Download Chapter 9 Geometry

ACCELERATED MATHEMATICS
CHAPTER 9
GEOMETRIC PROPERTIES
TOPICS COVERED:
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Geometry vocabulary
Similarity and congruence
Classifying quadrilaterals
Transformations (translations, reflections, rotations, dilations)
Measuring angles
Complementary and supplementary angles
Triangles (sides, angles, and side-angle relationships)
Angle relationships with transversals
Geometry is the area of mathematics that deals with the properties of points, lines, surfaces, and solids. It
is derived from the Greek “geometra” which literally means earth measurement.
Dictionary of Geometry
Name:
Activity 9-1: Vocabulary Match
Name:
Section 1: Polygons
Word bank:
Triangle
Decagon
Nonagon
Circle
Octagon
Quadrilateral
Hexagon
Pentagon
Heptagon
Regular polygon
Polygon
A geometric figure with 3 or more sides and angles
1. polygon
A polygon with 3 sides
2. triangle
A polygon with 4 sides
3. quadrilateral
A polygon with 5 sides
4. pentagon
A polygon with 6 sides
5. hexagon
A polygon with 7 sides
6. heptagon
A polygon with 8 sides
7. octagon
A polygon with 9 sides
8. nonagon
A polygon with 10 sides
The set of all points in a plane that are the same distance
from a given point (hint: not a polygon)
A polygon with all sides congruent and all angles
congruent
9. decagon
Section 2: Four sided polygons (Quadrilaterals)
A parallelogram with 4 right angles and 4 congruent sides
Word bank:
A parallelogram with 4 right angles
(sides may or may not be congruent)
Trapezoid
A parallelogram with 4 congruent sides
Parallelogram
(any size angles)
Rectangle
A quadrilateral with exactly one pair of opposite sides
Rhombus
parallel (any size angles)
Square
A quadrilateral with opposite sides parallel and opposite
sides congruent
10. circle
11. regular polygon
12. square
13. rectangle
14. rhombus
15. trapezoid
16. parallelogram
Section 3: Shape movement
Word bank:
Transformation
Reflection
Rotation
Translation
Dilation
Any kind of movement of a geometric figure
17. transformation
A figures that slides from one location to another without
changing its size or shape
18. translation
A figure that is turned without changing its size or shape
19. rotation
A figure that is flipped over a line without changing its
size or shape
20. reflection
A figure that is enlarged or reduced using a scale factor
21. dilation
Section 5: Angles
Word bank:
Angle
Acute angle
Right angle
Straight angle
Obtuse angle
Vertex
Diagonal
An angle that is exactly 180°
22. straight angle
An angle that is less than 90°
23. acute angle
The point of intersection of two sides of a polygon
24. vertex
An angle that is between 90° and 180°
25. obtuse angle
An angle that is exactly 90°
26. right angle
A segment that joins two vertices of a polygon but is not a side
27. diagonal
A figure formed by two rays that begin at the same point
28. angle
Section 6: Figures and Angles
Word bank:
Angles that add up to 90°
29. complementary
Angles that add up to 180°
30. supplementary
Congruent figures
Figures that are the same size and same shape
31. congruent figures
Similar figures
Figures that are the same shape and may or may not
32. similar figures
Line of symmetry
have same size
Complementary angles
Place where a figure can be folded so that both
Supplementary angles
33. line of symmetry
halves are congruent
Section 7: Lines
Word bank:
Perpendicular line
Ray
Line
Intersecting lines
Parallel lines
Line segment
Point
Plane
An exact spot in space
A straight path that has one endpoint and extends
forever in the opposite direction
Lines that cross at a point
Lines that do not cross no matter how far they are
extended
A straight path between two endpoints
34. point
35. ray
36. intersecting lines
37. parallel lines
38. line segment
Lines that cross at 90°
39. perpendicular lines
A thin slice of space extending forever in all directions
40. plane
A straight path that extends forever in both directions
41. line
Section 8: Triangles
Word bank:
Acute triangle
Right triangle
Obtuse triangle
Scalene triangle
Isosceles triangle
Equilateral triangle
A triangle with one angle of 90°
42. right triangle
A triangle with all angles less than 90°
43. acute triangle
A triangle with no congruent sides
44. scalene triangle
A triangle with at least 2 congruent sides
45. isosceles triangle
A triangle with an angle greater than 90°
46. obtuse triangle
A triangle with 3 congruent sides
47. equilateral triangle
Activity 9-2: Geometry Vocabulary
Name:
Polygons
Triangles
Regular polygon
Equilateral triangles
Quadrilaterals
Scalene triangles
Pentagons
Isosceles triangles
Hexagons
Acute triangles
Heptagons
Right triangles
Octagons
Obtuse triangles
Nonagons
Rectangles
Decagons
Squares
Circles
Parallelograms
Ovals
Rhombuses
Lines
Trapezoids
Rays
Line segments
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
U
V
W
X
Y
Z
A1
B1
C1
D1
E1
F1
G1
H1
J1
K1
L1
M1
N1
O1
P1
Q1
R1
S1
T1
S
T
Activity 9-3: Size and Shape
Name:
Figures that have the same size and shape are congruent figures.
Figures that have the same shape but may be different sizes are similar figures.
The symbol ≅ means “is congruent to.” The symbol ∼ means “is similar to.”
Similar figures are proportional.
∆ABC ∼ ∆DEF
Corresponding angles are congruent,
E
B
∠A ≅ ∠D
∠B ≅ ∠E
∠C ≅ ∠F
A
C
F
D
Corresponding side lengths are proportional.
