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Lesson 15.0
UNIT 15
INTRODUCTION TO GEOMETRY
Review. . . . . . .
Lesson 15.1 Points, Lines, and Planes
Lesson 15.2 Congruent Segments
Lesson 15.3 Angles . . . .
Lesson 15.4 Parallel and Skew Lines
Lesson 15.5 Triangles
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112
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Lesson 15.0 Review
Radical Equations:
Unit 15
A radical equation is one which contains one or more radicals involving the variable in the radicand. To solve a radical equation:
'12x + 1 - 3 = 2
'12x + 1 = 5
2x + 1 = 25
2x = 24
X - 12
Solving x2 = a or (kx + b)2 = a:
5(4x - 1)2 = 20
(4x - 1)2 = 4
• Get a single rad.teal tennfor one side.
• Square both sides. (Repeat steps 1 and 2 lf a rad.teal still appears.)
• Solve the resulting (radical-free) equation.
• Answer. Check solutionf s) tn original equation.
• Simplify to get a perfect square member.
4x - 1 = 2 or 4x - 1 = -2 • Extract roots: then solve each equation.
4x = 3 or
x = 3 or 4
4x= -1
X = _ j_ 4
• Answer. Check both solutions in original
equation.
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Completing the Square:
x2 - 6x + 5 = 0
x2 - 6x = -5
x2 - 6x + 9 = 4
(x - 3)2 = 4
x - 3 = 2 or x - 3 = -2
x = 5 or X = 1
Quadratic Formula:
Lesson 15.0
• Left side ts not a perfect square.
• Keep variable tem1S on one side, constant on the other.
• Add 9 (square of half the linear term
coeffic(en.tJ=( {) 2 = (-3) 2 = 9.
• Write trinomtal as square of a binomial
• Extract roots: then solve each equation.
• Answer. Check both roots in original equation.
To solve equations of the form ax2 +bx+ c = 0, use the quadratic formula:
2x2 = 3x - 1
2x2 - 3x + 1 = 0
x = -b ± ...Jb2 - 4ac 2a
x = - (-3) ±. y(-3)2 - 4{2Hll 2(2)
x=3±y9-8 4
X ~ 3 ±-fl 4
x = 3 ± 1 = 1 or 1. 4 2
• Standardform: a= 2, b = -3, c = 1.
• Substi.tute values for "a", "b," and "c" tnto quadratic formula.
• Simplify.
• Check both answers.
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Lesson 15.0
The Parabola:
y = ax2 + bx + c, a '# 0 has a parabola for its graph.
y = x2 + 2x - 3
0 = x2 + 2x - 3
X = -2 ± ..J22 - 4(1)(-3) 2(1)
X = -2 ± fT6 = -2 ± 4 2 2
x = 1 or x = -3
X = 1 + (-3) 2
X = -1
y = (-1)2 + 2(-1) - 3
y=l-2-3
y = -4
Graph of y = x2 + 2x - 3:
• "a" is + (+ 1), so graph opens upward. If "a" is -. graph wlll open. downward.
• Let y = O: solve for "x' wf1Ild x-intercepts.
• x-tntercepts of lhe parabola.
• Find axis of symmetry.
• Axis of symmetry is x = -1 (vertical line half -way between x-fntercepts).
• Substitute -1 for "x' to fmd vertex.
• Vertex is at (-1, -4).
y
axis of symmetr
X=-1
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x-intercepts: -3 and 1
y-intercept: (0,-3) vertex: (-1,-4)
Review Problems
Solve each equation:
1. '12x - 6 = 4 + -./x
2. (2x + 4)2 = 81
Solve by completing the square:
5. x2 + 4x = -3
3. Y3x + 1 = Y5x - 9
4. 2(3x - 5)2 + 6 = 14
6. x2 - 1 Ox + 9 = 0
Solve by using the quadratic formula:
7. x2 - 7x + 12 = 0 8. 2x2 - 16x + 30 = 0
Write an equation and solve:
Lesson 15.0
9. Toe base of a triangle is three times the height. If the area of the triangle is 37.5m 2• find the base and height of the triangle.
Graph the equation. Find the x-intercepts, axis of symmetry, and vertex. Tell if the vertex is a maximum or mfnfm,un:
10. y = x2 + 6x - 5
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Lesson 15.1 Points, Lines, and Planes Unit ·15
Rules:
Example:
Rule:
The study of points is called geometry. The study of geometry must begin with three words that will remain undefined. These undefined terms are point, line, and plane. These three terms serve as the base for defming other geometric terms.
Shown in the chart below is the information needed to identify point, line and plane:
Undefined Pictoral Other Term Representation Label Facts
A point has no
POINT . • p length, width, or thickness.
A dot. Capital letter
•x v• A line has length • • .... .... 1out no width or
LINE Straight line XY YX lthickness. It is
with arrows Two capital infinite in length.
on each end. letters or a It is defined by '"'·'-...... ,...~~~ I? ~,~t,nrt .-.~ln+~
7 L 7 IA plane has length
L m 10ut no thickness . . PLANE Parallelogram Lower-case It is flat and in-
letter. inite. It is defined ov 3 ooints.
1. Give seven possible names for the line: ~ t t t t .b
C D E ~~._.-.-c-+._. CD, CE, DE, DC, EC, ED, orb.
C
In geometry, a good definition uses only previously accepted undefined terms or previously defined terms. Definitions:
1. Space is the set of all points. 2. A geometric figure is a set of points. 3. Points that lie on the same line are said to be collinear
points. Noncollinear points do not lie on the same line. 4. Points that lie in the same plane are said to be coplanar.
