Xem mẫu
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The slope of a line can be found if you know the coordinates of any two points that lie on the line. It does
not matter which two points you use. It is found by writing the change in the y-coordinates of any two points on
the line, over the change in the corresponding x-coordinates. (This is also known as the rise over the run.)
y –y
The formula for the slope of a line (or line segment) containing points (x1, y1) and (x2, y2): m = x2 – x1 .
2 1
Example
Determine the slope of the line joining points A(–3,5) and B(1,–4).
Let (x1,y1) represent point A and let (x2,y2) represent point B. This means that x1 = –3, y1 = 5, x2 = 1,
and y2 = –4. Substituting these values into the formula gives us:
m = x2 – y1
y
–x
2 1
–4 – 5
m= 1 – (–3)
–9
m= 4
Example
Determine the slope of the line graphed below.
y
5
4
3
2
1
x
1
–5 2 3 4 5
–4 –3 –2 –1
–1
–2
–3
–4
–5
Two points that can be easily determined on the graph are (3,1) and (0,–1). Let (3,1) = (x1, y1), and
let (0,–1) = (x2, y2). This means that x1 = 3, y1 = 1, x2 = 0, and y2 = –1. Substituting these values into
the formula gives us:
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–1 – 1
m= 0–3
2
–2
m= =
–3 3
Note: If you know the slope and at least one point on a line, you can find the coordinate point of other points
7
on the line. Simply move the required units determined by the slope. For example, from (8,9), given the slope 5 ,
move up seven units and to the right five units. Another point on the line, thus, is (13,16).
Determining the Equation of a Line
The equation of a line is given by y = mx + b where:
y and x are variables such that every coordinate pair (x,y) is on the line
■
m is the slope of the line
■
b is the y-intercept, the y-value at which the line intersects (or intercepts) the y-axis
■
In order to determine the equation of a line from a graph, determine the slope and y-intercept and substi-
tute it in the appropriate place in the general form of the equation.
Example
Determine the equation of the line in the graph below.
y
4
2
x
2 4
–4 –2
–2
–4
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In order to determine the slope of the line, choose two points that can be easily determined on the
graph. Two easy points are (–1,4) and (1,–4). Let (–1,4) = (x1, y1), and let (1,–4) = (x2, y2). This
means that x1 = –1, y1 = 4, x2 = 1, and y2 = –4. Substituting these values into the formula gives us:
–4–4 –8
m = 1 – ( – 1) = 2 = – 4.
Looking at the graph, we can see that the line crosses the y-axis at the point (0,0). The y-coordinate
of this point is 0. This is the y-intercept.
Substituting these values into the general formula gives us y = –4x + 0, or just y = –4x.
Example
Determine the equation of the line in the graph below.
y
6
4
2
x
–6 2 4 6
–4 –2
–2
–4
–6
Two points that can be easily determined on the graph are (–3,2) and (3,6). Let (–3,2) = (x1,y1), and
let (3,6) = (x2,y2). Substituting these values into the formula gives us:
4 2
m = 3 6 – 23) = 6 = 3 .
– (–
We can see from the graph that the line crosses the y-axis at the point (0,4). This means the
y-intercept is 4.
2
Substituting these values into the general formula gives us y = 3 x + 4.
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A ngles
N AMING A NGLES
An angle is a figure composed of two rays or line segments joined at their endpoints. The point at which the rays
or line segments meet is called the vertex of the angle. Angles are usually named by three capital letters, where
the first and last letter are points on the end of the rays, and the middle letter is the vertex.
A
B
C
This angle can either be named either ∠ABC or ∠CBA, but because the vertex of the angle is point B, letter
B must be in the middle.
We can sometimes name an angle by its vertex, as long as there is no ambiguity in the diagram. For exam-
ple, in the angle above, we may call the angle ∠B, because there is only one angle in the diagram that has B as its
vertex.
But, in the following diagram, there are a number of angles which have point B as their vertex, so we must
name each angle in the diagram with three letters.
C
D
A
B E
G F
Angles may also be numbered (not measured) with numbers written between the sides of the angles, on the
interior of the angle, near the vertex.
1
C LASSIFYING A NGLES
The unit of measure for angles is the degree.
Angles can be classified into the following categories: acute, right, obtuse, and straight.
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An acute angle is an angle that measures between 0 and 90 degrees.