AB BC
=
DE EF
AC AB
=
DF DE
BC AB
=
EF DF
Congruent figures are the exact same shape and size.
∠A ≅ ∠D
∠B ≅ ∠E
∠C ≅ ∠F
B
E
AB ≅ DE
AC ≅ DF
A
C
D
BC ≅ EF
Are these triangles similar?
58°
31°
F
Activity 9-4: Similar Polygons
Name:
Tell whether each pair of polygons is similar.
3cm
1.
2.
6 cm
10 ft
7 cm
3.5 cm
3.5 cm
15 ft
7 cm
8 ft
12 ft
3.
101 m
4.
100 m
4 ft
8 ft
8 ft
150 m
151 m
12 ft
5.
A
13 in
B
13 in
12 in
E
D
C
12 in
For each pair of similar figures write a proportion and use the proportion to find the length of x. Use a
separate sheet of paper.
6.
7.
x
9m
12 m
6m
15 cm
12 cm
20 cm
x
8.
9.
18 cm
30 cm
24 cm
x
10 in
35 in
6 in
x
Activity 9-5: Proportions with Similar Figures
Name:
For each pair of similar figures write a proportion and use the proportion to find the length of x. Use a
separate sheet of paper.
1.
2.
36 in
72 in
30 m
x
25 in
x
25 m
3.
15 m
4.
21 cm
20 cm
x
x
20 m
35 cm
60 m
14 m
5.
6.
6m
4.5 m
35 mm
x
25 mm
18 mm
x
9m
45 mm
7.
8.
x
21 m
x
8 km
3m
9m
12 km
39 km
9.
A flagpole casts a shadow 22 ft long. If a 4 ft tall casts a shadow 8.8 ft
long at the same time of day, how tall is the flagpole?
10.
A photograph is 25 cm wide and 20 cm high. It must be reduced to fit a
space that is 8 cm high. Find the width of the reduced photograph.
Michael wants to find the length of the shadow of a tree. He first
measures a fencepost that is 3.5 feet tall and its shadow is 10.5 feet long.
11.
Next, Michael measures the height of the tree, and finds it is 6 feet tall.
How long is the shadow of the tree?
Activity 9-6: Similar Shapes
Name:
Tell whether the shapes below are similar. Explain your answer.
1.
2.
3.
4.
Solve
5.
A rectangle made of square tiles measures 8 tiles wide and 10 tiles long. What
is the length in tiles of a similar rectangle 12 tiles wide?
6.
A computer monitor is a rectangle. Display A is 240 pixels by 160 pixels.
Display B is 320 pixels by 200 pixels. Is Display A similar to Display B?
Explain.
The figures in each pair are similar. Find the unknown measures.
7.
8.
9.
10.
AMBIGRAMS
A graphic artist named John Langdon began to experiment in the 1970s with a special way to write words
as ambigrams. Look at all the examples below and see if you can determine what an ambigram is.
Activity 9-7: Quadrilaterals
Choose ALL, SOME, or NO
1.
All
Some
Name:
No
rectangles are parallelograms.
2.
All
Some
No
parallelograms are squares.
3.
All
Some
No
squares are rhombi.
4.
All
Some
No
rhombi are parallelograms.
5.
All
Some
No
trapezoids are rectangles.
6.
All
Some
No
quadrilaterals are squares.
7.
All
Some
No
rhombi are squares.
8.
All
Some
No
parallelograms are trapezoids.
9.
All
Some
No
rectangles are rhombi.
10.
All
Some
No
squares are rectangles.
11.
All
Some
No
rectangles are squares.
12.
All
Some
No
squares are quadrilaterals.
13.
All
Some
No
quadrilaterals are rectangles.
14.
All
Some
No
parallelograms are rectangles.
15.
All
Some
No
rectangles are quadrilaterals.
16.
All
Some
No
rhombi are quadrilaterals.
17.
All
Some
No
rhombi are rectangles.
18.
All
Some
No
parallelograms are rhombi.
19.
All
Some
No
squares are parallelograms.
20.
All
Some
No
quadrilaterals are parallelograms.
21.
All
Some
No
parallelograms are quadrilaterals.
22.
All
Some
No
trapezoids are quadrilaterals.
Solve each riddle.
I am a quadrilateral with two pairs of parallel sides and four sides of the
23.
same length. All of my angles are the same measure, too. What am I?
I am a quadrilateral with two pairs of parallel sides. All of my angles are
24.
the same measure, but my sides are not all the same length. What am I?
25. I am a quadrilateral with exactly one pair of parallel sides. What am I?
26. I am a quadrilateral with two pairs of parallel sides. What am I?
Sketch each figure. Make your figure fit as FEW other special quadrilateral names are possible.
27. Rectangle
28. Parallelogram
29. Trapezoid
30. Rhombus
Activity 9-8: Classifying Quadrilaterals
Name:
QUADRILATERALS (Four sided figues)
Trapezoids
Parallelograms
Rectangles
Rhombuses
Squares
List all the names that apply to each quadrilateral. Choose from parallelogram, rectangle, rhombus,
square, and trapezoid.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Answer the following on a separate sheet of paper.
11.
Evan said, “Every rectangle is a square.” Joan said, “No, you are wrong. Every square is a
rectangle.” Who is right? Explain your answer on your graph paper.
12.
What is the fewest number of figures you would have to draw to display a square, a rhombus,
a rectangle, a parallelogram, and a trapezoid? What are the figures?
