Download , line segment from A to B: AB Notation: “length of segment AB ” is

Geometry, Chapter 2
Section 2.1: Axiomatic Systems; definitions and notation for geometric figures; types of angles
1.
An axiomatic system (also called a deductive reasoning system) consists of undefined terms, definitions,
axioms or postulates, and theorems.
a. Undefined terms – can be described but cannot be given precise definitions. The properties of undefined
terms are given by the postulates or axioms of the system.
Set – a collection of objects
Point – determines a position but that has no dimension (length, width, or height).
Line – set of points in a one-dimensional figure with no thickness that extends in opposite directions
without ending. It is usually straight, but that depends on the postulates.
Plane – set of points in a surface having two dimensions (length and width) that extend in all
directions without ending.
b. Definitions – statements that give precise meaning using undefined terms and other definitions. Here are
two examples:
Space – the set of all points.
Geometric Figure – any set of points, lines or planes in space.
c. Postulates or Axioms – statements about undefined terms and definitions that are accepted as true
without verification or proof.
d. Theorems – a statement that we can prove using definitions, postulates, previously proved theorems and
the rules of deduction and logic.
2.
Postulates from the Textbook
(2.1)
(2.2)
(2.3)
(2.4)
(2.5)
(2.6)
Every line contains at least two distinct points.
Two points are contained in one and only one line.
If two points are in a plane, then the line containing these points is also in the plane.
Given any three distinct points in space not on the same line (noncollinear), there is exactly one plane that
passes through them.
No plane contains all points in space.
(The Ruler Postulate) There is a one-to-one correspondence between the set of all points on a line and the set
of all real numbers. (This postulate implies every line contains an infinite number of points and that every line
is infinitely long.)
3.
Collinear points: two or more points that lie on the same line.
Coplanar points: three or more points that lie on the same plane.
4.
Line Segment: A portion of a line that consists of two points, A and B, and all points that are between them.
A and B are called the endpoints.
Notation: line through points A and B: AB ,
5.
line segment from A to B: AB
Congruent Line Segments: Two lie segments with the same length. The symbol for congruency is: ≅ .
Notation: “length of segment AB ” is AB (when it is written with the bar above the endpoints it
refers to the collection of points in the segment, without the bar it refers to the length of the
segment. So “ AB ≅ CD ” (the segments are congruent) means the same thing as “AB = CD”
(their lengths are equal).
6.
Midpoint: A point that divides a line segment into two congruent segments.
7.
Ray: a portion of a line that begins at some point, A, and continues forever in one direction. A is called the
endpoint. Notation: The ray from point A through point B: AB
Geometry, Chapter 2
8.
Angle: A figure formed by two rays with a common endpoint. The common endpoint is called the vertex of
the angle and the rays are called the sides.
9.
Congruent Angles: Two angles with the same measure.
10.
Types of Angles
Acute: An angle whose measure is between 0° and 90°.
Right: An angle whose measure is 90°.
Obtuse: An angle whose measure is between 90° and 180°.
Straight: An angle whose measure is 180°.
Reflex: An angle whose measure is between 180° and 360°.
11.
Angle Bisector: A ray that divides an angle into two congruent angles.
12.
Adjacent Angles: Two angles that have a common endpoint and a common side lying between them.
13.
Perpendicular Lines: Two lines that form a right angle.
(An alternate definition for perpendicular lines is “two lines that intersect and form congruent adjacent
angles.” Sketch two intersecting lines until this definition makes sense to you.)
14.
Parallel lines: Two lines in a plane that do not intersect.
15.
Complementary Angles: Two angles whose sum is 90°.
Supplementary Angles: Two angles whose sum is 180°.
Section 2.2: types of triangles and quadrilaterals; definitions associated with polygons and circles
1.
Types of Triangles:
Right Triangle: A triangle with one right angle.
Hypotenuse: The side opposite the right angle in a right triangle.
Legs: The perpendicular sides of a right triangle.
Acute Triangle: A triangle in which all three angles are acute or less than 90°.
Obtuse Triangle: A triangle in which one angle is greater than 90°.
Scalene Triangle: A triangle in which all three sides have different lengths.
Equilateral Triangle: A triangle in which all three sides are the same length (congruent).
Equiangular: A triangle in which all three angles have the same measure (congruent).
Isosceles Triangle: A triangle with at least two congruent sides.
Base Angles: The angles opposite the congruent sides of an isosceles triangle.
