) If the hypotenuse and a leg of one right triangle
are congruent to the hypotenuse and a leg of a second right triangle, then the triangles
are congruent.
Math 30 Geometry
Page 59
Section 3.2
Theorem 3.2.2: (Pythagorean Theorem)
In a right triangle, with legs a and b
and hypotenuse c
a2 + b 2 = c 2
Recall the Square Root Property: Let x represent the length of a line segment and p > 0
If
Example: Find the length of each side of the right triangle given:
Example 1
a = 12, b = 5
‘
Example 2
a = 6, c = 10
Math 30 Geometry
3.3
Page 60
Section 3.3
Isosceles Triangles
Angle Bisector
Median
Perpendicular Bisector
Altitude
Math 30 Geometry
Page 61
Theorem 3.3.1: Corresponding altitudes of congruent triangles are congruent.
Definition: An isosceles triangle is a triangle that has two congruent sides.
Section 3.3
Math 30 Geometry
Page 62
Section 3.3
Theorem 3.3.2: The bisector of the vertex angle of an isosceles triangle separates
the triangle into two congruent triangles.
Theorem 3.3.3: The base angles of an isosceles triangles are congruent.
Math 30 Geometry
Page 63
Section 3.3
Example: Find the measure of each angle in isosceles 4ABC if:
a) ]2 = 62◦
b) ]3 = (2x + 5)◦ and ]1 = x◦
Theorem 3.3.4: If two angles of a triangle are congruent, then the sides opposite
these angles are congruent.
Math 30 Geometry
Page 64
Corollary 3.3.5: A triangles is equilateral if and only if it is also equiangular.
Definition: The perimeter of a triangle is the sum of the length of its sides.
P =a+b+c
Section 3.3
Math 30 Geometry
3.4
Page 65
Basic Constructions Justified
Construct: ∠ABC congruent to given angle ∠DEF
Example: Prove the construction above makes a congruent angle.
Construct: Angle bisector of ∠ABC
Example: Prove the construction above bisects the angle.
Section 3.4
Math 30 Geometry
Page 66
Section 3.4
Construct: The perpendicular bisector of AB
A
B
Example: Prove that the construction of a segment produces a perpendicular bisector.
Math 30 Geometry
Example: Construct a 45◦ angle.
Example: Construct a regular hexagon.
Page 67
Section 3.4
Math 30 Geometry
3.5
Page 68
Section 3.5
Inequalities in a Triangle
Definition: Let a and b be real numbers, then a > b if and only if there is a positive number p for
which a = b + p
Definition: A lemma is a theorem that is proved so that a later (usually more important)
theorem can be proved.
Lemma 3.5.1: If B is between A and C on AC, then AC > AB and AC > BC.
−−→
Lemma 3.5.2: If BD separates ∠ABC into two angles ∠1 and ∠2, then ]ABC > ]1 and
]ABC > ]2.
Lemma 3.5.3: If ∠4 is an exterior angle of a triangle, and ∠1 and ∠2 are nonadjacent
interior angles, then ]4 > ]1 and ]4 > ]2.
A
1
2
B
4
3
C
D
Math 30 Geometry
Page 69
Section 3.5
Lemma 3.5.4: In 4ABC, if ∠C is a right angle or an obtuse angle, then ]C > ]A and
]C > ]B.
Lemma 3.5.5: (Addition Property of Inequality) If a > b and c > d, then a + c > b + d.
Example:
Given segment A-B-C-D-E and AB > CD and BC > DE
Prove: AC > CE
Theorem 3.5.6: If one side of a triangle is longer than a second side, then the measure of
the angle opposite the longer side is greater than the measure of the angle opposite the
shorter side.
Math 30 Geometry
Page 70
Section 3.5
Theorem 3.5.7: If the measure of one angle of a triangle is greater than the measure of a
second angle, then the side opposite the larger angle is longer than the side opposite the
smaller angle.
For the indirect proof use the following fact: Given real numbers a and b, only one of the following
can be true:
a > b, or a = b, or a < b
Given:
Prove:
Proof: Assume
Example:
a) List the side of 4ABC in order from smallest to largest.
b) List the angles of 4ABC in order from smallest to largest.
Math 30 Geometry
Page 71
Section 3.5
Corollary 3.5.8-9: The perpendicular line segment from a point to a line (or plane) is the
shortest line segment that can be drawn from the point to the line (or plane).
Theorem 3.5.10: (Triangle Inequality) The length of any side of a triangle must lie between
the sum and the difference of the lengths of the other two sides.
Example: Can a triangle have the following length:
• 3, 4, and 5
• 3, 4, and 7
• 3, 4, and 8
• 3, 4, and x
Chapter 4
Quadrilaterals
4.1
Properties of Parallelograms
Note: For this class we will only consider quadrilaterals with coplanar vertices and sides.
Definition: A parallelogram is a quadrilateral in which pairs of opposite sides are parallel.
Theorem 4.1.1: A diagonal of a parallelogram separates it into two congruent triangles.
Given:
Prove:
Proof:
72
Math 30 Geometry
Page 73
Corollary 4.1.2: The opposite angles of a parallelogram are congruent.
Corollary 4.1.3: The opposite sides of a parallelogram are congruent.
Corollary 4.1.4: The diagonals of a parallelogram bisect each other.
Corollary 4.1.5: Two consecutive angles of a parallelogram are
supplementary.
Section 4.1
Math 30 Geometry
Page 74
Example: In parallelogram ABCD if ]A = 110◦ , AB = 3.2cm, and AD = 7cm find:
• ]B
• ]C
• CD
• BC
Theorem 4.1.6: Two parallel lines are equidistant everywhere.
Given:
Prove:
Proof:
Section 4.1
Math 30 Geometry
Page 75
Section 4.1
Definition: An altitude of a parallelogram is a line segment drawn from one side so that it is
perpendicular to the nonadjacent side (or to an extension of that side).
Lemma 4.1.7: If two sides of one triangle are congruent to two sides of a second triangle
and the measure of the included angle of the first triangle is greater than the measure of
the included angle of the second, then the length of the side opposite the included angle
of the first triangle is greater than the length of the side opposite the included angle of
the second.