13. How are a square and a rectangle different?
14. How are a parallelogram and a rhombus different?
15. How are a square and rhombus alike?
Activity 9-9: Translations
Name:
Consider the triangle shown on the coordinate plane.
1.
Record the coordinates of the vertices of the triangle.
2.
Translate the triangle down 2 units and right 5 units. Graph the translation.
3.
A symbolic representation for the translated triangle would be: ( x, y ) → ( x + 5, y − 2)
4.
Write a verbal description of the translation.
5.
Describe the translation above using symbolic representation.
Activity 9-10: Translations
Name:
Determine the coordinates of the vertices for each image of trapezoid STUW after each of the following
translations in performed.
1.
3 units to the left and 3 units down
2.
( x, y ) → ( x, y − 4)
3.
( x, y ) → ( x − 2, y + 1)
4.
( x, y ) → ( x − 4, y )
5.
Find a single transformation that has the same effect as the
composition of translations ( x, y ) → ( x − 2, y + 1) followed by
( x, y ) → ( x + 1, y + 3) .
6.
Draw triangle ABC at points (−1, 6), (−4,1), (1,3) .
7.
Draw triangle A′B′C′ is located at (5, 2), (2, −3), (7, −1)
8.
Write a symbolic representation for the translated triangle compared to the original.
9.
Write a description (in words) of this translation.
Connie translated trapezoid RSTU to trapezoid R′S ′T ′U ′ .
10. Vertex S was at (-5,-7). If vertex S ′ is at (-8,5), write a
description of this translation.
Move each vertex ___ units to the
_________ and ___ units
_________.
Activity 9-11: Properties of Translations
Name:
Describe the translation that maps point A to point A′.
1.
2.
_____________________________________
_____________________________________
Draw the image of the figure after each translation.
3. 3 units left and 9 units down
4. 3 units right and 6 units up
5. a. Graph rectangle J′K′L′M′, the image of rectangle
JKLM, after a translation of 1 unit right and 9 units
up.
b. Find the area of each rectangle.
_____________________________________
c. Is it possible for the area of a figure to
change after it is translated? Explain.
_____________________________________
_____________________________________
Activity 9-12: Reflections
Name:
1.
Record the coordinates of the vertices of triangle PQR.
2.
Sketch the reflection of triangle PQR over the y-axis.
3.
A symbolic representation for the reflected triangle would be: ( x, y ) → (− x, y )
The line over which an object is reflected is called the line of reflection.
4.
What is the line of reflection for the trapezoid above?
5.
Write a verbal description of the translation.
6.
A symbolic representation for the reflected triangle would be: ( x, y ) → ( x, − y )
Activity 9-13: Reflections
Name:
Use a piece of graph paper to draw the following.
1.
Draw triangle ABC at points (−4,1), (−1,3), (−5, 6) .
2.
Reflect triangle ABC across the y-axis. List the new vertices A′B′C′ .
3.
Write a symbolic representation for the reflected triangle compared to the original.
4.
Reflect triangle ABC across the x-axis. List the new vertices A′′B′′C′′ .
5.
Write a symbolic representation for the reflected triangle compared to the triangle A′B′C′ .
6.
Draw triangle ABC at points (−1,1), (−3, −2), (2, −3) .
7.
Reflect triangle ABC across the y-axis. List the new vertices A′B′C′ .
8.
Write a symbolic representation for the reflected triangle compared to the original.
9.
Reflect triangle ABC across the x-axis. List the new vertices A′′B′′C′′ .
10.
Write a symbolic representation for the reflected triangle compared to the triangle A′B′C′ .
11.
If point Q(6,-2) is reflected across the x-axis, what will be the coordinates of
point Q′ ?
Activity 9-14: Properties of Reflections
Name:
Use the graph for Exercises 1–3.
1. Quadrilateral J is reflected across the x-axis. What is the
image of the reflection?
_______________________________________________
2. Which two quadrilaterals are reflections of each
other across the y-axis?
_______________________________________________
3. How are quadrilaterals H and J related?
_______________________________________________
Draw the image of the figure after each reflection.
4. across the x-axis
5. across the y-axis
6.
a. Graph rectangle K′L′M′N′, the image of
rectangle KLMN after a reflection across
the y-axis.
b. What is the perimeter of each rectangle?
_____________________________________________
c. Is it possible for the perimeter of a figure to change after
it is reflected? Explain.
_____________________________________________
_____________________________________________
Activity 9-15: Rotations
Name:
A rotation is a transformation that describes the motion of a figure about a fixed point.
In the table below record the vertices of each triangle.
Triangle
Three vertices
A
B
C
D
1.
Starting at A to get to B involves a rotation of 90° clockwise about the origin.
Make a conjecture about the changes in the x and y coordinates when a point is rotated clockwise
90° .
2.
Starting at A to get to C involves a rotation of 180° clockwise about the origin.
Make a conjecture about the changes in the x and y coordinates when a point is rotated clockwise
180° .
3.
Starting at A to get to D involves a rotation of 270° clockwise about the origin.
Make a conjecture about the changes in the x and y coordinates when a point is rotated clockwise
270° .
4.
What would happen if A is rotated 360° clockwise about the origin?
5.
What is true about the area of the triangle each time the shape is rotated?
Activity 9-16: Rotations
Name:
Starting with the triangle above, rotate the triangle about the origin:
A.
90° counterclockwise
B.
180° counterclockwise
C.
270° counterclockwise
Record the coordinates of the original triangle.
Record the coordinates of the 90° counterclockwise rotated circle.