Vertex Angle: The angle other than the base angles of an isosceles triangle. The side opposite the
vertex angle is called the base.
2.
Simple Closed Curve: A figure that lies in the plane and can be traced so that its starting and ending points
are the same and no part of the curve is crossed or retraced.
3.
Polygon: A simple closed curve composed of line segments. Some common polygons and number of sides:
Triangle: 3
Hexagon: 6
Nonagon: 9
Quadrilateral: 4
Heptagon: 7
Decagon: 10
Pentagon: 5
Octagon: 8
Dodecagon: 12
Geometry, Chapter 2
4.
5.
Circle: a simple closed curve consisting of the set of all points equidistant from a
given point, the center.
Radius: a line segment that joins the center with a point on the circle
Diameter: a segment that contains the center and has its endpoints on the circle
arc
B
chord
central
angle
radius O
diameter
A
C
Quadrilateral: A polygon with four sides.
semicircle
Square: A quadrilateral with all sides congruent; all angles are right angles.
Rectangle: A quadrilateral in which all angles are right angles.
Rhombus: A quadrilateral with all sides congruent.
Parallelogram: A quadrilateral with parallel opposite sides.
Trapezoid: A quadrilateral with exactly one pair of sides parallel. The parallel sides called bases and the
other sides are called legs.
Isosceles Trapezoid: A trapezoid in which the nonparallel sides are congruent.
Kite: A quadrilateral with exactly two distinct pairs of congruent adjacent sides. The angles between the two
congruent sides are called vertex angles.
Section 2.3: terminology and formulas for vertex angles for polygons
1.
Angle Sum in a Triangle The sum of the measures of the angles of a triangle is 180˚. (We will prove this
in Chapter 5.)
2.
Vertex of a Polygon – the point where two sides intersect.
Vertex Angle or Interior Angle of a Polygon – the angle formed by two adjacent sides.
3.
Diagonal of a Polygon – a segment connecting two non-adjacent vertices.
4.
Angle Sum in a Polygon: The sum of the measures of the vertex angles in any polygon with n sides is
(n − 2) ⋅180O .
5.
Regular Polygon: All sides are congruent and all vertex angles are congruent. Since all vertex angles have
equal measures we can modify the formula above as follows:
(n − 2) ⋅180O
The measure of each vertex angle in a regular polygon is
n
6.
Center of a Regular Polygon - i. A point that is equidistant from all angles in the
regular polygon and equidistant from all sides.
ii. The center of the circle circumscribed about the regular polygon.
7.
Radius of a Regular Polygon – A line segment drawn from the
center of the polygon to one of its vertices.
8.
Central Angle of a Regular Polygon - The angle formed by radii
drawn to two consecutive vertices.
9.
Exterior Angle of a Polygon – The smaller angle that is adjacent to
an interior angle, formed by extending a side of the polygon.
B
A
C
O
F
c
D
E
S
A
exterior angle
B
interior angle
F
C
E
D
Geometry, Chapter 2
Section 2.4: 3-Dimensional Shapes
1. A polyhedron is a 3-dimensional shape composed of polygons (faces).
2. Euler’s formula: F + V = E + 2 where F = # faces, V = # vertices, E = # edges (sides of polygons)
Ex: for a cube or rectangular box: F = 6, V = 8, E = 12 and 6 + 8 = 12 + 2
3. Prism: Two congruent polygons form opposite, parallel bases connected by lateral faces which are
quadrilaterals. The height, h, is the distance between the bases.
h
h
Right Prism
Lateral faces are rectangles
Oblique Prism
Lateral faces are parallelograms
4. Pyramid: One polygon base, with triangular lateral faces meeting at the apex. The height, h, is the
perpendicular distance from the base to the apex.
l
h
h
Right Regular Pyramid
1. Base is a regular polygon
2. Lateral faces are isosceles triangles
3. Slant height, l, is the height of the lateral faces
Oblique Regular Pyramid
1. Base is a regular polygon
2. Lateral faces are not isosceles triangles
3. No slant height, lateral faces not congruent
5. Cylinders and Cones:
Right Circular Cylinder
Oblique Circular Cylinder
Right Circular Cone
l
h
h
h
Oblique Circular Cone
Geometry, Chapter 3
Sections 3.1, 3.2: Perimeters of polygons and circumferences of circles. Areas of rectangles, squares,
parallelograms, triangles, trapezoids, and the rhombus. Fill in the appropriate formulas.