Theorem 4.1.8: In a parallelogram with unequal pairs of consecutive angles, the longer
diagonal is opposite the obtuse angle.
Math 30 Geometry
Page 76
Example: In parallelogram ABCD:
(a) Find ]C if ]D = 72◦
(b) If ]D = 72◦ , which diagonal is longer?
(c) If ]D = (3x + 20)◦ and ]C = (8x − 5)◦ , which diagonal is shorter?
Section 4.1
Math 30 Geometry
4.2
Page 77
Section 4.2
The Parallelogram and the Kite
Theorem 4.2.1: If two sides of a quadrilateral are both parallel and congruent, then the
quadrilateral is a parallelogram.
Proof in book.
Theorem 4.2.2: If both pairs of opposite sides of a quadrilateral are congruent, then the
quadrilateral is a parallelogram.
Proof:
Given: Quadrilateral ABCD with
AB ∼
= CD and AD ∼
= BC
Prove: Quadrilateral ABCD is a parallelogram.
Theorem 4.2.3: If diagonals of a quadrilateral bisect each other, then the quadrilateral is
a parallelogram.
Math 30 Geometry
Page 78
Section 4.2
Definition: A kite is a quadrilateral with two distinct pairs of congruent adjacent sides.
Theorem 4.2.4: In a kite, one pair of opposite angles are congruent.
Math 30 Geometry
Page 79
Section 4.2
Theorem 4.2.5: The segment that joins the midpoints of two sides of a triangle is parallel
to the third side and has length equal to one-half the length of the third side.
Math 30 Geometry
Page 80
Example: In 4ABC, M and N are the midpoints of AB and AC respectively:
• If BC = 12.6, find M N .
• If M N = 3x + 2 and BC = 7x − 5, find x, M N and BC.
Section 4.2
Math 30 Geometry
4.3
Page 81
The Rectangle, the Square, and the Rhombus
Definition: A rectangle is a parallelogram that has a right angle.
Corollary 4.3.1: All angles of a rectangle are right angles.
Corollary 4.3.2: The diagonals of a rectangle are congruent.
Proof:
Definition: A square is a rectangle that has two congruent adjacent sides.
Section 4.3
Math 30 Geometry
Page 82
Corollary 4.3.3: All sides of a square are congruent.
Definition: A rhombus is a parallelogram with two congruent adjacent sides.
Corollary 4.3.4: All sides of a rhombus are congruent.
Theorem 4.3.5: The diagonals of a rhombus are perpendicular.
Section 4.3
Math 30 Geometry
Page 83
Section 4.3
The Pythagorean Theorem
In a right triangle with hypotenuse of length
c and legs of length a and b, then
c 2 = a2 + b 2
Example: In rectangle ABCD, if AB = 5 and BC = 12, find the length of a diagonal.
Example: Find the perimeter of the square shown.
Math 30 Geometry
4.4
Page 84
The Trapezoid
Definition: A trapezoid is a quadrilateral with exactly two parallel sides.
Example: In trapezoid P QRS, if ]S = 113◦ and ]R = 102◦
Find ]P and ]Q.
Section 4.4
Math 30 Geometry
Page 85
Section 4.4
Definition: An altitude of a trapezoid is a line segment from one vertex of one base of the trapezoid
perpendicular to the opposite base (or its extension).
The length of the altitude is the height of the trapezoid.
Theorem 4.4.1: The base angles of an isosceles trapezoid are congruent.
Corollary 4.4.2: The diagonals of an isosceles trapezoid are congruent.
Math 30 Geometry
Page 86
Example: Given isosceles trapezoid ABCD with AB k DC
a) Find ]B and ]D given that ]A = 3y − 12 and ]C = 5y − 8
b) Find the length of each diagonal if AC = 4x − 3 and BD = 3x + 5
Theorem 4.4.3: The length of the median of a trapezoid equals
one-half the sum of the length of the two bases.
Example: Quadrilateral ABCD is a trapezoid with AB k DC and median M N .
If AB = 2x + 3 and DC = 4x − 8 and M N = 21.5, find AB and CD
Section 4.4
Math 30 Geometry
Page 87
Section 4.4
Theorem 4.4.4: The median of a trapezoid is parallel to each base.
Theorem 4.4.5: If the two base angles of a trapezoid are congruent, then the trapezoid is
an isosceles trapezoid.
Theorem 4.4.6: If the diagonals of a trapezoid are congruent, then the trapezoid is an
isosceles trapezoid.
Theorem 4.4.7: If three (or more) parallel lines intercept congruent line segments on one
transversal, then they intercept congruent line segments on any transversal.
Chapter 5
Similar Triangles
5.1
Ratios, Rates, and Proportions
Definition:
a
• A ratio is a quotient with b 6= 0 that provides a comparison
b
between two quantities a and b.
• Two quantities are commensurable if they can be converted into the same units.
• Two quantities are incommensurable if no common unit of measure is possible.
• A rate is a quotient that compares two quantities that are incommensurable.
• A proportion is a statement that two ratios or two rates are equal
88
a
c
= .
b
d
Math 30 Geometry
Page 89
Example: Find the best form of each ratio or rate:
(a) 18 to 12
(b) 15 inches to 45 inches
(c) 24 in. to 3 ft.
(d) 7m to 21m
(e) 320 miles to 8 gallons
(f) 235 ft to 5 sec
(g) $43.08 to 12 gallons
Section 5.1
Math 30 Geometry
Page 90
Section 5.1
Property 5.1.1: (Means-Extremes Property) In a proportion the product of the
a
c
means equals the product of the extremes; that is, if = (where b 6= 0 and d 6= 0)
b
d
then a · d = b · c.
Example: Use the Means-Extremes Property to solve each proportion for x
(a)
x
8
=
7
9
(b)
x+2
3
=
7
x−2
Example: If a car can travel 150 miles on 6 gallons of gas, how far can it travel on 20 gallons of gas?
Math 30 Geometry
Page 91
Definition: The geometric mean of a and c is a number b such that
Section 5.1
a
b
= .
b
c
Example: In 4ABC, AD is the geometric mean
of BD and DC. If BC = 13 and BD = 3, find AD.
Definition: An extended ratio compares more than two quantities and is expressed in the form
a : b : c : d. Unknown quantities in the ratio a : b : c : d can be represented by ax, bx, cx, and dx.