Record the coordinates of the 180° counterclockwise rotated circle.
Record the coordinates of the 270° counterclockwise rotated circle.
Activity 9-17: Properties of Rotations
Name:
Use the figures at the right for Exercises 1–5. Triangle A has been rotated about the origin.
1. Which triangle shows a 90°
counterclockwise rotation?
____
2. Which triangle shows a 180°
counterclockwise rotation?
____
3. Which triangle shows a 270°
clockwise rotation?
____
4. Which triangle shows a 270°
counterclockwise rotation?
____
5. If the sides of triangle A have lengths of 30 cm,
40 cm, and 50 cm, what are the lengths of the
sides of triangle D?
_____________________________________
Use the figures at the right for Exercises 6–10. Figure A is to be rotated about the origin.
6. If you rotate figure A 90° counterclockwise,
what quadrant will the image be in?
____
7. If you rotate figure A 270° counterclockwise,
what quadrant will the image be in?
____
8. If you rotate figure A 180° clockwise,
what quadrant will the image be in?
____
9. If you rotate figure A 360° clockwise,
what quadrant will the image be in?
____
10. If the measures of two angles in figure
A are 60º and 120°, what will the measure
of those two angles be in the rotated figure?
_____________________________________
Use the grid at the right for Exercises 11–12.
11. Draw a square to show a rotation of 90° clockwise
about the origin of the given square in quadrant I.
12. What other transformation would result in the same
image as you drew in Exercise 11?
_____________________________________
Activity 9-18: Algebraic Representations of Transformations
Name:
Write an algebraic rule to describe each transformation of figure A to figure A′. Then describe the
transformation.
1.
2.
___________________________________
___________________________________
Use the given rule to graph the image of each figure. Then describe the transformation.
3. (x, y) → (−x, y)
___________________________________
4. (x, y) → (−x, −y)
___________________________________
Solve.
5.
Triangle ABC has vertices A(2, −1), B(−3, 0), and C(−1, 4). Find the vertices
of the image of triangle ABC after a translation of 2 units up.
6.
Triangle LMN has L at (1, −1) and M at (2, 3). Triangle L′M′N′ has
L′ at (−1, −1) and M′ is at (3, −2). Describe the transformation.
Activity 9-19: Dilations
Name:
When two figures satisfy at least one of the following conditions, then they are similar:
• The corresponding angles are congruent.
• The corresponding sides are proportional.
The ratio of the corresponding sides of similar triangles is called the scale factor.
A dilation is a transformation where the image is similar to the preimage (image before the
transformation). The center of dilation is a fixed point in the plane about which all points are expanded
or reduced. It is the only point under a dilation that does not move.
Figure 1 – Example of a dilation
Figure 2
When the origin is the center of dilation you can determine the new coordinates for a dilation by a factor
of 2 by: ( x, y ) → (2 x, 2 y )
When the scale factor is greater than 1, the dilation is called an enlargement.
When the scale factor is between 0 and 1, the dilation is called a reduction.
Use Figure 2 above to answer the following questions.
1.
What is the area of HLTZ ?
2.
HLTZ is dilated by a factor of 3. Draw the new H ′L′T ′Z ′ .
3.
What is the area of H ′L′T ′Z ′ ?
4.
HLTZ is dilated by a factor of
5.
What is the area of H ′′L′′T ′′Z ′′ ?
6.
How does the dilation factor affect the new area of the rectangles?
1
. Draw the new H ′′L′′T ′′Z ′′ .
2
Activity 9-20: Properties of Dilations
Name:
Use triangles ABC and A'B'C' for Exercises 1–4.
1. Use the coordinates to find the lengths of the sides.
Triangle ABC: AB = ____ ; BC = ____
Triangle A'B'C': A'B' = ____ ; B'C' = ____
2. Find the ratios of the corresponding sides.
A′B′
=
AB
=
B′C ′
=
BC
=
3. Is triangle A'B'C' a dilation of triangle ABC? ______________
4. If triangle A'B'C' is a dilation of triangle
ABC, is it a reduction or an enlargement? _________________
For Exercises 5–8, tell whether one figure is a dilation of the other or not. If one figure is a
dilation of the other, tell whether it is an enlargement or a reduction. Explain your reasoning.
5.
Triangle R'S'T' has sides of 3 cm, 4 cm, and 5 cm. Triangle RST has sides
of 12 cm, 16 cm, and 25 cm.
6.
Quadrilateral WBCD has coordinates of W(0, 0), B(0, 4), C(−6, 4), and D(−6,
0). Quadrilateral W'B'C'D' has coordinates of W'(0, 0), B'(0, 2), C'(−3, 2), and
D'(−3, 0).
7.
Triangle MLQ has sides of 4 cm, 4 cm, and 7 cm. Triangle M'L'Q' has sides
of 12 cm, 12 cm, and 21 cm.
Does the following figure show a dilation? Explain.
8.
Activity 9-21: Algebraic Representations of Dilations
Name:
Use triangle ABC for Exercises 1–4.
1. Give the coordinates of each vertex of ∆ABC.
A_________________ B_________________ C_________________
2. Multiply each coordinate of the vertices of ∆ABC by 2
to find the vertices of the dilated image ∆A'B'C'.
A'_________________ B'_________________ C'_________________
3. Graph ∆A'B'C'.
4. Complete this algebraic rule to describe the dilation.
(x, y) → ___________________________
Use the figures at the right for Exercises 5−
−7.