1.
Perimeter of a Polygon – i. Sum of the lengths of the polygons sides. ii. Distance around the polygon.
2.
Formula for Perimeter of a Square -
Formula for Perimeter of a Rectangle –
3.
w
s
l
s
4.
Circumference of a Circle – The distance around the circle.
It has been known since about 2000 BC that the ratio of the distance “around” a circle to the distance
“across” the circle is a constant: C/d = π
Circumference Formula -
5.
r
1 in.
Area:
1
square
inch
Square Unit – The surface enclosed by a square whose side is 1 unit.
1 in.
Area of a Polygon (or any closed plane figure) – The number of square units contained
in the polygons surface. (So, area of a polygon is a measure of the amount of surface that is
enclosed by the polygon.) Here, a rectangle 7 units long and 5 units wide has been divided
into 35 unit squares. Therefore, its area is 35 square units.
5
7
Additive Property of Areas – If a given region is divided into smaller regions, then the area of the original
region equals the sum of the areas of the smaller regions.
6.
Area Formulas:
a. Rectangle –
w
s
b. Square –
s
l
c. Parallelogram –
h
The length of the altitude is called a height. The side to which it is drawn is called
the base.
b
(By rearrangement we can show that the areas of the parallelogram and rectangle are the same.)
h
b
d. Triangle –
h
b
=
h
b
Geometry, Chapter 3
e. Trapezoid –
b2
h
b1
f.
Circle –
r
Section 3.3: Pythagorean Theorem and Special Triangles.
1.
Pythagorean Theorem - (Named after Greek philosopher and mathematician Pythagoras of Samos, ca. 570490 B.C., who is credited with its first proof. There is evidence that the Babylonians knew and used this fact
some 1000 years earlier. Almost 400 proofs of the Pythagorean Theorem are known today.)
Theorem: The sum of the squares of the lengths of the legs of a right triangle is equal to the square of the
length of the hypotenuse.
c
a
a 2 + b2 = c 2
b
2.
Special Triangles
a. 30º-60º-90º:
s
3 30°
s
2
s
2
60°
b. 45º-45º-90º
45˚
s 2
s
s
45˚
Geometry, Chapter 3
Sections 3.4, 3.5: Volumes and Surface Areas
Polyhedron Type
Lateral Surface Area
Total Surface Area
Volume
(total area of lateral faces) (including bases)
Right Prism
h
P∙h
2A + P∙h
A∙h
A = Area of base,
P = Perimeter of base
Right Circular Cylinder
C∙h
h
= 2π rh
A = Area of base,
C = Circumference of base
2A + C∙h
A∙h
= 2π r + 2π rh
2
2
=π r h
= 2π r (r + h)
Right Regular Pyramids
l
h
1
Pl
2
A+
1
Ah
3
1
Pl
2
A = Area of base,
P = Perimeter of base
Right Circular Cone
l
h
1
Cl
2
= π rl
2
= π r + π rl
A = Area of base,
C = Circumference of base
Sphere
r = radius
1
Ah
3
1
A+ C l
2
4π r 2
=
1
π r 2h
3
4 3
πr
3
Geometry, Chapter 4
Section 4.1: Deductive reasoning; conditional statements; the converse, inverse and contrapositive of conditional
statements
1.
Statement – A group of words and/or symbols that can be classified collectively as true or false.
Examples: Classify the each statement as true, false, or neither.
a. 8 – 6 = 2
b. An isosceles triangle has at least two congruent sides.
2.
c. 10 < 3
d. Hello!
Conditional Statement (or implication) – a statement of the form “If P, then Q.”
A conditional statement is classified as true or false as a whole. A conditional statement can be written in
equivalent forms.
Vertical Angles: Two non-adjacent angles formed by two intersecting lines.
In the figure at right ∠1 and ∠2 are vertical angles as are ∠3 and ∠4.
1
4
Ex: Vertical Angle Theorem:
“If two lines intersect, then the vertical angles formed are congruent.”
3
2
3.
Deductive Reasoning – The type of reasoning in which a conclusion is demonstrated by a sequence of
4.
Laws of Detachment and Syllogism
logically valid statements based on a set of accepted assumptions.
Two Forms of Valid Argument used in Deductive Reasoning
Premise – A statement that will be presumed true for the purposes of that argument.
Law of Detachment:
1. P Þ Q
2. P
Premise 1
Premise 2
true.
