Download Measuring and Drawing Angles and Triangles

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DATE
Measuring and Drawing Angles and Triangles
Measuring an angle
30°
arm
origin
If the arms are
too short to reach
the protractor scale,
lengthen them.
Step 1: Place the
origin of the protractor
over the vertex of
the angle.
Step 2: Rotate the
protractor so the base
line is exactly along
one of the arms of
the angle.
0°
0° 180°
base line
Step 3: Look at that
arm of the angle and
choose the scale that
starts at 0°.
Step 4: Use that
scale to find the
measurement.
Drawing an angle
angle mark
angle mark
60°
Step 1: Draw a line
segment.
Step 2: Place the protractor
with the origin on one endpoint.
This point will be the vertex of
the angle.
Step 3: Hold the protractor in
place and mark a point at the
angle measure you want.
Step 4: Draw a line
from the vertex through
the angle mark.
Drawing lines that intersect at an angle
45°
P
Step 1: Draw a line. Mark a point P
on the line.
45°
P
Step 2: Draw an angle of the given
measure using P as vertex.
P
Step 3: Extend the arms of your angle
to form lines.
90° 30°
5 cm
90°
5 cm
5 cm
Step 1: Sketch the
Step 2: Use a ruler to draw
triangle you want to draw. one side of the triangle.
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90° 30°
5 cm
Step 3: Use a protractor to draw
the angles at each end of this side.
Extend the arms until they intersect.
90° 30°
5 cm
Step 2: Erase any extra
arm lengths.
Blackline Master — Geometry — Teacher’s Guide for Workbook 7.1
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Drawing a triangle
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Drawing Perpendicular Lines and Bisectors
Drawing a line segment perpendicular to AB through point P
Using a set square
P
A
P
B
A
P
B
P
A
A
B
Here point P is on AB.
B
Here point P is outside AB.
Using a protractor
P
A
B
P
A
P
B
P
B
A
Here point P is on AB.
A
B
Here point P is outside AB.
Drawing the perpendicular bisector of line segment AB
A
M
A
B
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A
Step 1: Use a ruler to
determine the midpoint of
the line segment. Label it M.
M
M
B
A
M
B
B
Step 2: Use a set square or a protractor to draw a
line perpendicular to AB that passes through M.
Blackline Master — Geometry — Teacher’s Guide for Workbook 7.1
The line you have drawn
is the perpendicular bisector
of AB.
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NAME
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Drawing Parallel Lines
Drawing a line parallel to AB through point P
Using a set square
P
A
P
P
A
B
Step 1: Line up one of the
short sides of the set square
with AB.
P
A
B
A
Step 2: Use the set square
and a straight-edge to draw
a perpendicular to AB.
B
B
Step 3: Draw a line
perpendicular to the new
line that passes through P.
Step 4: Erase the line you
no longer need.
Using a protractor
P
A
P
P
B
A
B
B
A
B
Step 2: Line up the 90° line on the protractor with the
line segment drawn in step 1, and the straight side of the
protractor with point P. Draw a line parallel to AB. Erase
the first perpendicular you drew.
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Step 1: Line up the 90° line on the protractor with AB.
Use the straight side of the protractor to draw a line
segment perpendicular to AB.
A
P
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Blackline Master — Geometry — Teacher’s Guide for Workbook 7.1
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Properties of Parallel Lines
Investigation What happens if two lines meet a third line at the
same angle, but it is not a right angle?
A. Draw a pair of parallel lines and a third line intersecting both at
an angle that is not a right angle.
B. ∠ABD = ∠ACE = 70°. Draw a perpendicular to BD through point A.
Extend it to meet CE. Is the line you drew perpendicular to CE? Check
using a protractor. What can you say about the lines BD and CE?
E
D
C
B
A
D
C. ∠ABD = ∠ACE = 70°. Are the lines BD and CE parallel?
E
B
A
C
D. Draw a pair of lines that intersect at a 40° angle. Draw a third line
that meets one of the lines at the same angle. Try to make the
third line parallel to one of the lines you started with. Check by
drawing a perpendicular.
E. Compare the pattern between the equal angles ∠ABD and ∠ACE
in parts B and D. Which one looks more like the angles marked
in the letter C and which one is more like angles in the letter F?
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Mathematicians have proved that if two lines meet with a third line
at the same angles creating a pattern like in the letter F, the lines
are parallel.
When the lines meet at a right angle, you do not have to worry about
the pattern of equal angles—they are all right angles.
Blackline Master — Geometry — Teacher’s Guide for Workbook 7.1
C
E
B
D
A
C
E
A
B
D
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Distance Between Parallel Lines
InveStIgatIon
Does the distance between parallel lines depend on where you measure it?
a. Measure the line segments with endpoints on the two parallel lines with a ruler.
Write the lengths of the line segments on the picture.
B. Use a square corner to draw at least three perpendiculars from one parallel line to
the other, as shown.
Measure the distance between the two parallel lines along the perpendiculars.
What do you notice?