■
Acute
Angle
A right angle is an angle that measures exactly 90°. A right angle is symbolized by a square at the vertex.
■
Right
Angle
Symbol
An obtuse angle is an angle that measures more than 90°, but less than 180°.
■
Obtuse Angle
A straight angle is an angle that measures 180°. Thus, both of its sides form a line.
■
Straight Angle
180°
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S PECIAL A NGLE PAIRS
Adjacent angles are two angles that share a common vertex and a common side. There is no numerical
■
relationship between the measures of the angles.
2
2
1 1
Adjacent angles ∠1 and ∠2 Non-adjacent angles ∠1 and ∠2
A linear pair is a pair of adjacent angles whose measures add to 180°.
■
Supplementary angles are any two angles whose sum is 180°. A linear pair is a special case of supplemen-
■
tary angles. A linear pair is always supplementary, but supplementary angles do not have to form a linear
pair.
110˚
70˚ 70˚
110˚
Supplementary angles (but not a linear pair)
Linear pair (also supplementary)
Complementary angles are two angles whose sum measures 90 degrees. Complementary angles may or
■
may not be adjacent.
40˚
40˚
50˚ 50˚
Non-adjacent complementary angles
Adjacent complementary angles
Example
Two complementary angles have measures 2x° and 3x + 20°. What are the measures of the angles?
Since the angles are complementary, their sum is 90°. We can set up an equation to let us solve for x:
2x + 3x + 20 = 90
5x + 20 = 90
5x = 70
x = 14
Substituting x = 14 into the measures of the two angles, we get 2(14) = 28° and 3(14) + 20 = 62°. We
can check our answers by observing that 28 + 62 = 90, so the angles are indeed complementary.
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Example
One angle is 40 more than 6 times its supplement. What are the measures of the angles?
Let x = one angle.
Let 6x + 40 = its supplement.
Since the angles are supplementary, their sum is 180°. We can set up an equation to let us solve for x:
x + 6x + 40 = 180
7x + 40 = 180
7x = 140
x = 20
Substituting x = 20 into the measures of the two angles, we see that one of the angles is 20° and its
supplement is 6(20) + 40 = 160°. We can check our answers by observing that 20 + 160 = 180, prov-
ing that the angles are supplementary.
Note: A good way to remember the difference between supplementary and complementary angles is that the
letter c comes before s in the alphabet; likewise “90” comes before “180” numerically.
A NGLES I NTERSECTING L INES
OF
Important mathematical relationships between angles are formed when lines intersect. When two lines intersect,
four smaller angles are formed.
4
3
1
2
Any two adjacent angles formed when two lines intersect form a linear pair, therefore they are supplemen-
tary. In this diagram, ∠1 and ∠2, ∠2 and ∠3, ∠3 and ∠4, and ∠4 and ∠1 are all examples of linear pairs.
Also, the angles that are opposite each other are called vertical angles. Vertical angles are angles who share
a vertex and whose sides are two pairs of opposite rays. Vertical angles are congruent. In this diagram, ∠1 and ∠3
are vertical angles, so ∠1 ≅ ∠3; ∠2 and ∠4 are congruent vertical angles as well.
Note: Vertical angles is a name given to a special angle pair. Try not to confuse this with right angle or per-
pendicular angles, which often have vertical components.
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Example
Determine the value of y in the diagram below:
3y + 5
5y
The angles marked 3y + 5 and 5y are vertical angles, so they are congruent and their measures are
equal. We can set up and solve the following equation for y:
3y + 5 = 5y
5 = 2y
2.5 = y
Replacing y with the value 2.5 gives us the 3(2.5) + 5 = 12.5 and 5(2.5) = 12.5. This proves that the
two vertical angles are congruent, with each measuring 12.5°.
PARALLEL L INES T RANSVERSALS
AND
Important mathematical relationships are formed when two parallel lines are intersected by a third, non-parallel
line called a transversal.
l
2
1
4
m
3
6
5
8
7
n
In the diagram above, parallel lines l and m are intersected by transversal n. Supplementary angle pairs and
vertical angle pairs are formed in this diagram, too.