∴ Q
Conclusion
In other words, if P Þ Q is a true
conditional statement and P is true, then Q
is
(Note: ∴ is a symbol which means ‘therefore’.)
Law of Syllogism:
5.
1. P Þ Q
Premise 1
2. Q Þ R
Premise 2
Using words rather than symbols,
1. if P, then Q
2. if Q, then R
∴ PÞ R
Conclusion
∴ if P, then R
Three Variations of the Conditional, “If P, then Q” or p Þ q
1. Converse – interchanges its hypothesis and conclusion. “If Q, then P.” or q Þ p.
The converse of a given statement is not necessarily true.
2. Inverse – the negation of the hypothesis and conclusion. “If not P, then not Q.” or ~p Þ ~q.
The inverse of a given statement is not necessarily true.
3. Contrapositive – the negation of the hypothesis and conclusion are interchanged. “If not Q, then not P.”
or ~q Þ ~p.
If the given statement is true, then its contrapositive is true.
Geometry, Chapter 4
Example. Write the converse, inverse and contrapositive of the given statement and state whether they are
true or false.
Statement: “If a person lives in San Luis Obispo, then that person lives in California.”
Converse:
Inverse:
Contrapositive:
6.
Biconditional Statement – combination of a conditional and its converse when they are both true.
“If P, then Q” combined with “If Q, then P” becomes “P if and only if Q”. p Þ q & q Þ p gives p Û q.
Example: “A triangle is equilateral if and only if it is equiangular.” is a biconditional statement.
Section 4.2: Congruent Triangles; Triangle Congruency Postulates
1.
Congruent Geometric Figures – figures with the same shape and same size.
2.
Corresponding Parts of Congruent Triangles - the parts of the triangles that coincide when one is placed
on top of the other.
3.
Congruent Triangles – Two triangles, ∆ABC and ∆DEF are congruent, written ∆ABC ≅ ∆DEF, whenever
∠A ≅ ∠D, ∠B ≅ ∠E, and ∠C ≅ ∠F, AB ≅ DE , BC ≅ EF , and AC ≅ DF .
B
B
⇔
A
4.
C
∆ABC ≅ ∆DEF
C
A
Triangle Congruences
E
B
a. SAS Congruence
C
A
b. ASA Congruence
∆ABC ≅ ∆DEF
⇒
∆ABC ≅ ∆DEF
⇒
∆ABC ≅ ∆DEF
F
D
E
B
C
A
⇒
F
D
c. SSS Congruence
B
A
E
C
D
F
Geometry, Chapter 4
Right Triangle Congruences - Two right triangle congruences follow from the SAS Congruence.
5.
a. LL Congruence
A
b. HL Congruence
D
A
⇒
C
B F
D
⇒
∆ABC ≅ ∆DEF
E
C
B
F
∆ABC ≅ ∆DEF
E
Section 4.3: Important theorems we can prove with congruent triangles.
1.
Once you have proved that two triangles are congruent using one of the triangle congruences, it necessarily
follows that all remaining corresponding parts are congruent. Reason: Corresponding Parts
2.
Median of a Triangle – a line segment drawn from a vertex to the midpoint of the side opposite the vertex.
3.
Theorem – If two sides of a triangle are congruent, then the angles opposite those sides are congruent.
B
⇒
Note: We are talking about an isosceles triangle here. This theorem can be
restated, as “the base angles of an isosceles triangle are congruent.”
4.
B
C
A
A
Theorem – If two angles of a triangle are congruent, then the sides opposite those angles are congruent.
B
B
C
⇒
5.
C
A
A
Theorem - If a ray bisects the vertex angle in an isosceles triangle, then the ray bisects the base and is
perpendicular to it.
B
B
C
⇒
C
A
6.
C
A
Perpendicular Bisector Theorem – A point is on the perpendicular bisector of a line segment if and only if
it is equidistant from the endpoint of the segment.
P
P
⇔
A
7.
B
A
B
The Exterior Angle Theorem (Version 1) – The measure of an exterior angle of a triangle is greater than the measure
of either of the nonadjacent interior angles.
2
1
3
⇒
∠3 > ∠1 and ∠3 > ∠2
Geometry, Chapter 5
Section 5.1: Types of angles created by two lines cut by a transversal. Properties of angles created by a
transversal cutting two parallel lines.