Example: Suppose the perimeter of a pentagon is 240 cm and the length of the sides are in the
extended ratio 2 : 3 : 4 : 5 : 6. Find the length of each side.
Example: The measure of two supplementary angles are in a ratio of 2 : 3. Find the measure of
each angle.
Math 30 Geometry
Page 92
Section 5.1
Property 5.1.2: In a proportion the means or the
extremes (or both) may be exchanged; that is, if
a
c
= (where a, b, c, and d are nonzero) then
b
d
a
b d
c
d
b
= , = , and = .
c
d b
a
c
a
a
c
Property 5.1.3: If = (where b 6= 0 and d 6= 0)
b
d
then
c+d
a−b
c−d
a+b
=
and
=
.
b
d
b
d
Definition: An extended proportion equates more than two ratios and is expressed in the form
c
e
a
= =
b
d
f
Example: In the triangles shown
AB
AC
BC
=
=
DE
DF
EF
Find DF and DE
Math 30 Geometry
5.2
Page 93
Section 5.2
Similar Polygons
Definition: Two geometric figures are similar if they have exactly the same shape (but maybe
different size). The symbol for “is similar to” is ∼
Notation:
• = “equals to”
• ∼
= “congruent to”
• ∼ “similar to”
When two figures have the same shape, i.e. they are similar (∼), and all corresponding parts have
equal (=) measures, then the two figures are congruent (∼
=).
Suppose that 4ABC ∼ 4DEF , then
Corresponding
angles
Corresponding
vertices
Corresponding
sides
Math 30 Geometry
Page 94
Section 5.2
Definition: Two polygons are similar if and only if the following two conditions are satisfied:
(1) all pairs of corresponding angles are congruent.
(2) all pairs of corresponding sides are proportional.
Example: Which of the following figures must be similar?
• Any two squares.
• Any two right triangles.
• Any two pentagons.
• Any two regular pentagons.
Example: If ABCD ∼ EF GH as shown.
Find F G, GH, and EH.
Math 30 Geometry
Page 95
Section 5.2
Example: Suppose 4ABC ∼ 4ADE.
If DE = 5, AD = 8, and DB = BC, find AB.
Example: At a certain time of day Kenny’s shadow is 4ft. long. A nearby tree casts a shadow 18
ft. long at the same time. If Kenny is 6 ft. tall. How tall is the tree?
Math 30 Geometry
5.3
Page 96
Section 5.3
Proving Triangles Similar
Postulate 15: (AAA)
If three angles of one triangle are congruent to three angles of another triangle, then the two triangles are similar.
Corollary 5.3.1: (AA) If two angles of one triangle are congruent to two angles of another
triangle, then the two triangles are similar.
Example:
Given: m k BC with transversal AB and AC
Prove: 4ADE ∼ 4ABC
CSSTP: Corresponding sides of similar triangles are proportional.
CASTC: Corresponding angles of similar triangles are congruent.
Example:
Given: DE k BC with transversal BE and CD
DE
BC
Prove:
=
DA
AC
Math 30 Geometry
Page 97
Section 5.3
Theorem 5.3.2: The lengths of corresponding altitudes
of similar triangles have the same ratio as the lengths of
any pair of corresponding sides.
Lemma 5.3.5: If a line segment divides two sides of a triangle proportionally, then this line
segment is parallel to the third side.
Theorem 5.3.3: (SAS∼) If an angle of one triangle is
congruent to an angle of a second triangle and the pairs
of sides including the angles are proportional, then the
triangles are similar. (This is not the same as SAS.)
Theorem 5.3.4: (SSS∼) If the three sides of one triangle are proportional to the three sides of
a second triangle, then the triangles are similar. (This is not the same as SSS).
Math 30 Geometry
Page 98
Section 5.3
Example: Under the given conditions,which method establishes that 4ABC ∼ 4DEF ?
a) AB = 7, AC = 9, BC = 12, DE = 21,
DF = 27, and EF = 36
b) ∠A ∼
= ∠D, DF = 18, DE = 10,
1
AC = 6, and AB = 3
3
Lemma 5.3.5: If a line segment divides two sides of a triangle proportionally, then this line
segment is parallel to the third side.
Math 30 Geometry
5.4
Page 99
The Pythagorean Theorem
Which of the following triangles are similar?
Theorem 5.4.1: If 4ABC is a right triangle with
altitude AD as shown, then
4ABC ∼ 4DBA ∼ 4DAC.
Section 5.4
Math 30 Geometry
Page 100
Section 5.4
Theorem 5.4.2: The length of the altitude to the hypotenuse
of a right triangle is the geometric mean of the lengths of the
segments of the hypotenuse.
Lemma 5.4.3: The length of each leg of a right triangle is
the geometric mean of the hypotenuse and the length of the
segment of the hypotenuse adjacent to that leg.
Theorem 5.4.4: The Pythagorean Theorem
The square of the length of the hypotenuse
of a right triangle is equal to the sum of the
squares of the legs.
Given: 4ABC with right angle C
Prove: c2 = a2 + b2
Theorem 5.4.5: (Converse of the Pythagorean Theorem) If a, b and c are the lengths of the
three sides of a triangle with c the longest side, and if c2 = a2 + b2 then the triangle is a right
triangle with the right angle opposite the side of length c.
Math 30 Geometry
Page 101
Section 5.4
Example: Which of the following could be the lengths of the sides of a right triangle?
a) a = 6, b = 8, and c = 10
√
b) a = 2 3, b = 4, and c = 7
Example: A rhombus with sides of length 14 cm has a diagonal that is 6 cm long.
Find the length of the other diagonal.
Definition: A Pythagorean Triple is a set of three natural numbers (a, b, c) for which c2 = a2 + b2
Generating Pythagorean Triples:
p
q
p2 − q 2
2pq
p2 + q 2
(a, b, c)
Math 30 Geometry
Page 102
Section 5.4
Theorem 5.4.7: Let a, b and c be the lengths of the sides of a triangle with c the longest side, then
a) If c2 > a2 + b2 then the triangle is obtuse.
b) If c2 < a2 + b2 then the triangle is acute.
Example: Determine the type of triangle given the lengths of its sides.