5. Give the coordinates of each vertex of figure JKLMN.
J_________________ K_________________ L_________________
M_________________, N_________________
6. Give the coordinates of each vertex of figure J'K'L'M'N'.
J'_________________ K'_________________ L'_________________
M'_________________, N'_________________
7. Complete this algebraic rule to describe the dilation.
(x, y) → ___________________________
Li made a scale drawing of a room. The scale used was 5 cm = 1 m. The scale drawing is the preimage and the room is the dilated image.
8.
What is the scale in terms of centimeters to centimeters?
9.
Complete this algebraic rule to describe the dilation from the scale
drawing to the room.
10.
The scale drawing measures 15 centimeters by 20 centimeters.
What are the dimensions of the room?
(x, y) →
Activity 9-22: Dilations and Measurement
Name:
Find the perimeter and area of the original figure and of the image after dilating each figure.
2. Scale factor =
1. Scale factor = 3
1
3
P = ____
P = ____
P′ = ____
P′ = ____
A = ____
A = ____
A′ = ____
A′ = ____
Solve.
3.
A rectangle is enlarged by a scale factor of 4. The original rectangle is 8.4
cm by 5.3 cm. What are the dimensions of the enlarged rectangle?
A poster is 16 inches wide by 20 inches long. You use a copier to create a
4.
reduction with a scale factor of
3
. Will the reduction fit into a frame that
4
is 11 inches by 17 inches? Explain your answer.
Lincoln School is having a fundraising contest. Each student is to design a school sticker. The
designs are to be made to fit a rectangle that is 12 cm by 16 cm. Each design is to have a border.
The winning designs will be made into stickers that are dilated by a scale
5.
factor of
1
. What will the dimensions of the stickers be? What will the
4
perimeter and the area of the stickers be?
6.
The winning designs will also be made into posters that are dilated by a
scale factor of 6. What will the dimensions of the posters be? What will
the perimeter and the area of the posters be?
Activity 9-23: Dilations
Name:
Draw a dilation of the figure using the given scale factor, k.
1. k = 2
3. k =
1
2
2. k =
1
4
4. k = 1
1
2
Determine whether the dilation from Figure A to Figure B is a reduction or an enlargement. Then find
the values of the variables.
Activity 9-24: Dilations
Name:
Determine if the following scale factor would create an enlargement or reduction.
1. 3.5
2.
2
5
3. 0.6
4.
4
3
5.
5
8
Given the point and its image, determine the scale factor.
6. A(3,6) A’(4.5, 9)
9.
7. G’(3,6) G(1.5,3)
8. B(2,5) B’(1,2.5)
The sides of one right triangle are 6, 8, and 10. The sides of another right triangle are 10, 24, and
26. Determine if the triangles are similar. If so, what is the ratio of corresponding sides?
Use a provided coordinate grid or a piece of graph paper to answer the following questions.
10.
Draw triangle A(5,4) B(-2,1) C(0,-3). Apply transformation ( x, y ) → (2 x, 2 y ) What happened?
What is the scale factor of triangle ABC to triangle A’B’C’?
11.
Draw a dilation of quadrilateral A(2,1) B(4,2) C(5,5) D(1,3). Use the origin as the center and use
a scale factor of 2. What are the coordinates of A’, B’, C’, and D’? Write a symbolic
representation for the reflected triangle compared to the original.
12.
Draw a dilation of triangle A(9,8) B(-10,6) C(4,5). Use the origin as the center and use a scale
factor of 1/2. What are the coordinates of A’, B’, and C’? Write a symbolic representation for
the reflected triangle compared to the original.
13.
Draw a dilation of quadrilateral A(3,2) B(0,3) C(2,0) D(-2,2). Use origin as the center and use a
scale factor of 3. What are the coordinates of A’, B’, C’, and D’? Write a symbolic
representation for the reflected triangle compared to the original.
14.
Draw a dilation of quadrilateral A(10,10) B(5,8) C(-6,-2) D(1,-7). Use the original as the center
and use a scale factor of 1/2. What are the coordinates of A’, B’, C’, and D’? Write a symbolic
representation for the reflected triangle compared to the original.
Activity 9-25: Estimating Angles
Name:
Reference Angles:
45°
135°
90°
180°
Determine the best estimate for each angle. Circle your answer.
35°
70°
1.
65°
30°
2.
120°
95°
170°
3.
150°
55°
25°
4.
140°
85°
D
C
A
7.
50°
80°
5.
25°
110°
6.
110°
8.
155°
9.
P
Q
40°
15°
65°
m∠CAD is about…
90°
100°
m∠BAD is about…
130°
m∠BAC is about…
Z
O
R
Y
X
10.
11.
12.
160°
120°
40°
m∠POQ is about…
15°
105°
m∠QOR is about…
140°
m∠POR is about…
13.
14.
15.
60°
35°
45°
m∠ZYX is about…
25°
75°
m∠YZX is about…
40°
m∠YXZ is about…
B
Activity 9-26: Measuring Angles
Name:
Activity 9-27: Measuring Angles
Name:
Measure Angles: Write what type of angle each is and then measure it.
Activity 9-28: Measuring Angles
Name:
Draw the following angles using a protractor on a separate sheet of paper.
1. 43 degree angle
2. 116 degree angle
3. 135 degree angle
4. 20 degree angle
5. 165 degree angle
If you play golf, then you know the difference between a 3 iron and a 9 iron. Irons in the game of golf
are numbered 1 to 10. The head of each is angled differently for different kinds of shots. The number 1
iron hits the ball farther and lower than a number 2, and so on. Use the table below to draw all the
different golf club angles on the line segment below. Please use the 0 degree line as your starting point.