Supplementary Angle Pairs Vertical Angle Pairs
∠1 and ∠2 ∠2 and ∠4 ∠1 and ∠4
∠4 and ∠3 ∠3 and ∠1 ∠2 and ∠3
∠5 and ∠6 ∠6 and ∠8 ∠5 and ∠8
∠8 and ∠7 ∠7 and ∠5 ∠6 and ∠7
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Other congruent angle pairs are formed:
Alternate interior angles are angles on the interior of the parallel lines, on alternate sides of the transver-
■
sal: ∠3 and ∠6; ∠4 and ∠5.
Corresponding angles are angles on corresponding sides of the parallel lines, on corresponding sides of
■
the transversal: ∠1 and ∠5; ∠2 and ∠6; ∠3 and ∠7; ∠4 and ∠8.
Example
In the diagram below, line l is parallel to line m. Determine the value of x.
l
4x + 10
m
8x – 25
n
The two angles labeled are corresponding angle pairs, because they are located on top of the parallel
lines and on the same side of the transversal (same relative location). This means that they are con-
gruent, and we can determine the value of x by solving the equation:
4x + 10 = 8x – 25
10 = 4x – 25
35 = 4x
8.75 = x
We can check our answer by replacing the value 8.75 in for x in the expressions 4x + 10 and 8x – 25:
4(8.75) + 10 = 8(8.75) – 25
45 = 45
Note: If the diagram showed the two angles were a vertical angle pair or alternate interior angle pair, the prob-
lem would be solved in the same way.
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A rea, Circumference, and Volume Formulas
Here are the basic formulas for finding area, circumference, and volume. They will be discussed in detail in the
following sections.
Rectangle Triangle
Circle
r w
h
l
b
C = 2πr
A = πr2 _
A = 1 bh
A = lw 2
Rectangular
Cylinder
Solid
r
h
h
w
l
V = π r 2h V = lwh
C = Circumference w = Width
A = Area h = Height
r = Radius v = Volume
l = Length b = Base
Triangles
The sum of the measures of the three angles in a triangle always equals 180 degrees.
b
a
c
a + b + c = 180°
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E xterior Angles
An exterior angle can be formed by extending a side from any of the three vertices of a triangle. Here are some
rules for working with exterior angles:
An exterior angle and an interior angle that share the same vertex are supplementary. In other words,
■
exterior angles and interior angles form straight lines with each other.
An exterior angle is equal to the sum of the non-adjacent interior angles.
■
The sum of the exterior angles of a triangle equals 360 degrees.
■
Example
1
2
6 5 3
4
m∠1 + m∠2 = 180° m∠1 = m∠3 + m∠5
m∠3 + m∠4 = 180° m∠4 = m∠2 + m∠5
m∠5 + m∠6 = 180° m∠6 = m∠3 + m∠2
m∠1 + m∠4 + m∠6 = 360°
C LASSIFYING T RIANGLES
It is possible to classify triangles into three categories based on the number of congruent (indicated by the sym-
bol: ≅) sides. Sides are congruent when they have equal lengths.
Scalene Triangle Isosceles Triangle Equilateral Triangle
no sides congruent more than 2 congruent sides all sides congruent
It is also possible to classify triangles into three categories based on the measure of the greatest angle:
Acute Triangle Right Triangle Obtuse Triangle
greatest angle is acute greatest angle is 90° greatest angle is obtuse
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A NGLE -S IDE R ELATIONSHIPS
Knowing the angle-side relationships in isosceles, equilateral, and right triangles is helpful.
In isosceles triangles, congruent angles are opposite congruent sides.
■
6 6
48° 48°
In equilateral triangles, all sides are congruent and all angles are congruent. The measure of each angle in
■
an equilateral triangle is always 60°.
60°
x
x
60° 60°
x
In a right triangle, the side opposite the right angle is called the hypotenuse and the other sides are called
■
legs. The box in the angle of the 90-degree angle symbolizes that the triangle is, in fact, a right triangle.
Hypotenuse
Leg
Leg
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P ythagorean Theorem
The Pythagorean theorem is an important tool for working with right triangles. It states: a2 + b2 = c2, where a
and b represent the legs and c represents the hypotenuse.
This theorem makes it easy to find the length of any side as long as the measure of two sides is known. So,
if leg a = 1 and leg b = 2 in the triangle below, it is possible to find the measure of the hypotenuse, c.
c
1
2
a2 + b2 = c2
12 + 22 = c2
1 + 4 = c2
5 = c2
5=c
P YTHAGOREAN T RIPLES
Sometimes, the measures of all three sides of a right triangle are integers. If three integers are the lengths of a right
triangle, we call them Pythagorean triples. Some popular Pythagorean triples are:
3, 4, 5
5, 12, 13
8, 15, 17
9, 40, 41
The smaller two numbers in each triple represent the length of the legs, and the largest number represents
the length of the hypotenuse.