(a) a = 10, b = 12, and c = 16
(b) a = 5, b = 7, and c = 8
(c) a = 1.5, b = 2, and c = 2.5
Math 30 Geometry
5.5
Page 103
Section 5.5
Special Right Triangles
Definition: The square root of a number a > 0, denoted as
√
( a)2 = a
√
a is a positive number for which
Properties of square roots:
• For a ≥ 0 and b ≥ 0,
√
√ √
a · b = ab
r
• For a ≥ 0 and b > 0,
√
a
a
= √
b
b
The 45◦ - 45◦ -90◦ Triangle
Theorem 5.5.1: In a triangle whose angles have
◦
◦
degree measures 45◦ , 45
√ , and 90 the length of
the hypotenuse equals 2 times either leg.
Example: Find the lengths of the missing sides of each triangle.
√
Theorem 5.5.3: If the length of the hypotenuse of a right triangle equals the product of 2 and
the length of one leg, then the angles of the triangle have degree measures 45◦ , 45◦ , and 90◦
Math 30 Geometry
Page 104
Section 5.5
The 30◦ - 60◦ - 90◦ Triangle
Theorem 5.5.2: In a triangle whose angles have
degree measures 30◦ , 60◦ , and 90◦ the length of the
hypotenuse equals twice the length of the√shorter
leg and the length of the longer leg equals 3 times
the length of the shorter leg.
Example: Find the lengths of the missing sides of each triangle.
Theorem 5.5.4: If the length of the hypotenuse of a right triangle equals twice the length of
one leg of the triangle, then the angle opposite that leg measures 30◦ .
Math 30 Geometry
5.6
Page 105
Segments Divided Proportionally
Example:
Suppose B and E divide AC and AD proportionally.
If AB = 7, AE = 11, BC = 3.5, find DE
Note: A property of fractions: If
a
c
a+c
a
c
= , then
= =
b
d
b+d
b
d
Example:
Given: AD and EH are divided proportionally.
Prove:
BD
AC
=
FH
EG
Section 5.6
Math 30 Geometry
Recall Property #3, Section 5.3: If
Page 106
a
c
a−b
c−d
a+b
c+d
= , then
=
and
=
b
d
b
d
b
d
Theorem 5.6.1: If a line is parallel to one side of a triangle and
intersects the other two sides, then it divides theses sides proportionally.
←→
Given: 4ABC with DE k BC,
←→
DE intersecting AB at D and AC at E.
Prove:
AD
AE
=
DB
EC
Corollary 5.6.2: When three (or more) parallel
lines are cut by a pair of transversals, then the
transversals are divided proportionally by the parallel lines.
Section 5.6
Math 30 Geometry
Page 107
Example:
Given: m1 k m2 k m3 k m4 with transversals t and s,
AB = 7, BC = 5, CD = 4, and EF = 6
Find F G, GH, and EH
Theorem 5.6.3: If a ray bisects one angle of a triangle, then it divides the opposite side into segments
whose lengths are proportional to the lengths of
the two sides that form the bisected angle.
Example:
−−→
Given: AD bisects ∠BAC, AB = 9, AC = 12, and BC = 9
Find BD
Section 5.6
Math 30 Geometry
Page 108
Theorem 5.6.4: (Ceva’s Theorem)
Let point D be any point in the interior of 4BAC, and let AE, BF , and
CG be the line segments determined by D and the vertices of 4ABC.
Then the product of the ratios of the lengths of the segments of each of the
three sides (taken in order from a given vertex of the triangle) equals 1; that is:
AG BE CF
·
·
=1
GB EC F A
Section 5.6
Chapter 6
Circles
6.1
Circles and Related Segments and Angles
Definition: A circle is the set of all points in a plane that are at a fixed distance from a given
point, known as the center of the circle.
109
Math 30 Geometry
Page 110
Definition: Congruent circles are two or more circles that have congruent radii.
Definition: Concentric circles are coplanar circles that have
the same center.
J
Definition: In
A the set of points on the circle
>
from B to C is an arc denoted as BC.
Definition: A central angle of a circle is an angle
whose vertex is the center of a circle and whose sides
are radii of the circle.
Section 6.1
Math 30 Geometry
Page 111
Section 6.1
Theorem 6.1.1: A radius that is perpendicular to
a cord bisects the cord.
Postulate 16: (Central Angle Postulate)
In a circle the degree measure of a central angle is equal to the degree measure of its intercept arc.
Math 30 Geometry
Page 112
Section 6.1
Definition: In a circle or congruent circles, congruent arcs are arcs with equal measures.
Postulate 17: (Arc Addition Postulate)
>
>
>
>
>
If AB and BC intersect only at point B, then mAB + mBC = mABC.
Math 30 Geometry
Page 113
J
Example: Given ST = 21 (SR) in
Q,
◦
SR is a diameter, and ]U QR = 30 ,
find each of the following:
>
a. mUR
>
d. mST
>
b. mUSR
>
e. mTR
c. ]S
>
f. mSTU
Definition: An inscribed angle of a circle is an angle
whose vertex is a point on the circle and whose sides
are chords of the circle.
Section 6.1
Math 30 Geometry
Page 114
Theorem 6.1.2: The measure of an inscribed angle of a
circle is one-half the measure of its intercepted arc.
Theorem 6.1.3: In a circle (or in congruent circles), congruent minor arcs have congruent central angles.
Theorem 6.1.4: In a circle (or in congruent circles), congruent central angles have congruent arcs.
Theorem 6.1.5: In a circle (or in congruent circles), congruent chords have congruent minor (major) arcs.
Theorem 6.1.6: In a circle (or in congruent circles), congruent arcs have congruent chords.
Section 6.1
Math 30 Geometry
Page 115
Theorem 6.1.7: Chords that are the same distance from
the center of a circle are congruent.
Theorem 6.1.8: Congruent chords are located at the
same distance from the center of the circle.
Theorem 6.1.9: An angle inscribed in a semicircle is a
right angle.
Theorem 6.1.10: If two inscribed angles intercept the
same arc, then these angles are congruent.
Section 6.1
Math 30 Geometry
6.2
Page 116
More Angle Measures in the Circle
Definition: A tangent is a line that intersects a circle
at exactly one point; the point of intersection is the
point of contact, or the point of tangency.
