Download Properties of Triangles

MODULE
STUDY GUIDE REVIEW
22
Properties of Triangles
Essential Question: How can you use the properties of triangles
to solve real-world problems?
KEY EXAMPLE
(Lesson 22.1)
Determine the measure of the fifth interior angle of a pentagon if
you know the other four measures are 100°, 50°, 158°, and 147°.
Sum = (5 - 2)180° = 540°
Apply the Polygon
Angle Sum Theorem.
100 + 50 + 158 + 147 + x = 540
Set the sum of the angle
measures equal to 540.
455 + x = 540
Solve for x.
x = 85
KEY EXAMPLE
(Lesson 22.2)
¯, AB = 4x + 3,
¯ ≅ AC
Given an isosceles triangle △ABC with AB
and AC = 8x − 13, find AB.
_ _
Given
AB ≅ AC
4x + 3 = 8x − 13
interior angle (ángulo interior)
auxiliary line (línea auxiliar)
exterior angle (ángulo
exterior)
remote interior angle (ángulo
interior remoto)
isosceles triangle (triángulo
isósceles)
legs (catetos)
vertex angle (ángulo del
vértice)
base (base)
base angles (ángulos de la
base)
equilateral triangle
(triángulo equilátero)
equiangular triangle
(triángulo equiangular)
Substitution
x=4
Solve for x.
AB = 4(4) + 3
Substitute the value of x into AB.
AB = 19
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Key Vocabulary
Simplify.
KEY EXAMPLE
(Lesson 22.3)
Given a triangle with sides 7, 12, and x, find the range of values for x.
According to the Triangle Inequality Theorem, the sum of any two side lengths of a triangle is greater
than the third side length
7 + 12 > x
19 > x
5 < x < 19
Module 22
7 + x > 12
x>5
x + 12 > 7 Apply the Triangle Inequality Theorem.
x > −5
Simplify.
Combine the inequalities together.
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Study Guide Review
EXERCISES
Find how many sides a polygon has with the given interior angle sum. (Lesson 22.1)
1.
2700°
2. 1800°
Find the sum of interior angles a polygon has with the given number of sides.
(Lesson 22.1)
3. 3
4. 19
DE ≅ ¯
DF, DE = 26, and m∠F = 45°, find
Given an isosceles triangle △DEF with ¯
the desired measurements. (Lesson 22.2)
5. DF
6. m∠D
Determine whether a triangle can have sides with the given lengths. (Lesson 22.3)
7. 5 mi, 19 mi, 15 mi
8. 4 ft, 3 ft, 10 ft
Find the range of the unknown side of a triangle with the given sides. (Lesson 22.3)
9. 5 mi, 19 mi, x mi
10. 4 ft, 3 ft, x ft
MODULE PERFORMANCE TASK
What’s Up in the Federal Triangle?
The diagram shows a schematic of the Federal Triangle, an area located in Washington, DC.
The area is bounded by Constitution Avenue on the south and Pennsylvania Avenue on the
th
north and extends from 12 th Street on the west to just past 6 Street on the east.
Federal Triangle
© Houghton Mifflin Harcourt Publishing Company
Is the shape of the Federal Triangle a triangle?
How many sides does the Federal Triangle have?
What is the actual shape of the Federal Triangle?
What is the sum of the internal angles of the
Federal Triangle? What portion of the area is
actually a triangle?
Do some research and find the lengths of each
side. Find the perimeter and area of the Federal
Triangle. Find the area of the portion of the
Federal Triangle that is a triangle.
Module 22
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Study Guide Review
Ready to Go On?
22.1–22.3 Properties of Triangles
Determine whether a triangle can be formed with the given side lengths. If the
side lengths can form a triangle, determine if they will form an isosceles triangle,
equilateral triangle, or neither. (Lessons 22.2, 22.3)
• Online Homework
• Hints and Help
• Extra Practice
1. 3 mi, 8 mi, 3 mi
2. 7 cm, 7cm, 7cm
3. 4 ft, 4 ft, 2 ft
4. 20 m, 30 m, 10 m
5. 3 m, 4 m, 5 m
7
4
8 2
6. 26 yd, 26 yd, 26 yd
© Houghton Mifflin Harcourt Publishing Company
6
1
3
Use the figure to answer the following. (Lesson 22.1)
5
7. Given m∠2 = 76°, m∠1 = 3 ⋅ m∠3, and ∠4 ≅ ∠8, find m∠1, m∠3, m∠4, m∠5, m∠6, m∠7,
and m∠8.
ESSENTIAL QUESTION
8. Is it possible for one angle of a triangle to be 180° ? If so, demonstrate with an example. If not,
explain why not.
Module 22
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Study Guide Review
MODULE 22
MIXED REVIEW
Assessment Readiness
1. Two angles in a triangle have measurements of 34° and 84°.
Select Yes or No for A–C.
A. Does the third angle measure 62°?
Yes
B. Could a triangle congruent to this
Yes
one contain an angle of 75°?
C. Is this triangle congruent to a
Yes
right triangle?
No
No
No
2. Consider the following statements about a seven-sided polygon. Choose True or
False for each statement.
A. Each interior angle measures 135°.
True
False
B. The sum of the measures of the interior
angles is 1260°.
C. The sum of the measures of the interior
angles is 900°.
True
False
True
False
3. △ABC is an equilateral triangle, AB = 4x + 45, and BC = 6x - 3. Choose True or
False for each statement.
A. x = 24
True
False
B. The length of one side of the triangle is
141 units.
C. The distance from one vertex of the
triangle to the midpoint of an adjacent
side is 12 units.
True
False
True
False
4. Given a triangle with a side of length 6 and another side of length 13, find the
range of possible values for the third side, x.
Module 22
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Study Guide Review
© Houghton Mifflin Harcourt Publishing Company
5. Given △DEF, with DE = 3EF and DF = 4DE, explain how to write the sides and
angles in order of least to greatest.