1 iron
15 degrees
6 iron
32 degrees
2 iron
18 degrees
7 iron
36 degrees
3 iron
21 degrees
8 iron
40 degrees
4 iron
25 degrees
9 iron
45 degrees
5 iron
28 degrees
Pitching wedge
50 degrees
0 degrees
90 degrees
Activity 9-29: Angles
Name:
Complete each statement.
C
B
A
D
O
1.
The figure formed by two rays from the same endpoint is an…
2.
The intersection of the two sides of an angle is called its…
3.
The vertex of ∠COD in the drawing above is point…
4.
The instrument used to measure angles is called a…
5.
The basic unit in which angles are measured is the…
6.
∠AOB has a measure of 90° and is called a ______ angle.
7.
An angle whose measure is between 0° and 90° is an ______ angle.
8.
Two acute angles in the figure are ∠BOC and _______.
9.
An angle whose measure is between 90° and 180° is an ______ angle.
10. An obtuse angle in the figure is _______.
Give the measure of each angle.
11 ∠RQS
12
∠RQT
13
∠RQU
14
∠RQV
15
∠RQW
16
∠XQW
17
∠XQT
18
∠UQV
19
∠VQT
20
∠WQS
U
T
V
W
X
S
Q
R
Activity 9-30: Complementary/Supplementary Angles
Name:
Complementary angles add up to 90° . Supplementary angles add up to 180° .
Find the measure of the angle that is complementary to the angle having the given measure.
1.
20°
2.
67°
3.
14°
4.
81°
5.
45°
6.
74°
Find the measure of the angle that is supplementary to the angle having the given measure.
7.
120°
8.
56°
9.
29°
10.
162°
11.
83°
12.
1°
Find the angle measure that is not given.
13.
14.
15.
47°
60°
16.
28°
17.
35°
116°
72°
19.
18.
20.
21.
67°
73°
34°
22. Name two complementary angles in the drawing at the right.
E
D
F
23. Name two supplementary angles in the drawing at the right.
A
B
C
Activity 9-31: Triangle Inequality Theorem
Name:
For any triangle, the sum of any two sides must be greater than the length of the third side.
Can a triangle be formed using the side lengths below? If so, classify the triangle as scalene, isosceles, or
equilateral.
1.
5, 5, 5
2.
1, 6, 4
3.
3, 2, 4
4.
6, 6, 4
5.
1, 4, 1
6.
4, 4, 8
7.
8, 6, 4
8.
3, 3, 7
9.
7, 4, 4
10.
8, 4, 5
11.
1, 2, 8
12.
12, 5, 13
13.
Two sides of a triangle are 9 and 11 centimeters long. What is the
shortest possible length in whole centimeters for the third side?
14. For the problem above what is the shortest possible length?
In each of the following you are given the length of two sides of a triangle. What can you conclude about
the length of the third side?
15.
10 m, 8 m
16.
14 in, 20 in
17.
6 cm, 9 cm
18.
12 ft, 7 ft
19.
11 cm, 3 cm
20.
9 mm, 13 mm
Activity 9-32: Angles In Triangles
Name:
Acute vision = sharp vision. Acute pain = a sharp pain.
An acute angle between 0 and 90 degrees has a fairly sharp vertex.
Triangle Sum Theorem
The sum of the three angles measures in any triangle is always equal to 180°.
Classify the triangles as right, acute, or obtuse, given the three angles.
1.
40°, 30°,110°
2.
60°,30°,90°
3.
50°, 60°, 70°
4.
90°, 46°, 44°
Classify each triangle as equilateral, isosceles, or scalene, given the lengths of the three sides.
5.
3 cm, 5 cm, 3 cm
6.
50 m, 50 m, 50 m
7.
2 ft, 5 ft, 6 ft
8.
x mm, x mm, y mm
Find the value of x. Then classify each triangle as acute, right, or obtuse.
9.
10.
12.
13.
11.
x°
60°
15.
16.
17.
14.
60°
18.
Activity 9-33: Angles In Triangles
Name:
The above triangle is equilateral. It is also an equiangular triangle since all angles are equal.
Use the figure at the right to solve each of the following.
15.
Find m∠1 if m∠2 = 30° and m∠3 = 55°.
16.
Find m∠1 if m∠2 = 110° and m∠3 = 25°.
17.
Find m∠4 if m∠1 = 30° and m∠2 = 55°.
18.
Find m∠4 if m∠1 = 45° and m∠2 = 60°.
19.
Find m∠4 if m∠1 = 35° and m∠2 = 45°.
2
3
1
Using an equation, find x and then find the measure of the angles in each triangle.
20.
21.
22.
x°
33°
5 x°
( x − 33)°
Using an equation, find x and then find the measure of the angles.
23.
3 x°
24.
x°
4
Activity 9-34: Sides of Triangles
Name:
In a triangle, the side opposite the angle with the greatest measure is the longest side.
Activity 9-35: Sides of Triangles
Name:
Which angle is the second-largest angle?
C
A landscaper wants to place benches in the two larger corners of the deck below. Which corners should she
choose?
Activity 9-36: Angles of Triangles
Name:
Angle-Angle Criterion for Similarity
We know that the angles of a triangle must add up to 180°. This means that if a triangle has two angle
measurements of 40° and 80°, then the third angle must be 60°. Now if a second triangle has two angle
measurements of 40° and 60°, we know the third angle must be 80°. This means the two triangles are the
same shape, but not necessarily the same size. Alternately we may think of one as a dilation of the other.