Noncoplanar points do not lie in the same plane.
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Lesson 15.1
Example: 2. Find four collinear points, three coplanar points, three noncollinear points, and four noncoplanar points:
F, E, D, and H are collinear.
D, E, and Hare coplanar.
G, E, and H are noncollinear.
F, H, and Gare noncoplanar.
HO:MEWORK
Identify the following points as collinear, coplanar, noncollinear, or noncoplanar:
1. BCE
2. ABC
3. DFE
4. ABE
5. CFB
Draw each figure as it is described. Label the points, lines and planes: ~
6. AB intersecting planed ~
7. CD lying on plane r <-7>
8. GH lying on plane l with point Ron the plane but noncollinear with <-> GH
9. tit intersecting plane eat point B, and DB coplanar to plane e
10. Point C on plane f with DE intersecting plane f
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Lesson 15.2 Congruent Segments Unit 15
Rule: If two segments are the same length, they are congruent segments. = is the symbol for congruent.
Example: 1. A B C D
Rule:
----- -----2 3 4 5 6
• AB~ CD because they both have the same length.
The midpoint of a segment is the point that bisects the segment.
Example: 2. _________ _ • Point M is the midpoint which bisects AB if AM~ MB.
Rule:
A M B
To construct a segment that is congruent to another segment, a compass must be adjusted to the distance from the two end points of the original segment. Then the compass is moved to the new segment that is then marked by the compass.
Example: 3. Construct a segment congruent to a given segment:
Given: LM Construct: NO = LM on line r
L M
Solution: NO = LM
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• Adjust the compass to the distance of IM.
• Move the compass to line "r," keeping the same opening.
• Use the compass to mark points N and O on line "r."
----- -- ----- - -- - --- -
Lesson 15.2
HOMEWORK
Which pairs of segments are congruent? Use a ruler to measure each pair of segments:
1. 2. 3. 4. 5.
Which pairs of segments are congruent? Use a compass to compare the lengths:
7. 8. 9.
Which pairs of segments are conanient? Use the coordinates of the number line to decide:
A B C D E F G H I J K L MN 0 p Q R ST UV W X Y
• I I I I I I I I I I I I I I I I I I I I I I I I I I I I I
-14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12
11. AC and IK 14. GR and WL 12. BG and MS 15. AX and YC 13. XM and AL
I I 14
Construct a segment congruent to the given segment. Next, construct a mi~point on each of the new segments:
16. A B
19. G H
17. C D 20.
18. E
J F
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•
Lesson 15.3 Angles Unit 15
Rule: The union of two rays with a common end point is called an angle. The symbol L is used for angle.
Example: 1. Draw an angle:
Rules:
Acute Angle
• Air and 'Xe have a common end point A.
A C Every angle has a vertex and two sides:
A There are four basic types of angles:
1. Acute angles have a degree measure that is greater than o0 and less than goo.
2. Right angles have a goo measure. 3. Obtuse angles have a degree measure that is greater
than goo and less than 180°. 4. Straight angles have a degree measure of 180°.
• Right Angle Obtuse Angle Straight Angle
Example: 2. Label each angle as an acute, right, obtuse or straight angle:
A. 8. C. D.
Obtuse Angle
Acute Angle
Right Angle Obtuse Angle
E.
•
Straight Angle •
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Lesson 15.3
Rule: Angles can be labeled in several different ways, using points on the rays:
Example: 3. L BAC LCAB LA ~x
Draw the given angle:
1. acute angle
2. straight angle
3. right angle
4. obtuse angle
A C
HOMEWORK
Label all the angles in each drawing. Use all four methods:
8 . 5. C • I
A L
B C
9.
6.
d
b R
D E
10. 7.
s
b
y z L D
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I • R
s
Lesson 15.4 Parallel and Skew Lines Unit 15
Rules: Some lines intersect while other lines do not.
Lines that do not intersect may either be in the same plane or in different planes.
Lines that lie in the same plane (coplanar) that do not intersect are parallel lines.
Lines that do not lie in the same plane (noncoplanar) are skew lines.
Examples: 1.
Rule:
s u
T V
2. A
B
D
C
3.
._,..~ ++ • ST II UV means ST is parallel to1.N.
~ ~-> ~ • AC-Ir BD Tr]£_ans AC is not
parallel to 13-a
• 11tese lines are skew because they do not lie in the same plane.
Segments of lines and rays may also be parallel if the lines which contain them are parallel.
Example: 4. Which segments are parallel?
A. B.
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4.
1 o.
Lesson 15.4
Example: 5. Which rays are parallel?
E.~
F .
• HOMEWORK
Which pairs of lines, rays or segments are parallel? Which are skew? Which are neither parallel nor skew?
3.
5. 6.
8.
t
1/ 11 .
• 4
12.
13. Define parallel lines.
14. Define skew lines.
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Lesson 15.5 Triangles Unit 15
Rules:
Rules:
In geometry, one of the basic figures that is studied is a triangle. A triangle is a figure with three segments that have end points in common. The three end points must be non collinear.
The sides of the triangle are formed by the line segments. The vertices are the three end points.
vertex
vertex vertex side
Triangles are either classified by the length of their sides or the measure of their angles.
Classifications by Length qf Sides
1. Equilateral triangle: All three sides are congruent.
2. Isosceles triangle:
3. Scalene triangle:
At least two sides are congruent.
There are no congruent sides.
Example: 1. Draw an equilateral triangle, an isosceles triangle, and a scalene triangle: