Download Ch 3 Worksheets L2 S15 Key

Ch 3 Worksheets S15 KEY
LEVEL 2
Name _________________________
3.1 Duplicating Segments and Angles [and Triangles]
Warm up:
Directions: Draw the following as accurately as possible. Pay attention to any problems you may be having.
You draw a figure using measuring tools, such as a protractor and a ruler. The lengths and angle measures
need to be fairly precise. Mark the measures in the diagrams.
Duplicate will mean to make an exact copy.
Draw a duplicate line segment.
Draw a duplicate angle.
Draw AB ≅ CD .
Draw m∠ A = m∠ B .
A
B
B
A
Do the length of the sides matter? Why?
No, they are rays.
Duplicate the triangle.
Draw an Equilateral Triangle.
Draw ∆ TRI ≅ ∆ ABC by measuring only the side
lengths.
Draw equilateral triangle ∆ EQU , with sides all
equal to AB.
MUST
match up
vertices!
I
A
B
U
R
T
C
Can’t do
with out an
angle
measure or a
compass!
Can’t do
with out an
angle
measure or a
compass!
Need to
locate where
the 2 sides
meet!
E
A
Q
B
Did you have any problems with making the drawings accurate? What were the issues?
Yes, you can’t figure out how the sides meet. You would have to just guess at it,
because you don’t have angle measures.
S. Stirling
Page 1 of 16
Ch 3 Worksheets S15 KEY
LEVEL 2
Name _________________________
EXERCISES Lesson 3.1 Below is Page 147-148 #1 – 3, 7, 8, 17
Use only a compass and a straight edge unless the instructions say to draw or measure!
1) Use compass to measure AB .
1. Duplicate the line segments below them. 2) With point on A, use compass to make an arc, label intersection B.
Repeat for the other segments.
B
A
C
E
D
F
1) Use compass to measure AB .
2) With point on X, use compass to make an arc.
3) Use compass to measure CD .
4) Add on the length, use compass to make an
arc, label intersection Y.
2. Construct line segment XY with length AB + CD.
3. Construct line segment XY with length AB + 2 EF – CD.
7. Duplicate triangle ∆ACB by copying the three sides, SSS method.
1) Make a segment equal to
AB + 2 EF as you did before.
2) With point on the last arc, make
a segment equal to CD toward the
left. Label intersection Y.
C
C
Follow the Notes page 1.
A
B
B
A
I
8. Construct an equilateral triangle ∆TRI .
Each side should be the length of this segment.
Follow the Notes page 1.
R
T
17. Use your ruler to draw a triangle with side lengths 8 cm, 10 cm, and 11 cm. Explain your
C
Labels were provided to make
method!
the explanation clearer.
1) Draw a segment on the ray,
AB = 11 cm.
2) Measure 10 cm with your
compass. Swing an arc with
center A.
3) Measure 8 cm with your
compass. Swing an arc with
center B. Label the intersection
of the two arcs C.
4) Construct sides AC and
CB .
B
A
S. Stirling
Page 2 of 16
Ch 3 Worksheets S15 KEY
LEVEL 2
Name _________________________
3.2 Constructing Perpendicular Bisectors
Investigation 1:
(A) Using the definitions from your notes, draw the following. Remember that when you draw,
you are measuring, so remember to write your measures in your diagrams! Write down your
steps too.
Draw at least 3 bisectors of
A
Draw a perpendicular bisector
AB .
Draw CD , the perpendicular bisector of AB .
A
B
B
How many bisectors can you draw through one
point?
Is it possible to draw more than one perpendicular
bisector?
Infinite
No
(B) In the drawing AB is the perpendicular bisector of
Using your ruler measure the distance of point A, B,
C and D from the endpoints of
What do you notice?
EG .
EG .
A
Label them.
G
The distance from any one point on the
perpendicular bisector is the same distance
from endpoints E and G.
B
E
C
D
Is this true for any segment bisector? Draw a counterexample.
Must be a perpendicular bisector. A is closer to G than E.
S. Stirling
Page 3 of 16
Ch 3 Worksheets S15 KEY
LEVEL 2
Name _________________________
EXERCISES Lesson 3.2 Below is Page 151-153 #1 – 3 and Exercise #12.
Use only a compass and a straight edge on this page!
Use only a compass and a straight edge on 1 – 3!
1 & 3. Construct the perpendicular bisector of AB then construct the perpendicular
Since EF is too close to the edge, you
bisector of EF at the right.
Follow the Notes page 2.
E
need to make two points, C and D, that
are equidistant from E and F that were
produced from different sized radii.
B
D
C
A
2. Construct perpendicular bisectors to divide QD into four congruent segments.
F
1) With compass set longer
than 1 QD , construct a
2
Q
Y
Z
X
D
perpendicular bisector.