Either way we know that the triangles are similar. We call this the angle-angle criterion for similarity.
Decide if the following triangles are similar and explain why using the angle-angle criterion.
Triangle 1
Triangle 2
Triangle 1
Triangle 2
1.
45°, 45°
45°,90°
2.
50°,30°
30°,100°
3.
60°, 20°
40°,100°
4.
40°,30°
90°,30°
5.
25°,115°
25°, 40°
6.
5°,15°
120°,15°
7.
5°,15°
160°,15°
8.
45°,55°
55°,90°
9.
45°,30°
30°,100°
10.
80°, 40°
40°, 60°
11.
105°, 35°
40°,105°
12.
50°,50°
50°,90°
13.
80°,30°
70°, 30°
14.
72°, 23°
85°, 23°
Explain whether the triangles are similar.
15.
16.
The diagram below shows a Howe roof truss, which is used to frame the roof of a building.
17.
Explain why ∆LQN is similar to
∆MPN .
18. What is the length of support MP?
Using the information given in the
19. diagram, can you determine whether
∆LQJ is similar to ∆KRJ ? Explain.
Activity 9-37: Angles
Name:
Use the figure below to answer questions 1 through 7.
W
1
2
U
Q
S
R
3 4
V
1.
2.
3.
4.
5.
T
6
5
8
7
Describe how QR and ST are related.
A. They are perpendicular lines.
B. They are parallel lines.
Describe how WX and UV are related.
A. They are perpendicular lines.
B. They are parallel lines.
Describe how UV and ST are related.
A. They are perpendicular lines.
B They are parallel lines.
Which are complementary angles?
A. ∠1 and ∠2
C.
B. ∠5 and ∠6
D.
Which are supplementary angles?
A. ∠1 and ∠2
C.
B. ∠5 and ∠8
D.
X
C. They are intersecting lines.
D. They are complementary.
C. They are intersecting lines.
D. They are supplementary.
C. They are complementary.
D. They are right angles.
∠3 and ∠4
∠7 and ∠8
∠4 and ∠5
∠7 and ∠8
6.
If the measure of ∠5 is 45° , what is the measure of ∠6?
7.
What is the measure of ∠3?
What is the measure of x in the parallelogram?
124
56
8.
x
Write an equation to find x. Then find the measure of the missing angles in each triangle.
Angle 1
Angle 2
Angle 3
Angle 1
Angle 2
Angle 3
9.
x°
x + 20°
x + 70°
10.
2 x − 40°
x + 10°
3 x − 60°
11.
x°
120 − x°
100 − x°
12.
3 x°
2 x°
5 x°
Activity 9-38: Angles in Polygons
Name:
Find the value of x.
1.
90°
90°
x°
90°
3.
2.
75°
105°
4.
x°
62°
75°
x°
80°
114°
93°
x°
70°
103°
Write an equation to find x and then find all the missing angles.
5. A trapezoid with angles 115°, 65°, 55°, and x° .
6.
A quadrilateral with angles 104°, 60°, 140°, and x° .
7.
A parallelogram with angles 70°, 110°, (x+40)°, and x° .
8.
A quadrilateral with angles x°, 2x °, 3x °, and 4x° .
A quadrilateral with angles ( x + 30)°, (x-55)°, x °, and (x − 45)° .
Which of the following could be the angle measures in a parallelogram (all
numbers are in degrees):
10.
a) 19, 84, 84, 173
b) 24, 92, 92, 152
c) 33, 79, 102, 146
d) 49, 49, 131, 131
9.
For any polygon with n sides, the following formula can be used to calculate the sum of the angles:
180(n − 2)
Find the sum of the measures of the angles of each polygon.
11. quadrilateral
12.
pentagon
13.
octagon
14.
12-gon
15.
18-gon
16.
30-gon
17.
75-gon
18.
100-gon
For any polygon with n sides, the following formula can be used to calculate the average angle of size:
180(n − 2)
n
Find the measure of each angle of each regular polygon (nearest tenth).
19.
regular octagon
20.
regular pentagon
21.
regular heptagon
22.
regular nonagon
23.
regular 18-gon
24.
regular 25-gon
Activity 9-39: Angles in Lines
Name:
A transversal is a line that intersects two or more other lines to form eight or more angles.
1. Name three pairs of angles above that are supplementary.
2. Which angles appear to be acute?
3. Which angles appear to be obtuse?
4. If ∠1 = ( x + 25)° and ∠2 = 85° find the size of all the other listed angles.
5. If ∠1 = 5 x° and ∠2 = 65° find the size of all the other listed angles.
Alternate angles are on opposite sides of the transversal and have a different vertex. There are two pairs
of angles in the diagram that are referred to as alternate exterior angles and two pairs of angles that are
referred to as alternate interior angles.
Activity 9-40: Angles in Lines
Name:
Adjacent angles
have a common side
and vertex but no
common interior
points.
Vertical angles are pairs of
nonadjacent angles formed
when two lines intersect.
They share a vertex but have
no common rays.
1
8
2
7
3
6
4
5
1.
Name all pairs of vertical angles in the figure.
2.
Name all pairs of alternate interior angles in the figure.
3
Name all pairs of alternate exterior angles in the figure.
4.
Name all pairs of corresponding angles in the figure.
5.
Name two pairs of adjacent angles.
6.
Name all of the angles that are supplementary to ∠8.
7.
If m∠2 = 57°, find m∠3 and m∠4.
8.
If m∠6 = (5 x + 1)° and m∠8 = (7 x − 23), find m∠6 and m∠8.