1. Transversal – a line that intersects two distinct lines in two distinct points.
l
l
Transversal
Not a Transversal
l
2. Angles Formed by a Transversal
x
Alternate Interior Angles - ∠ w and ∠ r, ∠ z and ∠ s
y
z w
r
Alternate Exterior Angles - ∠ y and ∠ t, ∠ x and ∠ v
Corresponding Angles - ∠ x and ∠ r, ∠ z and ∠ t, ∠ y and ∠ s, ∠ w and ∠ v
m
t
s
v
Cointerior Angles (or Same Side Interior) - ∠ w and ∠ s, ∠ z and ∠ r
Coexterior Angles (or Same Side Exterior) - ∠ x and ∠ t, ∠ y and ∠ v
3. Exterior Angle Theorem: In a triangle, an exterior angle is greater than either nonadjacent interior angle.
sketch:
4. Euclidean Parallel Postulate: Given a line l and a point P, not on l, there can be only one line parallel to l
through the point P.
∙
P
l
5. Theorem: Two lines are cut by a transversal are parallel if and only if a pair of alternate interior angles are
congruent.
sketch:
6.
Corollary: Two lines cut by a transversal are parallel if and only if a pair of corresponding angles are
congruent.
sketch:
7.
Corollary: Two lines are cut by a transversal are parallel if and only if a pair of same side interior angles
are supplementary.
sketch:
8.
Corollary: Two lines are perpendicular to a third line if and only if they are parallel.
sketch:
n
Geometry, Chapter 5
9. Indirect Proof or Proof by Contradiction
A theorem usually consists of two parts: the hypotheses (given statements) and a conclusion (to prove.)
Method of Direct Proof: We start by assuming all of the hypotheses are true and then produce a sequence of
statements, each of which follows logically from previous statements, the hypotheses, postulates, definitions,
or other proven theorems. The final statement should be the conclusion of the theorem.
Method of Indirect Proof: We assume the hypotheses are true as before, but in addition we assume that the
conclusion of the theorem is false. Our goal is then to produce a contradiction which is caused by these
assumptions so that it is clear that whenever the hypotheses are true it is impossible for the conclusion to be
false. Therefore, whenever the hypotheses are true the conclusion must also be true, which proves the
theorem.
Example of an Indirect Proof:
Theorem: If two lines are cut by a transversal and a pair of alternate interior angles are congruent, then
the lines are parallel.
(Note: This is one half of the “if and only if” theorem from number 5 above. Our proof will also use
the Exterior Angle Theorem from number 3 above.)
Proof: Let’s identify the hypotheses and the conclusion of this theorem.
Hypotheses: Two lines are cut by a transversal and a pair of alternate interior angles are congruent.
Conclusion: The two lines are parallel.
We begin by assuming the hypotheses are true, which we can show using the following diagram:
m
1
(note: we don’t know that l is parallel to m yet)
2
l
t
Next we assume the conclusion is false, that the lines are not parallel and so must intersect somewhere:
1
2
3
l
m
t
This immediately contradicts the Exterior Angle Theorem because one of the angles becomes an
exterior angle to the triangle formed when the lines intersect, let’s say its ∠1 as shown. Then the
Exterior Angle Theorem requires that ∠1 > ∠2 , which is incompatible with our hypotheses that
∠1 =∠2 . So we conclude that whenever a pair of alternate interior angles are congruent the two lines
cannot intersect and therefore must be parallel.
Geometry, Chapter 5
Section 5.2: Sum of the angles in a triangle. Related theorems.
1. Theorem: The sum of the angles of a triangle is 180°.
2.
Corollary: (Ext Angle Theorem 2) An exterior angle of a triangle equals the sum of the two nonadjacent
interior angles.
3.
Corollary: (AAS Congruence) If two angles and a side of one triangle are congruent respectively to two
angles and a side of another triangle, then the two triangles are congruent.
4.
Corollary: (HA Congruence) If the hypotenuse and one acute angle of a right triangle are congruent
respectively to the hypotenuse and one acute angle of another right triangle, then the triangles are
congruent.
5.
Angle Bisector Theorem: A point is on the bisector of an angle if and only if it is equidistant from the
sides of the angle.
Section 5.3: Properties of parallelogram and rhombus.
1. Parallelogram
Definition: A parallelogram is a quadrilateral whose opposite sides are parallel.
B
Theorem:
A diagonal divides the parallelogram into two congruent triangles.
Corollary:
Corollary:
Opposite sides of a parallelogram are congruent.