Label the intersection X.
2) Construct a perpendicular
bisector of QX . Label the
intersection Y.
3) Construct a perpendicular
bisector of XD . Label the
intersection Z.
QY = YX = XZ = ZD
7. Construct perpendicular bisectors of each side of ∆ALI . (Make your lines long!) Anything
interesting happen?
L
They all intersect
in one point.
I
A
Next page please!
S. Stirling
Page 4 of 16
Ch 3 Worksheets S15 KEY
LEVEL 2
Name _________________________
EXERCISES Lesson 3.2 Review Problems
Page 148 Exercises #12. Make sure you can state the conjectures (properties) you are using.
50
50
50
65
130
130
50
50
155
50
130
115
90
a = 50, b = 130, c = 50, d = 130, e = 50, f = 50, g = 130, h = 130, k = 155,
m = 115, n = 65
Page 153 #15 – 20
15. ___F_____
________
___ E_____
16. ________
___ B_____
17. ________
___ A _____
18. ________
____ D ____
19. ________
____ C ____
20. ________
Page 158 #13 show your table, and #16. Write your answers below.
#13
Rectangle, n
1
Calc. Value
# of shaded
triangles
2
5
6
(1)(2) (3)(3) (5)(4) (7)(5)
(9)(6)
(11)(7) (2 n – 1)(n + 1)
(69)(36)
2
54
77
2484
9
3
20
4
35
…n …
35
(2 n – 1)(n + 1)
#16 Sketch AB ⊥ CD and EF ⊥ CD .
S. Stirling
Page 5 of 16
Ch 3 Worksheets S15 KEY
LEVEL 2
Name _________________________
3.3 Constructing Perpendiculars to a Line
Warm up:
(A) You are standing at point A and need to run to Church Hill Road as quickly as possible. How
would you determine the shortest distance to the road? Try some different measures and use
centimeters, cm, for convenience. Draw them in the figure below. What geometric figure will
give you the shortest distance from point A to the line (road)?
A
The shortest distance from the point to the line
must be measured along the perpendicular
segment from the point to the line.
X
So the shortest distance is AX = 2.4 cm.
All of the other distances are longer than AX.
(B) How could you measure the distance from point B to each of the sides of ∠DCP ? Think about
how you measured your distance from you and the road. (Treat each side of the angle as a
road.) Find these distances in cm.
P
Need to make a perpendicular
from the point B to each side.
B
Y
D
X
C
B is closer to CP than to CD .
Or BY < BX.
(C) Hans, H, is a mountain climber and Jose, J, is a cliff diver. (You can see them on their
mountains pictured below.) What would you need to measure to determine Hans’ altitude at
the top of his mountain? Show it in the drawing. How far will Jose dive before hitting the
water surface? Show it in the drawing, and you may need to draw some water first. Also, label
“sea level” in each drawing.
J
H
1.5 cm
1.5 cm
S. Stirling
Page 6 of 16
Ch 3 Worksheets S15 KEY
LEVEL 2
Name _________________________
EXERCISES Lesson 3.3 Below is Page 156-158 #1 – 3, 8, 10, 12, 18, 20.
B
1. Draw perpendiculars from the point P to both sides of
∠BIG . Which side is closer to point P?
To find the distances from P to each
side, you must first draw the
perpendiculars to each side through
point P (see notes page 3). P is closer
P
to side IB , but not by much!
I
G
2 & 3. Draw altitudes from all three vertices of each triangle below. Observe where the altitudes
are located (inside, outside or on). Also identify the type of triangle (acute, right or obtuse).
A
Acute triangle.
All altitudes are
inside the triangle.
T
C
R
Right triangle.
One altitude is inside; the
other two are on the triangle,
RG and GT .
T
G
See notes page 4!!!
B
Obtuse triangle. One altitude is
inside; the other two are outside
the triangle, so you need to extend
the sides.
O
S. Stirling
T
Page 7 of 16
Ch 3 Worksheets S15 KEY
LEVEL 2
Name _________________________
8. Draw an altitude CM from the vertex angle of the
isosceles right triangle.
What do you notice about this segment?
Write at least 3 statements!
CM bisects AB . M is the midpoint of AB .
CM is the perpendicular bisector of AB .
CM bisects ∠ ACB . CM ⊥ AB .
m∠ A = m∠ B = 45° .
m∠ CMA = m∠CMB = 90° .
∆AMC ≅ ∆BMC both are isosceles right triangles.
10. Draw and/or construct a square ALBE given AL as a side.
Explain how did it and support your
reasoning with properties we’ve learned.
C
B
A
3.4 cm
A
3.4 cm
3.4 cm
Need to make right angles at A and L and
then make AL = LB = AE.
Draw BE .