Definition: A secant is a line (or segment or ray) that
intersects a circle at exactly two points.
Definition: A polygon is inscribed in a circle if
its vertices are points on the circle and its sides are
chords of the circle. Equivalently, the circle is said to be
circumscribed about the polygon. A polygon
that can be inscribed in a circle is called a cyclic polygon.
Theorem 6.2.1: The opposite angles of a cyclic quadrilateral are supplementary.
Section 6.2
Math 30 Geometry
Page 117
Definition: A polygon is circumscribed about a circle if
all the sides of the polygon are line segments tangent to
the circle; also, the circle is said to be inscribed in the polygon.
Theorem 6.2.3: The radius (or any other line through
the center of a circle) drawn to a tangent at the point of
tangency is perpendicular to the tangent at that point.
Example: In the
J figure shown, suppose AC
is tangent
to
B at C. If AD = 5 and the radius
J
of
B is 12, find AC.
Section 6.2
Vertex of angle is on
the circle
Corollary 6.2.4: The
measure of an angle
formed by a tangent
and a chord drawn to
the point of tangency
is one-half the
measure of the
intercepted arc.
Vertex of angles is in
the interior of the
circle
Theorem 6.2.2: The
measure of an angle
formed by two chords
that intersect within
a circle is one-half the
sum of the measures
of the arcs intercepted
by the angle and its
vertical angle.
Vertex of angle is at
the center of the circle
Postulate 16: In a
circle the degree
measure of a central
angle is equal to the
degree measure of its
intercept arc.
Page 118
Theorem 6.2.5: The measure of an angle formed
when two secants intersect at a point outside the
circle is one-half the difference of the measures of
the two intercepted arcs.
Theorem 6.2.6: If an angle is formed by a secant
and a tangent that intersect in the exterior of a
circle, then the measure of the angle is one-half
the difference of the measures of its intercepted
arcs.
Theorem 6.2.7: If an angle is formed by two
intersecting tangents, then the measure of the
angle is one-half the difference of the measures of
the intercepted arcs.
Vertex of angles is in the exterior of the circle
Angle Measurements Related to the circle
Math 30 Geometry
Section 6.2
Math 30 Geometry
Page 119
Example: Use the figure to answer each question:
>
>
a. Find ]2 if mAC = 107◦ and mBD = 81◦
>
>
b. If ]1 = 105◦ and mBD = 76◦ , find mAC
>
>
Example: If ]G = (3x + 4)◦ , mEI = (61 − x)◦ , mFH = 18◦ ,
>
find mEI
Section 6.2
Math 30 Geometry
Page 120
Theorem 6.2.8: If two parallel lines intersect a circle,
the intersected arcs between these lines are congruent.
Section 6.2
Math 30 Geometry
6.3
Page 121
Line and Segment relationships in the Circle
Theorem 6.3.1: If a line is drawn through the center of a
circle perpendicular to a chord, then it bisects the chord
and its arc.
Theorem 6.3.2: If a line through the center of a circle
bisects a chord other than a diameter, then it is perpendicular to a chord.
Theorem 6.3.3: The perpendicular bisector of a chord
contains the center of the circle.
Section 6.3
Math 30 Geometry
Page 122
Definition: Two circles that touch at only one point are called tangent circles.
Definition: For two circles with different centers, the
line of centers is the line (or line segment)
containing the centers of both circles.
Definition: A common tangent is a line tangent to more than one circle.
Section 6.3
Math 30 Geometry
Page 123
Theorem 6.3.4: The tangent segments to a circle from
an external point are congruent.
Theorem 6.3.5: If two chords intersect within a circle,
then the product of the lengths of the segments (parts)
of one chord is equal to the product of the lengths of the
segments (parts) of the other chord.
Example:
In the figure above, suppose AE = 2x − 3, EC = x − 1, DE = x, and EB = x − 1.
Find AE and EB.
Section 6.3
Math 30 Geometry
Page 124
Theorem 6.3.6: If two secant segments are drawn to a
circle from an exterior point, then the products of the
lengths of each secant with its external segment (part)
are equal.
Theorem 6.3.7: If a tangent segment and a secant segment are drawn to a circle from an exterior point, then
the square of the length of the tangent equals the product of the length of the secant with the length of its
external segment (part).
Example:
If AB = 7, AD = 11, and AC = 12.
Find EC and F D.
Section 6.3
Math 30 Geometry
6.4
Page 125
Section 6.4
Some Constructions and Inequalities in the Circle
Theorem 6.4.1: The line that is perpendicular to the
radius of a circle at its endpoint on the circle is tangent
to the circle.
Example:
J
J
Given
O with point A on
O
J
←−→
Construct XW tangent to
O at A.
Example:
J
Given
O and external point E
←
→
Construct Tangent ET with point
of tangency T .
Math 30 Geometry
Page 126
Theorem 6.4.2: In a circle (or congruent circles) containing two unequal central angles, the larger angle corresponds to the larger intercepted arc.
Theorem 6.4.3: In a circle (or congruent circles) containing two unequal arcs, the larger arc corresponds to
the larger central angle.
Theorem 6.4.4: In a circle (or congruent circles) containing two unequal chords, the shorter chord is at a
greater distance from the center.
Theorem 6.4.5: In a circle (or congruent circles) containing two unequal chords, the chord nearer the center
of the circle has the greater length.
Theorem 6.4.6: In a circle (or congruent circles) containing two unequal chords, the longer chord corresponds to
the greater minor arc.
Theorem 6.4.7: In a circle (or congruent circles) containing two unequal minor arcs, the greater minor arc
corresponds to the longer of the chords related to these
arcs.
Section 6.4
Chapter 7
Locus and Concurrance
7.1
Locus of Points
Definition: A locus is the set of all points and only those points that satisfy a given condition (or
set of conditions).
Example:
Find the locus of points that are equidistant
from a fixed point in a plane.
