MODULE
STUDY GUIDE REVIEW
Tools of Geometry
Essential Question: How can you use tools of geometry to solve
real-world problems?
KEY EXAMPLE
(Lesson 1.1)
Find the midpoint of (5, 6) and (1, 3).
6+3
5+1 _
,
(_
2
2 )
Apply the midpoint formula.
( )
( )
9
6 , __
= __
2 2
9
= 3, __
2
Simplify the numerators.
Simplify.
KEY EXAMPLE
(Lesson 1.2)
→
‾ is the angle bisector of ∠ABC and m∠ABC = 40°.
The ray BD
Find m∠ABD.
→
‾ is the angle bisector of ∠ABC so it divides the angle into two
BD
angles of equal measure.
Then m∠ABD + m∠DBC = m∠ABC and m∠ABD = m∠DBC.
So, 2 · m∠ABD = m∠ABC.
m∠ABD = 20°
Substitute the angles and simplify.
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KEY EXAMPLE
(Lesson 1.3)
Use the rule (x, y) → (x + 1, 2y) and the points of a triangle,
A(1, 2), B(2, 4), and C(2, 2) to draw the image. Determine
whether this is a rigid motion.
A'(1 + 1, 2(2)), B'(2 + 1, 2(4)),
C'(2 + 1, 2(2))
A'(2, 4), B'(3, 8), C'(3, 4)
___
A'B' = √(3 - 2) + (8 - 4)
2
2
_
= √17 ≈ 4.1
___
AB = √ (2 - 1) + (4 - 2)
2
_
= √ 5 ≈ 2.2
2
Use the transformation
rule.
Simplify.
Use the distance formula
to find the distance
between A’ and B’.
Simplify.
Key Vocabulary
1
point (punto)
line (línea)
plane (plano)
line segment (segmento de
línea)
endpoints (punto final)
ray (rayo)
coplanar (coplanares)
parallel (paralelo)
collinear (colineales)
postulate (postulado)
midpoint (punto medio)
segment bisector (segmento
bisectriz)
angle (ángulo)
vertex (vértice)
side (lado)
degrees (grados)
angle bisector (bisectriz de un
ángulo)
transformation
(transformación)
preimage (preimagen)
image (imagen)
rigid motion (movimiento
rígido)
conjecture (conjetura)
inductive reasoning
(razonamiento inductivo)
deductive reasoning
(razonamiento deductivo)
theorem (teorema)
counterexample
(contraejemplo)
conditional statement
(sentencia condicional)
linear pair (par lineal)
Use the distance formula
to find the distance
between A and B.
Simplify.
The image is not a rigid motion because the side lengths are not equal.
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Study Guide Review
EXERCISES
Find the midpoint of the pairs of points. (Lesson 1.1)
(4, 7) and (2, 9)
1.
2.
(5, 5) and (-1, 3)
Find the measure of the angle formed by the angle bisector. (Lesson 1.2)
→
‾ is the angle bisector of ∠ABC and m∠ABC = 110°. Find m∠ABD.
3. The ray BD
Use the rule (x, y) → (3x, 2y) to find the image for the preimage defined by the
points. Determine whether the transformation is a rigid motion. (Lesson 1.3)
4. A(3, 5), B(5, 3), C(2, 2)
.
The points of the image are
The image
a rigid motion.
Determine whether the conjecture uses inductive or deductive reasoning. (Lesson1.4)
5. The child chose Rock in all four games of Rock-Paper-Scissors. The child always chooses Rock.
MODULE PERFORMANCE TASK
How Far Is It?
Many smartphone apps and online search engines will tell you the distances to nearby
restaurants from your current location. How do they do that? Basically, they use
latitude and longitude coordinates from GPS to calculate the distances. Let’s explore
how that works for some longer distances.
•
•
Module 1
City
Which of the state capitals do you think
is nearest to you? Which is farthest away?
Use the distance formula to calculate your
distance from each of the cities in degrees.
Then convert each distance to miles.
Latitude
Austin, TX
30.31° N
97.76° W
Columbus, OH
39.98° N
82.99° W
Nashville, TN
36.17° N
86.78 ° W
Sacramento, CA
38.57° N
121.5° W
Use an app or search engine to find the
distance between your location and
Your Location
each of the capital cities. How do these
distances compare with the ones you calculated? How might you account for any
differences?
58
Longitude
Study Guide Review
© Houghton Mifflin Harcourt Publishing Company
The table lists latitude and longitude for four state capitals. Use an app or search
engine to find the latitude and longitude for your current location, and record them
in the last line of the table.
Ready to Go On?
1.1–1.4 Tools of Geometry
• Online Homework
• Hints and Help
• Extra Practice
Use a definition, postulate, or theorem to find the value desired.
1. Point M is the midpoint between points A(-5, 4) and B(-1, -6). Find the location of M.
(Lesson 1.1)
Given triangle EFG, graph its image E'F'G' and confirm that the transformation
preserves length and angle measure. (Lesson 1.1)
2. (x, y) → (x - 1, y + 5)
8
y
4
-8
-4
0 E
-4
© Houghton Mifflin Harcourt Publishing Company
-8
4
x
F
G
Find the measure of the angle formed by the angle bisector. (Lesson 1.2)
⇀
3. The ray ‾ GJ is the angle bisector of ∠FGH and m∠FGH = 75°. Find m∠FGJ.
⇀
4. The ray XZ
is the angle bisector of ∠WXY and m∠WXY = 155°. Find m∠YXZ.
‾
ESSENTIAL QUESTION
5. When is a protractor preferred to a ruler when finding a measurement?
Module 1
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Study Guide Review
MODULE 1
MIXED REVIEW
Assessment Readiness
1. For two angles, ∠ABC and ∠DBC, m∠ABC = 30° and ∠DBC is its
⇀
complement. Ray ‾ BE is the angle bisector of ∠ABD. Consider each angle.
Does the angle have a measure of 45°?
Select Yes or No for A–C.
A. ∠DBC
Yes
No
B. ∠ABE
C. ∠DBE
Yes
Yes
No
No
_
_
2. The line y = √ x is transformed into y = √5x . Choose True or False for each
statement.
A. A dilation can be used to obtain this
True
False
transformation.
B. A rotation can be used to obtain this
True
False
transformation.
C. A translation can be used to obtain this
True
False
transformation.
3. Triangle ABC is given by the points A(1, 1), B(3, 2), and C(2, 3).
Consider each rule of transformation. Does the rule result in an image with
points A'(2, 2), B'(6, 3), and C'(4, 4)?
Select Yes or No for A–C.
A. (x, y) → (x, y + 1)
Yes
No
B. (x, y) → (2x, 2y)
Yes
No
C. (x, y) → (2x, y + 1)
Yes
No
4. Find the midpoint of (4, 5) and (–2, 12). Show your work.
© Houghton Mifflin Harcourt Publishing Company
Module 1
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Study Guide Review