Download 3-7 Geometry

3-7-1 Geometry Vocabulary
It is important to know the vocabulary to be able to communicate and understand the questions you
come across. Geometry is the study of the size, shape and positions of object in space. Plane
geometry studies these objects in flat surfaces.
Terms
A point is a position in space represented by a
dot and usually named with a capital letter. A
point can’t be measured because it has no length
and no height.
Diagram
A
C
B
A line, represented on a diagram by a straight
line with arrows on both ends, has no width and
extends infinitely in both directions. It take two
points to define a line, but an infinite number of
points are on the line. It is named with two of
the points on the line or with a lowercase
A
C
k
B
D
italicized letter. AB , DA and line k are the same
in the diagram.
A ray is part of a line. It starts at one point and
goes infinitely in one direction. BD starts at point
B and goes through point D and continues
forever in that direction. DA is not the same
ray.
A segment is between two points. AB is the
segment between point A and point B. Notice
the part of ray BD that is the same as part of
ray DA is segment BD .
An intersection is where geometric objects
share space. Two lines intersect at a point. Line
k and line m intersect at point B. BD is the
A
C
intersection of DA and BD .
m
k
B
Two rays with a common starting point form an
angle. The common point is called the vertex.
An angle may be named by it's vertex, B when
the vertex alone is not ambiguous. Three points
can also designate an angle with the vertex as
the middle point, ABC . Angles might also be
numbered, 1 .
D
A
B
1
C
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Perpendicular lines intersect at 90 degree
A
angles. CE  BD is read line CE is
perpendicular to line BD.
C
m
E
Parallel lines don't intersect, even if they are
extended much farther than shown in the
diagram. Line k is parallel to line m but is not
parallel to line n.
k
B
D
n
An acute angle is less than 90 . DBE is
acute. Think "a cute little angle" to remember the
term.
E
An obtuse angle is greater than 90 . CBD is
obtuse. Think "obtuse angles are obese" to
remember.
D
A
B
A right angle is 90 . EBC and ABE are
right angles. The little square at the vertex
indicates the angle is 90 .
C
Complementary angles add to 90 .
ABD and DBE are complementary angles.
Supplementary angles add to 180
ABD and DBC are supplementary angles.
Vertical angles are across the intersection of two
lines from each other. 8 and 5 are vertical
angles. Vertical angles are equal. There are two
pairs of vertical angles in one intersection.
2
A transversal crosses a set of parallel lines. CD
is a transversal to the parallel lines m and k. This
grouping makes two sets of four equal angles.
m2  m3  m8  m5 and
m1  m4  m7  m6 The "m" stand for
"the measure of".
C
m
1
4
D
8
3
6
5
7
k
Alternate interior angles are equal and between the parallel lines and across the transversal from
one another. 8 and 3 are one set of alternate interior angles.
Alternate exterior angles are equal and outside the parallel lines and across the transversal from one
another. 4 and 6 are one set of alternate interior angles.
Corresponding angles are on the same side of the transversal and either both above each parallel
line or both below. 4 and 7 are one set of corresponding angles.
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3-7-2 Geometry Vocabulary Practice
T
Practice:
a) Color point E green. Points have no width, but we represent them with a dot.
b) Highlight line l yellow. It takes two points to make a line. Lines go on forever in both directions.
c) Highlight AD brown. Lines also have no thickness.
The symbol is read “line AD.”
d) Color ray IQ red. A ray starts at one point and goes forever in one direction.
e) Color the intersection BI of ED and purple. The intersection of two lines is where they cross.
f) Highlight two lines that never intersect orange.
Lines that don’t intersect are parallel.
g) Find and color black a line that crosses the parallel lines. This line is called a transversal.
h) Color ADE blue. An angle is two rays that start at the same point.
i) Color the vertex of ADE yellow. The vertex is the point on an angle.
j) Fill in an acute angle with vertex at point C red. An acute angle is less than 90 .
k) Fill in an obtuse angle with vertex C blue. An obtuse angle is greater than 90 .
l) Fill in a right angle purple. A right angle is 90 .
m) Measure each angle with vertex I.
n) Color a set of vertical angles orange. Vertical angles are equal. Vertical angles are the angles
opposite each other when two lines intersect.