Example:
Find the locus of points that are equidistant from
two fixed points in a plane. (Theorem 7.1.2)
127
Math 30 Geometry
Page 128
Section 7.1
Example:
Find the locus of points that are equidistant from
the sides of an angle in a plane. (Theorem 7.1.1)
When proving a locus theorem, two statements must be verified:
1. If a point is in the locus, then it satisfies the conditions.
2. If a point satisfies the condition, then it is a point of the locus.
Theorem 7.1.1: The locus of points in a plane and equidistant from the
sides of an angle is the angle bisector.
Proof:
Part 1: If a point is on the angle bisector, then it is equidistant from the sides of the angle.
Math 30 Geometry
Page 129
Section 7.1
Part 2: If a point is equidistant from the sides of the angle, then it is on the angle bisector.
Theorem 7.1.2: The locus of points in a plane that are equidistant from the endpoints
of a line segment is the perpendicular bisector of that line segment.
Math 30 Geometry
Page 130
Example:
Construct a rhombus ABCD given the diagonals AC and BD
Section 7.1
Math 30 Geometry
Page 131
Example:
Construct an isosceles right triangle that has a hypotenuse of AB
A
B
Section 7.1
Math 30 Geometry
7.2
Page 132
Section 7.2
Concurrence of Lines
Definition: Two or more lines are concurrent if they have exactly one point in common.
Incenter
Circumcenter
Theorem 7.2.1: The three angle bisectors of
the angles of a triangle are concurrent.
Theorem 7.2.2: The three perpendicular bisectors
of the sides of a triangle are concurrent.
Orthocenter
Centroid
Theorem 7.2.3: The three altitudes of a
triangle are concurrent.
Theorem 7.2.4: The three medians of the sides of
a triangle are concurrent at a point that is
two-thirds the distance from any vertex to the
midpoint of the opposite side.
Math 30 Geometry
Page 133
Section 7.2
Theorem 7.2.4: The three medians of the sides of a triangle are concurrent at a point
that is two-thirds the distance from any vertex to the midpoint of the opposite side.
Example:
In isosceles 4RST , RS = RT = 10 and ST = 16.
Medians RZ, T X, and SY are concurrent at
centroid Q. Find each of the following:
1. SZ
4. QZ
2. RZ
5. SQ
3. RQ
6. SY
Math 30 Geometry
7.3
Page 134
More About Regular Polygons
Example:
J
Given regular pentagon ABCDE, construct inscribed
O.
Example:
J
Given regular octagon F GHIJKLM , construct circumscribed
Q.
Section 7.3
Math 30 Geometry
Page 135
Section 7.3
Theorem 7.3.1: A circle can be circumscribed about (or inscribed in) any regular polygon.
Definition: A center of a regular polygon is
the common center for the inscribed and circumscribed
circles of the polygon.
Definition: A radius of a regular polygon is
any line segment that joins the center of the polygon
to any one of its vertices.
Definition: An apothem of a regular polygon is
any line segment from the center of the polygon
perpendicular to one of its sides.
Definition: A central angle of a regular polygon is
an angle formed by two consecutive radii of the polygon.
Theorem 7.3.2: The measure of a central angle of a regular polygon with n sides is given by:
c=
360
n
Example: Find:
1. The measure of the central angle of a regular polygon with 15 sides.
2. The number of sides of a regular polygon whose central angle measures 30◦ .
Math 30 Geometry
Page 136
Theorem 7.3.3: Any radius of a regular polygon bisects
the angle at the vertex to which it is drawn.
Theorem 7.3.4: Any apothem of a regular polygon
bisects the side of the polygon to which it is drawn.
Example: Given that each side of a regular hexagon is 10 cm, find
the length of the radius and the apothem.
Section 7.3
Chapter 8
Area
8.1
Area and Initial Postulates
Definition: A region is a closed or bounded portion of the plane.
Area is used to measure a region. The units are square units.
Postulate 18: (Area Postulate)
Corresponding to every bounded region is a unique positive number A, known as the area of that
region.
Postulate 19:
If two closed plane figures are congruent, then their areas are equal.
Postulate 20: (Area-Addition Postulate)
Let R and S be two enclosed regions that do not overlap. Then AR∪S = AR + AS .
137
Math 30 Geometry
Figure
Rectangle
Square
Parallelogram
Triangle
Right Triangle
Page 138
Area Formula
Section 8.1
Math 30 Geometry
Page 139
Example: Find the area of the figure shown
Example: In parallelogram M N P Q, QP = 12 and QM = 9.
The length of altitudes QR (to side M N ) is 6
Find the length of altitude QS from Q to P N .
Section 8.1
Math 30 Geometry
8.2
Page 140
Section 8.2
Perimeter and Area of Polygons
Definition: The perimeter of a polygon is the sum of the lengths of all sides of the polygon.
Example: Find the perimeter of the figure shown.
Math 30 Geometry
Page 141
Section 8.2
Theorem 8.2.1: Heron’s Formula If the three sides of a triangle have lengths a, b, and c,
then the area A of the triangle is given by
A=
p
s(s − a)(s − b)(s − c)
where s = 12 (a + b + c) is called the semiperimeter .
Example: Find the area of the triangle shown.
Theorem 8.2.2: Brahmagupta’s Formula For a cyclic
quadrilateral with sides of lengths a, b, c, and d, the area A
of the quadrilateral is given by
A=
p
(s − a)(s − b)(s − c)(s − d)
where s = 21 (a + b + c + d) is called the semiperimeter .
Math 30 Geometry
Page 142
Theorem 8.2.3: The area A of a trapezoid whose
bases have lengths b1 and b2 , and whose altitude
has length h is given by
1
A = h (b1 + b2 ) .
2
Theorem 8.2.4: The area A of any quadrilateral
with perpendicular diagonals of lengths d1 and d2
is given by
1
A = d1 d2 .
2
Section 8.2
Math 30 Geometry
Page 143
Section 8.2
Because the diagonals of a kite and a rhombus are perpendicular, we can find the area of each using
A = 12 d1 d2 .
Example: Find the area of the kite shown.
Theorem 8.2.7: The ratio of the areas of two
similar triangles equals the square of the ratio
of the lengths of any two corresponding sides;
that is,
2 2 2
A1
a1
b1
c1
=
=
=
A2
a2
b2
c2
This theorem can be extended to any pair of
similar polygons.