Opposite angles of a parallelogram are congruent.
Theorem:
Theorem:
The diagonals of a parallelogram bisect each other.
Consecutive angles of a parallelogram are supplementary.
Theorem:
Parallel lines are the same distance apart everywhere.
Definition:
The distance between a point and a line is the length of the segment from the point
perpendicular to the line.
A
C
D
B
A
C
D
2. Proving a quadrilateral is a parallelogram
Prove that both pairs of opposite sides are parallel. (Definition)
Theorem:
If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a
parallelogram.
Theorem:
If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a
parallelogram.
Theorem:
If one pair of opposite sides of a quadrilateral are both parallel and congruent, then the
quadrilateral is a parallelogram.
Theorem:
If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.
3. Rhombus
Definition:
Theorem:
Theorem:
Theorem:
Theorem:
Theorem:
A rhombus is a quadrilateral with all sides congruent.
A rhombus is a parallelogram.
The diagonals of a rhombus are perpendicular to each other.
The diagonals of a rhombus bisect the angles.
If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus.
If the diagonals of a parallelogram bisect the angles, then the parallelogram is a rhombus.
Geometry, Chapter 6
Section 6.1: Ratios; Properties of Proportions; Geometric Mean
1.
2.
Ratio – A fraction relating two quantities. A ratio can be represented in four ways:
a
a to b
a:b
ab
b
Proportion – A statement that two ratios are equal.
a c
or a : b = c : d where a, b, c and d are called terms.
=
b d
a is the 1st term, b is the 2nd term, c is the 3rd term and d is the 4th term.
a and d (1st and last terms) are called the extremes and b and c (2nd and 3rd terms) are called the means.
3.
Properties of Proportions
a.
Means-Extremes Property: The product of the means equals the product of the extremes.
a c
x x+2
if and only if ad = bc. Example:
=
⇒ 2( x) =
3( x + 2)
=
3
2
b d
b.
Exchange the Extremes
a c
d c
3 6
10 6
⇒
=
= if and only if
= . Example: =
5
10
5 3
b d
b a
c.
Exchange the Means
a c
a b
3 6
⇒
= if and only if
= . Example: =
5 10
b d
c d
3 5
=
6 10
d.
Invert each Ratio
a c
b d
3 6
⇒
= if and only if
= . Example: =
5 10
b d
a c
5 10
=
3 6
NOTE: This last property is very useful when the variable is in the denominator.
Example:
4.
1 3
=
⇒
x 5
x 5
5
=
⇒ x=
1 3
3
Geometric Mean or Mean Proportional of Two Numbers
a.
Definition – b is the geometric mean of a and c if and only if b = ac . Written as a proportion, we
a b
have = . Then b 2 = ac or b = ac . (thus the name “mean proportional”)
b c
b.
Relating a geometric sequence to the geometric mean.
a1 , a2 , a3 , ..., an is a geometric sequence if and only if the ratio of any 2 consecutive terms is
constant.
This is equivalent to saying: The 2nd of any 3 consecutive terms in a geometric sequence is the
geometric mean between the first and third term.
Consider the 3 consecutive terms a4 , a5 , a6 . Here
c.
a6 a5
= , which gives a5 = a4 a6 .
a5 a4
Example: What is the geometric mean of 3 and 12? of 4 and 7? For each of these, write the three
numbers as a geometric sequence.
Geometry, Chapter 6
Section 6.2: Similar Triangles; Requirements for Similarity
1.
Two polygons are similar if and only if two conditions are satisfied:
(a)
(b)
All pairs of corresponding angles are congruent.
All pairs of corresponding sides are proportional.
Similar Triangles
2.
a)
b)
All three pairs of corresponding angles are congruent.
All pairs of corresponding sides in proportion.
E
B
A
3.
Definition of Similar Triangles
∆ABC  ∆DEF if and only if
C
D
F
∠A ≅ ∠D, ∠B ≅ ∠E , ∠C ≅ ∠F
AB BC AC
and = =
DE EF DF
Ways to Establish Two Triangles are Similar
a. AA Similarity - Two angles in one triangle are congruent respectively to two angles of the other
triangle.
b. SAS Similarity – Two sides of one triangle are proportional to two sides of another triangle and the
included angles are congruent.
c. SSS Similarity – Three sides of one triangle are proportional to three sides of another triangle.
d. A Similarity in a Right Triangle – An acute angle of one right triangle is congruent to an acute angle of
another triangle.
e. LL Similarity in a Right Triangle – The legs of one right triangle are proportional respectively to the
legs of another right triangle.