9.
Suppose ∠9 , which is not shown in the figure, is complementary to ∠4 .
Given that m∠1 = 153°, what is m∠9 ?
Activity 9-41: Parallel Lines Cut by a Transversal
Name:
Use the figure below for Exercises 1–6.
1.
Name both pairs of alternate interior angles.
2.
Name the corresponding angle to ∠3 .
3.
Name the relationship between ∠1 and ∠5 .
4.
Name the relationship between ∠2 and ∠3 .
5.
Name the interior angles that is supplementary to ∠7 .
6.
Name the exterior angles that is supplementary to ∠5 .
Use the figure at the right for problems 7–10. Line MP || line QS.
Find the angle measures.
7. m∠KRQ when m∠KNM = 146° ____
8. m∠QRN when m∠MNR = 52° ____
If m∠
∠RNP = (8x + 63)°° and m∠
∠NRS = 5x°°, find
the following angle measures.
9. m∠RNP = _________________
10. m∠NRS = _________________
In the figure at the right, there are no parallel lines. Use the figure for problems 11–14.
11. Name both pairs of alternate exterior angles.
________________________________________
12. Name the corresponding angle to ∠4 ____
13. Name the relationship between ∠3 and ∠6.
________________________________________
14. Are there any supplementary angles? If so, name two pairs. If not, explain why
not.
Activity 9-42: Angles in Lines
Name:
An exterior angle of a triangle is formed by extending a side of the triangle.
Describe the relationship between ∠ZYX and ∠XYM .
Find the measure of each of the three exterior angles.
For each exterior angle of a triangle, the two nonadjacent interior angles are its remote interior angles.
Complete the table below.
Exterior Angle
Size of each remote interior
Sum of the two remote
Exterior Angle
Size
angle
interior angles
∠A
∠B
∠C
Use the diagram at the right to answer each question below.
9. What is the measure of ∠DEF?
________________________________________
10. What is the measure of ∠DEG?
________________________________________
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 9-43: Angle Relationships/Parallel Lines
Name:
Find the value of x in each figure.
1.
2.
3.
Each pair of angles is either complementary or supplementary. Find the measure of each angle.
4.
5.
6.
( x + 20)°
In the figure at the right m n . If the measure of ∠3 is 95° , find the measure of each angle.
7.
∠1
8.
∠2
9.
∠3
10.
∠4
11.
∠5
12.
∠6
13.
∠7
14.
∠8
2
1
6
3
4
7
5 8
In the figure at the right m n . Find the measure of each angle.
15.
∠1
16.
∠2
17.
∠3
18.
∠4
19.
∠5
20.
∠6
21.
∠7
22.
∠8
23.
∠9
24.
∠10
8
1
2
5
6
9
m
3
4 7
10
n
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 9-44: Angles
Name:
Figure 1
Figure 2
A
C
A
B
E
1.
2.
C
B
D
E
D
Use Figure 1 to find the following:
Find x. Find the measure of ∠ABC . Find the
measure of ∠ABE .
Use Figure 2 to find the following:
Find x. Find the measure of ∠ABC . Find the
measure of ∠ABE .
For Exercises 1, use the figure at the right.
1.
Name a pair of vertical angles. Name a pair of complementary
angles. Name a pair of supplementary angles.
Use the diagram to find each angle measure.
4. If m∠1 = 120° , find m∠3 .
5.
If m∠3 = 110° , find m∠2 .
6. If m∠2 = 13° , find m∠4 .
7.
If m∠4 = 65° , find m∠1 .
Find the value of x in each figure.
8.
9.
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 9-45: Names of Polygons
Name:
The word “gon” is derived from the Greek word “gonu”. Gonu means “knee”, which transferred to the
word “angle” in English.
SIDES
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
NAME
monogon
digon
trigon or triangle
tetragon or quadrilateral
pentagon
hexagon
heptagon or septagon
octagon
enneagon or nonagon
decagon
hendecagon
dodecagon
triskaidecagon
tetrakaidecagon or
tetradecagon
pentakaidecagon or
pentadecagon
hexakaidecagon or
hexadecagon
heptakaidecagon
octakaidecagon
enneakaidecagon
icosagon
SIDES
21
22
23
24
25
26
27
28
29
30
31
40
41
NAME
icosikaihenagon
icosikaidigon
icosikaitrigon
icosikaitetragon
icosikaipentagon
icosikaihexagon
icosikaiheptagon
icosikaioctagon
icosikaienneagon
triacontagon
tricontakaihenagon
tetracontagon
tetracontakaihenagon
50
pentacontagon
60
hexacontagon
70
heptacontagon
80
90
100
1000
10000
octacontagon
enneacontagon
hectogon or hecatontagon
chiliagon
myriagon
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 9-46: Photographs
Name:
Kaitie moves five pictures around on the table and then steps back to look at the arrangement. She is
arranging the photographs for a page in the school yearbook. The five pictures that she is working with
are all the same size square. If each photo has to share at least one side with another photo, what are all
the different ways she could place the pictures on the page?
(Rotations are considered the same shape. A reflection which produces a different shape is considered a
new shape.)
To determine all the arrangement it makes perfect since to draw out pictures. One strategy when
drawing pictures is to still create them in an organized way. Otherwise, you won’t have any idea when
you have completed all of your pictures.
To start with the easiest is to place all 5 in a straight line.
Continue drawing pictures starting with all the 4 in a straight line options and so on.
Created by Lance Mangham, 6th grade math, Carroll ISD