E
B
12. Draw the complement of ∠A (without measuring ∠A ).
Explain how did it and support your
reasoning with properties.
Use a protractor to draw a 90º angle at ∠A .
So m∠CAD + m∠BAC = 90°
S. Stirling
L
Remember to
mark 90º angles!
A
Page 8 of 16
Ch 3 Worksheets S15 KEY
LEVEL 2
Name _________________________
18. Draw a triangle with a 6 cm side and an 8 cm side and the angle between them measuring 40º. Draw a
second triangle with a 6 cm side and an 8 cm side and exactly one 40º angle that is not between the
two given sides. Are the two triangles congruent? Hint: start each triangle with the 8 cm segment on
the rays below.
Two triangles ∆DEF are possible, because side DF can
intersect EF in two different places!
F
Only one triangle ∆ABC is
possible. Your triangle should be
congruent to ∆ABC .
C
F
A
B
D
E
20. Draw two triangles. Each should have one side measuring 5 cm and one side measuring 7 cm,
but they should not be congruent. Start with the 7 cm segments.
Many different triangles are
possible with only two
given sides.
S. Stirling
Page 9 of 16
Ch 3 Worksheets S15 KEY
LEVEL 2
Name _________________________
EXERCISES Lesson 3.4 Below is Page 161-162 #6 – 8, 12.
6. Draw and/or construct an isosceles right triangle with z the length of each of the two legs.
A
B
z
1) Make a 90º angle at either endpoint, I made
m∠A = 90° .
2) Either use a compass or a ruler to measure length = z.
Make sides of ∠A , the legs of ∆ABC , equal length z.
Label the intersections B and C.
3) Draw CB .
C
Draw and/or construct ∆RAP with angle bisector RB and the perpendicular bisector of RP .
Place your answer on the ray below.
7.
A
P
R
A
R
P
6.6 cm
1) With a compass, duplicate ∆RPA as on Notes page 1.
You cannot locate point A without a compass!
2) Measure m∠ARP = 26° , and bisect it.
3) Find the midpoint of RP and draw the perpendicular
bisector.
Note: since ∆RPA is scalene, B is not the midpoint of AP
and the perpendicular bisector does not pass through A.
8.
Draw and/or construct ∆MSE with angle bisector SA and altitude SB . Place your answer on
the ray below.
1) With a protractor, duplicate m∠M = 29° .
2) With a compass or a ruler, duplicate MS and ME .
3.7 cm
M
5.7 cm
M
3) Measure m∠MSE
and bisect it.
E
= 38° ,
S
4) With a protractor, draw altitude
SB . You will need to extend side
M
S. Stirling
29º
ME first because
∆ MES is obtuse!
Page 10 of 16
Ch 3 Worksheets S15 KEY
LEVEL 2
Name _________________________
12. Construct a linear pair of angles (that are not congruent). Carefully bisect each angle in the
linear pair. What do you notice about the two angle bisectors? Can you make a conjecture? Can
you prove that it is always true?
The angle bisectors of a linear pair of
angles will be perpendicular (or will form
C
a 90° angle).
X
Prove:
Label equal angles x and y.
Y
y + y + x + x = 180 form a straight
y
x
y
angle.
x
2 y + 2 x = 180 simplify
D
A
B
y + x = 90 divide both sides by 2.
So XA ⊥ AY .
EXERCISES Lesson 3.4 Review Problems
Do page 162 #14 – 16 Put the info. into the drawings! Must use algebra to solve!. Also do #19 &
20.
14. Given that the lines are parallel. Find y.
68 – x
55º
15. If AE bisects ∠CAR and
m∠CAR = 84° , find m∠R .
y 110
º
C
55º
5x – 10
55º
E
70º
42º
A
7x + 4
46º
If parallel, alternate interior
angles =.
R
68 − x = 5 x − 10
Angle bisector
2 ( 4 x + 18 ) = 84
78 = 6 x
x = 13
If parallel, alternate interior
angles =.
5 (13 ) − 10 = 55
55i2 = 110 = y
S. Stirling
4x + 18
42º
Or linear pairs supp.
180 − 2 • 55 = 70
and If parallel, Same
side interior angles supp.
y = 180 − 70 = 110
4 x + 18 = 42
4 x = 24
x=6
m∠R = 7 ( 6 ) + 4 = 46
Page 11 of 16
Ch 3 Worksheets S15 KEY
LEVEL 2
16. Given BX bisects ∠ABC . Which angle is
largest, ∠A , ∠B or ∠C ?
A
66º
6x + 36
B
7x – 3
32º
57 – 5x
32º
Name _________________________
Angle bisector (congruent angles)
7 x − 3 = 57 − 5 x
12 x = 60
x=5
m∠A = 6 ( 5 ) + 36 = 66
X
m∠B = 2  7 ( 5 ) − 3 = 64
8x + 10
50º
m∠C = 8 ( 5 ) + 10 = 50
C
Largest ∠A .