4.
Prove that the diagonals of a trapezoid divide each other into proportional segments.
Given:
A
Prove:
B
E
D
C
Geometry, Chapter 6
Section 6.3: Four theorems involving similar triangles
1.
B
Midsegment Theorem
A line segment that joins the midpoints
of two sides of a triangle is
(i) parallel to the third side and
(ii) is half the length of the third side.
D
⇒
E
A
C
This line segment is called a midsegment of the triangle.
2.
Geometric Mean (Mean Proportional) in a Right Triangle
A
In a right triangle, the altitude to the
hypotenuse is the geometric mean of the
two segments on the hypotenuse.
D
AD CD
=
CD BD
⇒
B
C
3.
DE C AC and DE = 12 AC
Side Splitting Theorem
B
A line parallel to one side of a
triangle divides the other two sides
into proportional segments.
D
If DE C AC , then
E
A
BD BE
.
=
DA EC
C
Proof: Since DE C AC , ∠ BDE ≅ ∠ BAC because corresponding angles are congruent.
∠ B ≅ ∠ B by the reflexive property. So, ∆ BDE ~ ∆ BAC by AA-Similarity. Therefore,
BD BE
. Then
=
BA BC
BA BC
by the Inversion Property of Proportions.
=
BD BE
BD + DA BE + EC
Since BA = BD + DA and BC = BE + EC, we have
by substitution. This proportion
=
BD
BE
DA
EC
BD DA BE EC
DA EC
or 1 +
. Thus
, which
can be written as the equation
=
1+
+
=
+
=
BD
BE
BD BD BE BE
BD BE
BD BE
by inversion becomes
.
=
DA EC
4.
Midquad Theorem – The line segments connecting the
midpoints of the sides of any quadrilateral form a parallelogram.
If E, F, G and H are midpoints of quadrilateral ABCD as shown,
Then EFGH is a parallelogram.
(Be able to prove this. Hint: Draw a diagonal of ABCD and use the
Midsegment Theorem.)
B
G
C
F
H
A
E
D
Geometry, Chapter 7
Sections 7.1: Parts of a circle; Central angles; Length of an arc; Area of a sector; Inscribed Angle Theorem
1.
Circle – The set of all points in a plane that are a fixed distance from a given point called the center.
A circle is named by its center point.
arc
B
•
Radius – A segment that joins the center of the circle to a point on the circle.
•
Chord – A segment that joins two points on a circle.
•
Diameter – A chord that contains the center of the circle.
•
Arc – The part of a circle between two points including the two points.
•
Semicircle - An arc whose endpoints are the endpoints of a diameter.
•
Major arc: An arc that is longer than a semicircle. Arc ACB is a major arc.
•
Minor arc: An arc that is shorter than a semicircle. Arc AB is a minor arc.
Note: If 2 letters are used to name an arc, it is assumed that the name refers to the minor arc and
not the major arc.
chord
A
central
angle
radius O
diameter
C
semicircle
2.
Congruent Circles have congruent radii. Concentric Circles are coplanar circles with a common center.
3.
Central Angle of a Circle – An angle whose vertex is the center of the circle and whose sides are radii of the
circle. ∠BOC and ∠AOB are examples of central angles.
4.
Intercepted Arc of an Angle – The two points of intersection of the angle with the circle and all points of the
arc in the interior of the angle.
 is the intercepted arc of central angle ∠BOC.
In the figure above, BC
6.
Degree Measure of an Arc (Central Angle Postulate) – In a circle, the degree measure of an arc is equal to
the degree measure of the central angle which intercepts it.
7.
=
Arc-Addition Postulate – If point B lies between points A and C on a circle, then m 
AB + mBC
m
ABC .
8.
Length of an Arc – Informally, this is the distance between the endpoints of the arc as though it were
measured along a straight line. Length of an arc is measured in linear units.
Postulate: The ratio of the degree measure m of the arc to 360 (degree measure of the
entire circle) is the same as the ratio of the length l of the arc to the circumference.
l
m
m
Practice: Find the length of 
=
⇒ =
AB .
l
⋅ (2π r )
C 360
360
9.
A
C
12 in.
O
l
80°
B
Sector of a Circle – A region bounded by two radii of a circle and their intercepted arc.