Math 30 Geometry
Page 144
Section 8.2
Example: Find the ratio of areas of two similar rectangles if:
a) The ratio of corresponding sides is
s1
s2
=
2
5
b) The length of the first rectangle is 6 m and the length of the second rectangle is 4 m.
Math 30 Geometry
Page 145
Proof of the Pythagorean Thm by 20th USA President James A. Garfield.
Section 8.2
Math 30 Geometry
8.3
Page 146
Section 8.3
Regular Polygons and Area
Recall:
• The
of a regular polygon is the common center for the inscribed and circumscribed circles of the polygon.
of a regular polygon is any line segment that joins the center of the polygon
• A
to one of its vertices.
• An
of a regular polygon is any line segment drawn from the center of the
polygon perpendicular to one of its sides.
• A
radii.
of a regular polygon is any angle formed by two consecutive
• 7.3.3: Any radius of a regular polygon bisects the angle at the vertex to which it is drawn.
• 7.3.4: Any apothem of a regular polygon bisects the side of the polygon to which it is drawn.
Example: Find the area of an equilateral triangle in which each side has length 7 inches.
Example: Find the area of a square with apothem of length 5 meters.
Math 30 Geometry
Page 147
Section 8.3
Example: Find the area of an equilateral triangle in which the length of an apothem is 9 cm.
Example: Find the area of a regular polygon:
Math 30 Geometry
Page 148
Section 8.3
Theorem 8.3.1: The area A of a regular polygon with perimeter P and apothem
of length a is given by
1
A = aP
2
.
Example: Use A = 21 aP to find the area of a square with apothem of length 5 meters.
Example: Use A = 21 aP to find the area of an equilateral triangle with apothem of length 9 cm.
Example: Find the area of a regular pentagon with apothem of length a = 9.8 in and each side of
length s = 8.1 in
Math 30 Geometry
8.4
Page 149
Section 8.4
Circumference and Area of a Circle
Definition: The number π is the ratio between the circumference C and the diameter length d of
any circle. That is:
C
π= .
d
π ≈ 3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 . . .
Theorem 8.4.1: The circumference of a circle is given by
C = 2πr
Example: In
J
or C = πd
Q, if QD = 21 cm:
a) then the approximate the circumference using π ≈
>
b) the approximate length of DE
22
.
7
Math 30 Geometry
Page 150
Section 8.4
Degree measure vs. Arc Length
Theorem 8.4.2: In a circle with circumference C, the length ` of an arc whose degree measure
is m is given by
m
`=
·C
360
>
>
AB · C
So for each arc AB, we have ` = m360
>
Example: Find the exact length of minor RS if OS = 5 m
Math 30 Geometry
Page 151
Definition: A limit is a number that a sequence of numbers approaches.
What number does the following sequence approach?
1 1 1 1 1 1 1 1 1 1
, , , , , , , , ,
, ···
1 2 3 4 5 6 7 8 9 10
Example: Find the upper and lower limits of
the length of a chord in a circle or radius 7 cm.
Theorem 8.4.3: The area A of a circle whose radius has length r is given by
A = πr2
Section 8.4
Math 30 Geometry
Page 152
Section 8.4
A = πr2
Example: Find the exact area, and the approximate area of a circle whose radius has length 5 in.
(Use π ≈ 3.14)
Note: For this class, all solutions should be in terms of π, unless otherwise stated.
Math 30 Geometry
8.5
Page 153
Section 8.5
More Area Relationships in a Circle
Definition: A sector of a circle is a region bounded by two radii of the circle and an arc intercepted
by those two radii.
Theorem 8.5.1: In a circle with radius r, the area A of a sector whose arc or central angle
has degree measure m is given by
m
A=
· πr2
360
Example:
For both find the exact value and use your calculator value of π to approximate each answer to the
nearest hundredth.
a) Find the area of the sector shown.
b) Find the perimeter of the sector shown.
Math 30 Geometry
Page 154
Example: Suppose each vertex of the square shown
is the center of a circle. The circles are congruent and
tangent to each other on the square at the midpoint of
each side as shown. Find the area of the shaded region.
The sides of the square measure 10 cm.
Definition: A segment of a circle is a region bounded
by a chord and its minor (or major) arc.
Section 8.5
Math 30 Geometry
Page 155
Example:
J
>
In
Q of radius 16 inches with chord AB and mAB = 90◦ find:
a) the exact area of the segment.
b) the exact perimeter of the segment.
Theorem 8.5.3: Let P represent the perimeter of a triangle
and r represent the length of the radius of its inscribed
circle, the area of the triangle is given by
1
A = rP
2
Picture Proof:
Section 8.5
Chapter 9
Surfaces and Solids
9.1
Prisms, Area, and Volume
Definition: A polyhedron is a three-dimensional solid which consists of a collection of polygons,
usually joint at their edges.
Definition: A prism is a polyhedron consisting of two parallel, congruent faces called the bases,
whose vertices are connected by line segments.
Definition:
• The bases are two congruent parallel faces.
• The base edges sides of the base figures.
• The lateral edges connect corresponding vertices.
• The lateral faces are quadrilaterals formed by lateral edges.
• The vertices are vertices of the bases.
• The altitude of a prism is the distance between parallel planes.
156
Math 30 Geometry
Example:
Page 157
Section 9.1
Math 30 Geometry
Page 158
Section 9.1
Definition: A right prism is a prism in which the lateral edges are perpendicular to the base edges
at their points of intersection.
Definition: An oblique prism (aka slanted) is a prism in which the parallel lateral edges are oblique
to the base edges at their points of intersection.
Example: Name each type of prism:
Definition: The lateral area L of a prism is the sum of the areas of all lateral faces.
Theorem 9.1.1: The lateral area L of any prism whose altitude has measure h and whose
base has perimeter P is given by
L = hP
Definition: For any prism the total area T is the sum of the lateral area and the areas of the bases.
Theorem 9.1.2: The total area T of any prism with lateral area L and
base area B is given by
T = L + 2B
Example: Find the total area of the right triangular prism shown:
Math 30 Geometry
Page 159
Section 9.1
Definition: A regular prism is a right prism whose bases are regular polygons.
Example: Find the total area of the regular hexagonal prism shown:
Definition: The volume of a solid is a number that represents the amount of space enclosed by a
solid.
Note: Volume is measured in cubic units.