C. Explain why all the perpendiculars you drew in part B are parallel.
e. Measure the line segments you drew between the two given parallel lines in part D.
What do you notice?
F. To measure the distance between two parallel lines, draw a line segment
perpendicular to both lines and measure it. Does the distance between parallel lines
depend on where you measure it?
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Blackline Master — Geometry — Teacher’s Guide for Workbook 7.1
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D. A parallelogram is a 4-sided polygon with opposite sides parallel. You can draw
parallelograms by using anything with parallel sides, like a ruler. Place a ruler across
both of the parallel lines and draw a line segment along each side of the ruler. Use
this method to draw at least 3 parallelograms with different angles.
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Sum of the Angles in a Triangle
Investigation
What is the sum of the angles in a triangle?
A. Circle the combinations of a 70° angle and another angle that will make a triangle.
(Hint: Imagine the sides of the triangle extended—will they ever intersect?)
70°
80°
70°
90°
70°
100°
70°
110°
70°
120°
Circle the combinations of a 50° angle and another angle that will make a triangle.
50°
100°
50°
110°
50°
120°
50°
130°
50°
140°
Circle the combinations of a 90° angle and another angle that will make a triangle.
90°
70°
90°
80°
90°
90°
90°
100°
90°
Make a prediction:
To make a triangle, the total measures of any two angles must be less than °.
110°
B. List the sum of the measures of the angles in each triangle.
90°
° + ° + ° = °
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118°
55°
68°
42°
57°
20°
70°
° + ° + ° = °
25°
20°
25°
130°
° + ° + ° = ° ° + ° + ° = °
What do you notice about the sums of the angles? Do you think this result will be true for all triangles? Make a conjecture: The sum of the three angles in any triangle will always be °.
Blackline Master — Geometry — Teacher’s Guide for Workbook 7.1
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C. Calculate the sum of the angles.
70°
°+
90°
°+
42°
57°
20°
°=
°
°+
68°
55°
°+
°=
25°
25°
118°
°
130°
20°
°+
°+
°=
°
°+
°+
°=
°
What do you notice about the sums of the angles?
D. Cut out a paper triangle and fold it as follows:
C
A
B
Step 1: Find the midpoints of the
sides adjacent to the largest angle
(measure or fold). Draw a line
between the midpoints.
A
B
C
Step 2: Fold the triangle along the
new line so that the top vertex meets
the base of the triangle. You will get
a trapezoid.
A
C
B
Step 3: Fold the other two vertices
of the triangle so that they meet the
top vertex.
The three vertices folded together add up to a straight angle.
What is the sum of the angles in a straight angle?
°
e. Could you fold the vertices of any triangle along a line and get a straight angle as
you did in part D? Do the results of the paper folding activity support your conjecture
in part B? Explain.
F. In fact, it has been mathematically proven that…
the sum of the angles in a triangle is
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°.
Blackline Master — Geometry — Teacher’s Guide for Workbook 7.1
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So ∠A + ∠B + ∠C =
°
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Straw Quadrilaterals
1. Take 6 straws.
• Leave 2 straws whole.
• Cut 2 straws in half.
• Cut 2 straws into a quarter straw and a three-quarter straw.
2. Make as many quadrilaterals as you can with the combinations of 4 straws below.
• Try placing the straws at different angles.
• Try placing the straws in different orders.
Sketch the quadrilaterals you make.
a) 2 whole straws and 2 quarter straws
b) 4 half straws
c) 1 whole straw, 1 three-quarter straw, and 2 half straws
d) 1 whole straw, 1 three-quarter straw, 1 half straw, and 1 quarter straw
3. Check off the correct ending for the statement.
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With any of the four given side lengths above,
only one possible quadrilateral can be made
exactly two different quadrilaterals can be made
many different quadrilaterals can be made
Blackline Master — Geometry — Teacher’s Guide for Workbook 7.1
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Protractors
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Blackline Master — Geometry — Teacher’s Guide for Workbook 7.1
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Circles
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Quadrilaterals (1)
2
3
4
5
6
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1
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Blackline Master — Geometry — Teacher’s Guide for Workbook 7.1
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Quadrilaterals (2)
8
9
10
11
12
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7
Blackline Master — Geometry — Teacher’s Guide for Workbook 7.1
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2-D Shapes Sorting Game (1)
13
14
More than
one line
of symmetry
15
No lines
of symmetry
No right
angles
Two or more
right angles
17
Two or more
acute angles
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16
18
Equilateral
Blackline Master — Geometry — Teacher’s Guide for Workbook 7.1
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NAME
DATE
2-D Shapes Sorting Game (2)
19
20
Not
equilateral
No obtuse
angles
21
One or more
acute angles
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23
At least
two pairs of
parallel sides
22
At least one
reflexive angle
24
All angles
equal
Blackline Master — Geometry — Teacher’s Guide for Workbook 7.1
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Regular Polygons
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Blackline Master — Geometry — Teacher’s Guide for Workbook 7.1