M ULTIPLES P YTHAGOREAN T RIPLES
OF
Whole-number multiples of each triple are also triples. For example, if we multiply each of the lengths of the triple
3, 4, 5 by 2, we get 6, 8, 10. This is also a triple.
Example
If given a right triangle with sides measuring 6, x, and a hypotenuse 10, what is the value of x?
3, 4, 5 is a Pythagorean triple, and a multiple of that is 6, 8, 10. Therefore, the missing side length is 8.
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C OMPARING T RIANGLES
Triangles are said to be congruent (indicated by the symbol: ≅) when they have exactly the same size and shape.
Two triangles are congruent if their corresponding parts (their angles and sides) are congruent. Sometimes, it is
easy to tell if two triangles are congruent by looking at them. However, in geometry, it must be able to be proven
that the triangles are congruent.
There are a number of ways to prove that two triangles are congruent:
Side-Side-Side (SSS) If the three sides of one triangle are congruent to the three corresponding
sides of another triangle, the triangles are congruent.
Side-Angle-Side (SAS) If two sides and the included angle of one triangle are congruent to the cor-
responding two sides and included angle of another triangle, the triangles
are congruent.
Angle-Side-Angle (ASA) If two angles and the included side of one triangle are congruent to the cor-
responding two angles and included side of another triangle, the triangles
are congruent.
Used less often but also valid:
Angle-Angle-Side (AAS) If two angles and the non-included side of one triangle are congruent to
the corresponding two angles and non-included side of another triangle,
the triangles are congruent.
Hypotenuse-Leg (Hy-Leg) If the hypotenuse and a leg of one right triangle are congruent to the
hypotenuse and leg of another right triangle, the triangles are congruent.
9'
7'
7' 9'
7' 7'
≅ ≅
≅
50˚
30˚ 50˚ 30˚
30˚ 30˚
5' 5'
5'
5' 5'
5'
SAS ≅ SAS SSS ≅ SSS
ASA ≅ ASA
7'
7' 10'
10'
≅ ≅
50˚
30˚
50˚
30˚
6'
6'
Hy-Leg ≅ Hy-Leg
AAS ≅ AAS
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Example
Determine if these two triangles are congruent.
8"
8"
150˚
150˚
6"
6"
Although the triangles are not aligned the same, there are two congruent corresponding sides, and
the angle between them (150°) is congruent. Therefore, the triangles are congruent by the SAS pos-
tulate.
Example
Determine if these two triangles are congruent.
11" 11"
8" 8"
150˚ 150˚
Although the triangles have two congruent corresponding sides, and a corresponding congruent
angle, the 150° angle is not included between them. This would be “SSA,” but SSA is not a way to
prove that two triangles are congruent.
Area of a Triangle
Area is the amount of space inside a two-dimensional object. Area is measured in square units, often written as
unit2. So, if the length of a triangle is measured in feet, the area of the triangle is measured in feet2.
A triangle has three sides, each of which can be considered a base of the triangle. A perpendicular line seg-
ment drawn from a vertex to the opposite base of the triangle is called the altitude, or the height. It measures how
tall the triangle stands.
h
h
h
b
b
b
Acute Triangle Right Triangle Obtuse Triangle
It is important to note that the height of a triangle is not necessarily one of the sides of the triangle. The cor-
rect height for the following triangle is 8, not 10. The height will always be associated with a line segment (called
an altitude) that comes from one vertex (angle) to the opposite side and forms a right angle (signified by the box).
In other words, the height must always be perpendicular to (form a right angle with) the base. Note that in an obtuse
triangle, the height is outside the triangle, and in a right triangle the height is one of the sides.
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10
10
8
12
1
The formula for the area of a triangle is given by A = 2 bh, where b is the base of the triangle, and h is the
height.
Example
Determine the area of the triangle below.
5"
10"
1
A = 2 bh
1
A = 2 (5)(10)
A = 25 in2
V OLUME F ORMULAS
A prism is a three-dimensional object that has matching polygons as its top and bottom. The matching top and
bottom are called the bases of the prism. The prism is named for the shape of the prism’s base, so a triangular
prism has congruent triangles as its bases.