Postulate 24: (Volume Postulate)
Corresponding to every solid is a unique positive number V , known as the volume of that solid.
Postulate 25:
The volume of a right rectangular prism is given by
V = lwh
where l measures the length, w the width, and h the altitude of the prism.
Example: Find the volume of the rectangular box (right rectangular prism) shown:
Math 30 Geometry
Page 160
Postulate 26:
The volume of a right prism is given by
V = Bh
where B is the area of the base, and h is the length of the altitude of the prism.
Example: Find the volume of the regular hexagonal prism shown:
Section 9.1
Math 30 Geometry
9.2
Page 161
Section 9.2
Pyramids, Area, and Volume
Definition: A pyramid is a three-dimensional solid which is formed by joining the vertices of a
planar base and a non-coplanar point.
Definition:
• The base is the figure in the plane (polygon).
• The base edges are the sides of the base.
• The lateral edges are from a vertex of the base to the non-coplanar point.
• The lateral faces are triangles formed by a base edge and vertex P .
• The vertex or appex of the pyramid is the non-coplanar point (P ), the tip.
• The altitude of a pyramid is the perpendicular distance from P to the plane of the base,
altitude h.
Definition: A regular pyramid is a pyramid whose base is regular polygon and whose lateral edges
are congruent.
Definition: The slant height of a regular pyramid is the altitude from the vertex of the pyramid
to the base of any of the congruent lateral faces.
Math 30 Geometry
Page 162
Section 9.2
Example: Find the length of the slant height l of the regular square pyramid with altitude 4 inches
and base length 6 inches.
Theorem 9.2.1: In a regular pyramid, with altitude h, slant
height l and apothem of the base a
l2 = a2 + h2
Theorem 9.2.2: The lateral area L of a regular pyramid
with slant height of length l and perimeter P of the base
is given by
1
L = lP
2
Example: Find the lateral area of a regular hexagonal pyramid if the sides of the base measure
10 cm and the lateral edges measure 17 cm each.
Math 30 Geometry
Page 163
Theorem 9.2.3: The total surface area T of a pyramid with
lateral area L and base area B is given by
T =L+B
Example: Find the total area of the regular pyramid shown.
Theorem 9.2.4: The volume V of a pyramid with base area
B and altitude of length h is given by
1
V = Bh
3
Example: Find the volume of the pyramid shown.
Section 9.2
Math 30 Geometry
Page 164
Theorem 9.2.5: In a regular pyramid the length of the
altitude h, the radius r of the base and the length of the
lateral edge e are related by
e2 = h2 + r2
Example: Find the volume of the pyramid shown.
Section 9.2
Math 30 Geometry
9.3
Page 165
Section 9.3
Cylinders and Cones
Definition: A cylinder is a three-dimensional solid which is formed by joining two congruent circles
in parallel planes.
Definition: The line segment joining the centers of the two circular bases is known as the axis of
the cylinder.
Theorem 9.3.1: The lateral area L of a right cylinder with altitude of length h and
circumference C of the base, or with radius r of the base is given by
L = hC
= 2πrh.
Theorem 9.3.2: The total area T of a right cylinder with base area B and lateral area
L, or with altitude of length h and radius r of the base is given by
T = L + 2B
= 2πrh + 2πr2 .
Example: Find the exact lateral area and the exact total area of the cylinder shown.
Math 30 Geometry
Page 166
Section 9.3
Theorem 9.3.3: The volume V of a right cylinder with base area B and altitude of
length h and radius r of the base is given by
V = Bh
= πr2 h.
Example: Find the approximate volume of the cylinder shown (use
22
7
for π).
Example: In a right circular cylinder, suppose the volume is 9π in3 and the diameter of the base
is 43 the length of the altitude, find the dimensions of the cylinder.
Math 30 Geometry
Page 167
Section 9.3
Cones
Theorem 9.3.4: The lateral area L of a right circular cone with slant height l and
circumference C of the base, or radius r of the base is given by
1
L = lC
2
= πrl.
Theorem 9.3.5: The total area T of a right circular cone with base area B and lateral
area L, or with with slant height l and radius r of the base is given by
T =L+B
= πrl + πr2 .
Example: For a right circular cone with r = 5 in and h = 10 in, find the exact lateral and total area.
Math 30 Geometry
Page 168
Section 9.3
Theorem 9.3.6: The volume V of a right circular cone with base area B and altitude
of length h and radius r of the base is given by
1
V = Bh
3
1
= πr2 h.
3
Example: For a right circular cone with r = 5 in and h = 10 in, find the exact volume.
Type of solid
Prism
Cylinder
Pyramid
Cone
Lateral Area
Total Area
Volume
Math 30 Geometry
Solids of Revolution
Page 169
Section 9.3
Math 30 Geometry
9.4
Page 170
Section 9.4
Polyhedrons and Spheres
Definition: A polyhedron is a solid bounded by plane regions. Polygons form the faces of the
solid, and the segments common to these polygons are the edges of the polyhedron. Endpoints of
the edges are the vertices of the polyhedron.
Theorem 9.4.1: (Euler’s Equation) The number of vertices V , the number of edges E,
and the number of faces F of a polyhedron are related by the equation
V +F =E+2
Definition: A regular polyhedron is a convex polyhedron whose faces are congruent regular
polygons arranged in such a way that adjacent faces form congruent dihedral angles.
There are exactly five regular polyhedrons. They are shown in the figure above.
1.
2.
3.
4.
5.
Tetrahedron: 4 faces made from congruent regular triangles.
Hexahedron (or cube): 6 faces made from congruent squares.
Octahedron: 8 faces made from congruent regular triangles.
Dodecahedron: 12 faces made from congruent regular pentagons.
Icosahedron: 20 faces made from congruent regular triangles.
Math 30 Geometry
Page 171
Section 9.4
Definition: A sphere is the locus of points in spaces that are at a fixed distance r from a given
point O. Point O is known as the center of the sphere.
Theorem 9.4.2: The surface area S of a sphere with radius of length r is given by
S = 4πr2
Theorem 9.4.3: The volume V of a sphere with radius of length r is given by
4
V = πr3
3
Example: For a sphere with radius r = 6 cm, find the exact surface area and volume.