0) Color a set of supplementary angles green. Supplementary angles sum to 180
p) Color a set of complementary angles brown. Complimentary angles sum to 90 .
q) Find a set of alternate interior angles. They are equal. Alternate angles are on opposite sides
of a transversal. Interior angles are inside parallel lines.
r) Find a set of alternate exterior angles. The are equal. Exterior angles are outside the parallel
lines.
s) Find a set of corresponding angles. Angles formed with parallel lines and a transversal and
located in the same position compared to the transversal are corresponding angles They are equal.
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3-7-3 Geometry Vocabulary Puzzle
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2
3
4
6
5
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
Across
1 Units angles are
measured in
3 A location in space; has
no dimension
4 Angles that are formed
when two lines intersect
8 Angles that have a
measure equal to 90
degrees
9 Angles on opposite sides
of a transversal
11 angles formed by the
transversal and parallel
lines on the inside of
the parallel lines
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12 Angle greater than 90
degrees
14 Angle measuring less
than 90 degrees
16 Two angles that add
up to 90 degrees
17 Extending endlessly in
two directions in space.
Takes two points to
Define
19 A tool used to measure
angles
21 A line that intersects
two parallel lines
22 Two lines in the same
plane that never meet
23 Angles on the outside
of parallel lines
Down
2 When the sum of the
measures of two angles
is 180 degrees, the
angles are________.
5 Equal angles located in
the same position
compared to the
transversal
6 When two lines cross
7 Part of a line between
two points
10 two rays starting at
a common point
13 The study of flat
surfaces
15 two lines that intersect
form 90 degree angles
18 The study of size,
shape, positions, of
objects in space
20 A part of a line that
starts at one point and
extends endlessly in
the other direction
3-7-4 Geometry with Algebra
The following will provide practice with both geometry vocabulary and algebra. Think about what the
picture or vocabulary means and make the equation accordingly.
3x-7
2x-3
Examples: Find x. The two angles are complementary and must
add to 90 degrees.
2x-3 + 3x-7 = 90
When this is worked out x is 20. The angles could be found by
plugging the 20 into the expression for each angle.
E
D
A
B
C
The angles in a triangle add to 180
degrees. Cut out any triangle. Tear off
the angles and arrange them next to
each other as shown. This works with
any triangle.
Example: Find the measure of the angles in an equilateral triangle. First draw a
picture. An equilateral triangle has all sides the same length and all angles the
same size. x+x+x=180
3x = 180
x= 60
The angles in an equilateral triangle are each 60 degrees.
x
x
x
Practice:
2x-5
a) Find x. Remember the complementary angles add to
ninety degrees.
E
D
x
A
B
C
b) One angle in a set of supplementary angles is 5 more
than twice the other. Find the measurement of the
angles.
c) Solve for x in each of the following diagrams. Assume lines that appear parallel are parallel.
3x+85
2x-8
3(x-8)
5(2x-15)
3(2x -5)
x+15
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8x+22
2(3x -5)
3(x+15)
5(x-2)
3(4x -12)
3(x-4)
d) The angles on any triangle add up to 180 degrees. Find the measure of each angle if the
triangle is isosceles with one 40-degree angle.
e) What is the measure of the angles in an equilateral triangle? (all sides and all angles equal)
f) One angle of a triangle is three less than twice another. The other is 4 times the sum of the
small angle and 2. What are the angle measurements?
g) One vertical angle is a number plus 12. The other 40 less than is three times the same
number. What is the number?
h) What is the measure of the other two angles in a right isosceles triangle? (Angles in any triangle
add to 180.)
2x
2(5(x-3)+14)
i)
3x+15
5(x-3)+14
2x
Find x.
Use a protractor to measure angles. A protractor has a notch or hole to mark where the vertex of the
angle must be. One ray of an angle is along the line marked 0 degrees. Most times this is not along
the bottom of the protractor, but in line with the hole for the vertex. The size of the angle is read from
the numbers on the curved part of the protractor.
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Students sometimes have a hard time knowing which number to read of the two that the angle
touches. There is an easy way to decide. If the angle is acute it is the measurement that is less than
90 degrees. If the angle is obtuse the measurement is more than 90 degrees.
The angle is positioned with the vertex on
the hole and one ray along the zero
degree line.
The numbers the other side of the angle
touches are 60 degrees and 120 degrees.
Which measurement is correct?
Sometimes you must use a straight edge
to extend the angel farther than is
presented in the original problem to see
exactly where the line hits the protractor.
Practice: Measure the following angles.
a)
b)
c)
d) Draw a 65 angle.
Draw a 32 angle.
Draw a 15 angle.
e) Draw a 120 angle.
Draw a 155 angle.
Draw a 175 angle.
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