• Area of a Sector
The ratio of the degree measure m of the central angle of a sector to 360º is the same as the ratio of
the area of the sector to the area of the circle.
area of sector
m
=
area of circle 360
⇒
A=
m
π r2
360
for a circle with radius r.
Practice: Find the shaded area in the diagram if m = 60° and the radius is 3 cm.
B
O
mº
C
Geometry, Chapter 7
B
Inscribed Angle – An angle whose vertex is a point on the circle and sides are chords of
the circle.
10.
D
O
Inscribed Angle Theorem – The measure of an inscribed angle in a circle is equal to half
1 
the measure of its intercepted arc. ∠BCD =
BD
11.
C
2
A
C
O
Corollary – Inscribed angles that intercept the same arc or congruent arcs
are congruent. Here, ∠ABC = ∠ADC
12.
B
D
A
B
Corollary – An angle is inscribed in a semicircle if and only if it is a right angle.
13.
C
D
O
Section 7.2: Perpendicular bisectors of chords; Measures of angles and segments formed by intersecting chords.
Theorem: The perpendicular bisector of a chord contains the center of the circle.
1.
Corollary: The perpendicular bisectors of any two nonparallel chords intersect at the center of the circle.
Using construction techniques and the second theorem, locate the center of the circle.
Theorem:
2.
C
(
1  
Measures of Angles Formed by Intersecting Chords: =
∠1
AC + BD
2
a.
b.
)
B
1
Measures of Segments formed by Intersecting Chords: (AE)(BE) = (CE)(DE)
A
E
D
To show part a. Start with ∠1 = ∠C + ∠B (exterior angle thm 2).
Apply Inscribed Angle Thm to ∠C and ∠B.
C
C
B
1
A
For part b. Hint: Show ∆AEC ∼ ∆DEB.
Then corresponding sides are proportional.
E
A
D
B
E
D
Geometry, Chapter 7
Section 7.3: Secant lines and tangent lines; Theorems involving intersecting secants, tangents, and chords
1.
Secant Line – A line that intersects a circle (or, in general, any curve)
at two points.
2.
Tangent Line – A line that intersects a circle at one point.
Point of
Tangency
Tangent line
The point of intersection is called a point of tangency.
Secant line
3.
Theorems for two secants intersecting outside the circle.
=
E
a. ∠
(
1  
AC − BD
2
)
B
A
3
1
D
2
b. (AE)(BE) = (CE)(DE)
C
4.
Theorems for two intersecting tangents.
a. =
∠B
(
1  
ADC − AEC
2
)
6.
E
O
b. BA = BC
5.
B
A
C
D
Theorem for intersecting tangent and secant.
(
1  
AC − AD
2
a.
∠ABC =
b.
BD AB
=
AB BC
B
A
D
)
C
Theorems for intersecting tangent and chord.
1
a. ∠ABC =
AB
2
C
B
A
b. A radius or diameter of a circle is perpendicular to a tangent line at its point of tangency
E
Geometry, Chapter 7
Section 7.4: , Constructing incenter, orthocenter, circumcenter, centroid. Constructing an incircle and circumcircle.
Constructing a center of a circle. Triangle Inequality.
1.
Triangle Centers
(a) Incenter – The intersection of the angle bisectors of a triangle.
(b) Circumcenter – The intersection of the perpendicular bisectors of the sides of a triangle.
(c) Orthocenter – The intersection of the altitudes of a triangle.
(d) Centroid – The intersection of the medians of a triangle.
Note: The Centroid of a triangle divides the median into segments with
ratio 1:2. In other words, the length of the segment from the vertex to the
centroid = 2/3 the length of the median. Or, BP = (2/3) BF, AP = (2/3) AE,
CP = (2/3) CD.
D
A
2.
B
E
P
F
C
Construct the circle that passes through the points.
These theorems will be helpful when we are solving triangles in Trigonometry with the Law of Sines and the Law of
Cosines.
3.
Theorem: If two sides of a triangle are not congruent, the angle opposite the longer side is greater than the
angle opposite the shorter side.
4.
Theorem: If two angles of a triangle are not congruent, the side opposite the larger angle is longer than the
side opposite the smaller angle.
5.
The Triangle Inequality. In a triangle, the sum of the lengths of any two sides is greater than the third side.
B
AB + BC > AC
AC + BC > AB
AB + AC > BC
Example: Suppose that AB = 8 cm, BC = 5cm, and AC = 7 cm
Order the angles from least to greatest.
A
C