Height of prism
Base of prism
Note: This can be confusing. The base of the prism is the shape of the polygon that forms it; the base of a
triangle is one of its sides.
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Volume is the amount of space inside a three-dimensional object. Volume is measured in cubic units, often
written as unit3. So, if the edge of a triangular prism is measured in feet, the volume of it is measured in feet3.
The volume of ANY prism is given by the formula V = Abh, where Ab is the area of the prism’s base, and h
is the height of the prism.
Example
Determine the volume of the following triangular prism:
15'
40'
20'
1
The area of the triangular base can be found by using the formula A = 2 bh, so the area of the base is
1
A = 2 (15)(20) = 150. The volume of the prism can be found by using the formula V = Abh, so the
volume is V = (150)(40) = 6,000 cubic feet.
A pyramid is a three-dimensional object that has a polygon as one base, and instead of a matching polygon
as the other, there is a point. Each of the sides of a pyramid is a triangle. Pyramids are also named for the shape
of their (non-point) base.
1
The volume of a pyramid is determined by the formula 3 Abh.
Example
Determine the volume of a pyramid whose base has an area of 20 square feet and stands 50 feet tall.
Since the area of the base is given to us, we only need to replace the appropriate values into the formula.
1
V = 3 Abh
1
V = 3 (20)(50)
1
V = 33 3
1
The pyramid has a volume of 33 3 cubic feet.
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P olygons
A polygon is a closed figure with three or more sides, for example triangles, rectangles, pentagons, etc.
A B
C
F
E D
Shape Number of Sides
Circle 0
Triangle 3
Quadrilateral (square/rectangle) 4
Pentagon 5
Hexagon 6
Heptagon 7
Octagon 8
Nonagon 9
Decagon 10
T ERMS R ELATED P OLYGONS
TO
Vertices are corner points, also called endpoints, of a polygon. The vertices in the above polygon are:
■
A, B, C, D, E, and F and they are always labeled with capital letters.
A regular polygon has congruent sides and congruent angles.
■
An equiangular polygon has congruent angles.
■
Interior Angles
To find the sum of the interior angles of any polygon, use this formula:
S = 180(x – 2)°, where x = the number of sides of the polygon.
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Example
Find the sum of the interior angles in the polygon below:
The polygon is a pentagon that has 5 sides, so substitute 5 for x in the formula:
S = (5 – 2) 180°
S = 3 180°
S = 540°
E XTERIOR A NGLES
Similar to the exterior angles of a triangle, the sum of the exterior angles of any polygon equals 360 degrees.
S IMILAR P OLYGONS
If two polygons are similar, their corresponding angles are congruent and the ratios of the corresponding sides
are in proportion.
Example
A = V = 140°
W
B = W = 60°
C = X = 140°
B 6
6
D = Y = 100°
3 3
E = Z = 100°
V X
A C
5 5
10 10 AB BC CD DE EA
VW WX XY YZ ZV
D
E 2
3 3 5 5 2
= = = =
Z Y
4 6 6 10 10 4
These two polygons are similar because their angles are congruent and the ratios of the correspon-
ding sides are in proportion.
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Q uadrilaterals
A quadrilateral is a four-sided polygon. Since a quadrilateral can be divided by a diagonal into two triangles, the
sum of its interior angles will equal 180 + 180 = 360 degrees.
1 2
6
3
5
4
m∠1 + m∠2 + m∠3 + m∠4 + m∠5 + m∠6 = 360°
Parallelograms
A parallelogram is a quadrilateral with two pairs of parallel sides.
B
C
A
D
In the figure above, AB || CD and BC || AD . Parallel lines are symbolized with matching numbers of trian-
gles or arrows.
A parallelogram has:
opposite sides that are congruent (AB = CD and BC = AD)
■
opposite angles that are congruent (m∠A = m∠C and m∠B = m∠D)
■
consecutive angles that are supplementary (m∠A + m∠B = 180°, m∠B + m∠C = 180°,
■
m∠C + m∠D = 180°, m∠D + m∠A = 180° )
diagonals (line segments joining opposite vertices) that bisect each other (divide each other in half)
■
S PECIAL T YPES PARALLELOGRAMS
OF
A rectangle is a parallelogram that has four right angles.
■
x
y y
x
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