19. Accurate size!
20. Accurate size!
S. Stirling
Page 12 of 16
Ch 3 Worksheets S15 KEY
LEVEL 2
Name _________________________
EXERCISES Lesson 3.5 & 3.6 Page 164-166 #1, 2, 4, 5, 17; Page 172 #6
On a separate sheet of paper: Page 165 #14, 15; Page 173 Review #15, 16
1. Draw a line parallel to n through P using
alternate interior angles. Label what you
measured and state the property you used.
Various answers, but the alternate interior angles
need to be labeled as equal.
2. Draw a line parallel to n through P using
corresponding angles. Label what you
measured and state the property you used.
Various answers, but the corresponding angles
need to be
labeled as equal.
n
n
P
P
4. Draw and/or construct a rhombus with x as the length of each side and ∠A as one of the acute
angles. Place your answer on the ray below.
x
1) Use a protractor to draw m∠A = 35° .
2) Either use a compass or a ruler to
measure length = x. Make sides of ∠A
equal length x.
3) With a compass set
to length x, locate the
intersection of the other
two congruent sides.
A
A
5. Draw and/or construct trapezoid TRAP with TR and AP as the two parallel sides and with AP
as the distance between them. (There are many solutions.) Place your answer on the ray below.
T
R
A
Y
P
1) Either use a compass or a ruler to measure length
of
TR and duplicate it.
2) Draw a perpendicular line anywhere on TR .
Make XY = 4.3 and draw a perpendicular line at Y.
3) Draw PA anywhere on this line and draw the
two remaining sides of TRAP.
S. Stirling
X
Page 13 of 16
Ch 3 Worksheets S15 KEY
LEVEL 2
Name _________________________
3.5 Page 165 Review Exercise #17
k = 90 – 72 = 18
90
90
72
108
72
72
108
108
108
108
72
Vertical angles
m = 108 ÷ 2
= 54
62
Vertical angles
62
q = 118 ÷ 2 = 59
118
a = 72, b = 108, c = 108, d = 108, e = 72, f = 108, g = 108, h = 72, j = 90,
k = 18, l = 90, m = 54, n = 62, p = 62, q = 59, r = 118
Page 172 # 6. Draw and/or construct isosceles triangle CAT with perimeter y and length of the base
equal to x. Place your answer on the ray below.
A
C
y
X
Y
T
x
1) Use a compass or a ruler
to measure length x = CA
and subtract it from
Also construct
given ray.
CY .
CA on the
3) Use a compass,
set to AX or XY to
construct the two
2) Since y represents the
perimeter of the
isosceles triangle, use a
compass or a ruler to
bisect AY . The
resulting segments,
AX = XY , are the
lengths of the remaining
sides of the triangle,
sides
CT and
AT .
C
A
CT and AT .
S. Stirling
Page 14 of 16
Ch 3 Worksheets S15 KEY
LEVEL 2
Name _________________________
Review Problems
Page 165 #14, 15; Page 173 Review #15, 16
Page 165 #14 Sketch trapezoid ZOID with
ZO ID , point T the midpoint of OI and point
Page 165 #15 Draw rhombus ROMB with
m∠R = 60° and diagonal OB .
R the midpoint of ZD . Sketch TR .
Page 173 #15 If a polygon has 500 diagonals from each vertex, how many
sides does it have?
500 = n − 3
503 = n
Page 173 #16. Must use actual measures! Draw parallelogram CARE so that
CA = 5.5 cm , CE = 3.2 cm and m∠A = 110° .
S. Stirling
Page 15 of 16
Ch 3 Worksheets S15 KEY
LEVEL 2
Name _________________________
3.8 Page 190 Review Exercise #14
h = 90 – 52 = 38
38
52
128
52 128
128
142 = 180 – 38
38
52
71
128
52
52
38
142
n = 142 ÷ 2
= 71
38
52 52
a = 128, b = 52, c = 128, d = 128, e = 52, f = 128, g = 52, h = 38, k = 52,
m = 38, n = 71, p = 38
3.R Page 197-198 Review Exercise #62 & 64
38
38
106
74
a = 38,
b = 38,
c = 142,
d = 38,
e = 50,
f = 65,
g = 106,
h = 74
142
38
65 65
50
f = 130 ÷ 2 = 65
30
30
S. Stirling
One possible explanation:
Since linear pairs are
supplementary
m∠ FAD = 30 ° .
m∠ ADC = 30 ° because
AB CD and alternate
interior angles =.
But its vertical angle has a
measure of 26º. This is a
contradiction!
Page 16 of 16