Congruence, Triangles, and Quadrilaterals

100 Pages·2010·5.46 MB·English

Unit
Congruence, 2
Triangles, and
Quadrilaterals
Unit Overview
Essential Questions
Properties and characteristics of triangles, quadrilaterals
and polygons are the focus of study in this unit. Using
?? How does proving theorems
a variety of methods you will prove theorems related to extend your understanding of
these fi gures. Geometry?
Academic Vocabulary
?? How are patterns, algebra,
Add these words to the academic vocabulary portion of and geometry related?
your math notebook.
altitude of a triangle indirect proof
EMBEDDED
circumscribe isosceles triangle
ASSESSMENTS
equiangular regular polygon
The three embedded assessments
exterior angle
in this unit allow you to demonstrate
ed. your understanding of the measures
v
ser of interior and exterior angles of
e
hts r polygons and your knowledge
g
All ri of quadrilaterals. By using
d. several methods of proof you will
ar
o demonstrate your ability to present
B
e
g convincing mathematical arguments.
e
oll
C
0
01 Embedded Assessment 1
2
©
Angles and Sides of Polygons
p. 123
Embedded Assessment 2
Congruence, Triangles, and
Proof p. 157
Embedded Assessment 3
Quadrilaterals p. 183
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UNIT 2
Getting
Ready
Write your answers on notebook paper. 5. Find the slope of a line that passes through
(-2, 5) and (1, 7).
1. Simplify.
___
a. √64 6. Write the equation of a line that contains the
___
b. √98 points (3, 8) and (-4, -6).
2. Solve the following equations. 7. Find the midpoint and length of a line segment
a. x2 - 8x + 15 = 0 that has endpoints (3, 5) and (9, -3).
b. x2 = 27
8. a. Find the solution of the system of equations:
3. If f (x) = 2x - 5, fi nd x + 3y = 4
a. f (4) 5x - 2y = 3
b. f (-3) b. Explain the method you used to fi nd the
solution.
4. Write the equation of the line shown below.
y 
9
8
7
6
5
4
3
2
1
x
–9–8–7–6–5–4–3–2 –1 1 2 3 4 5 6 7 8 9
–2
–3
–4
–5
–6
–7
–8 d.
e
–9 v
er
s
e
s r
ht
g
All ri
d.
ar
o
B
e
g
e
oll
C
0
1
0
2
©
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Interior Angles of Polygons
ACTIVITY
Plenty of Polygons 2.1
SUGGESTED LEARNING STRATEGIES: Summarize/Paraphrase/
Retell, Activating Prior Knowledge, Think/Pair/Share
My Notes
It is impossible to measure lengths and angles exactly; however, tools such
as rulers and protractors allow you to measure with reasonable accuracy.
In the diagram below, the sum of the measures of the fi ve indicated angles
should be exactly 360°.
1. Measure the fi ve angles and record each measure in the table below.
Th en calculate the sum of the angle measures.
2 3
1
5 4
m ∠ 1 m ∠ 2 m ∠ 3 m ∠ 4 m ∠ 5 Sum
d.
e
v
er
s
e
s r
ght 2. Compare your sum to the results of other students in your class. Are
All ri the results always the same? If not, explain why diff erences can occur
d.
ar and which answers are reasonably close.
o
B
e
g
e
oll
C
0
1
0
2
©
Unit 2 • Congruence, Triangles, and Quadrilaterals 93
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Interior Angles of Polygons
ACTIVITY 2.1
continued PPlleennttyy ooff PPoollyyggoonnss
SUGGESTED LEARNING STRATEGIES: Think/Pair/Share,
Create Representations, Look for a Pattern, Self/Peer Revision
My Notes
Your teacher will provide your group with a page containing polygons.
3. Measure, as precisely as possible, each interior angle, and record your
results below. Complete the table by calculating the indicated sums.
MATH TERMS 1st 2nd 3rd 4th 5th 6th
Total
A polygon is a closed Angle Angle Angle Angle Angle Angle
geometric fi gure with
Triangle
sides formed by three or
more coplanar s egments
Quadrilateral
that intersect exactly two
other segments, one at
Pentagon
each endpoint. The angles
formed inside the polygon
Hexagon
are interior angles.
4. Compare your results in the table with those of other groups in your
class. What similarities do you notice?
5. Write a conjecture about the sum of measures of the interior angles of
each of the polygons.
d.
e
Triangle: erv
s
e
s r
ht
g
Quadrilateral: All ri
d.
ar
o
B
e
g
e
Pentagon: oll
C
0
1
0
2
©
Hexagon:
Knowing that the sum of the measures of the interior angles of a triangle
is a constant and that the sum of the measures of non-overlapping angles
around a single point is always 360°, you can determine the sum of the
measures of any polygon without measuring.
94 SpringBoard®Mathematics with Meaning™ Geometry
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Interior Angles of Polygons
ACTIVITY 2.1
PPlleennttyy ooff PPoollyyggoonnss continued
SUGGESTED LEARNING STRATEGIES: Summarize/Paraphrase/
Retell, Think/Pair/Share, Create Representations, Look for a
Pattern, Quickwrite
My Notes
6. Use auxiliary segments to determine a way to predict and verify
the exact sum of the angles of any quadrilateral and any pentagon. MATH TERMS
Describe your methods so that another group would be able to To replicate means to duplicate
replicate your results for the pentagon. or imitate. Notice the word
replica contained within the
word. In science, experiments
are designed so that they can be
replicated by other scientists.
7. Use the method you described in Item 6 to answer the following.
a. Explain how to determine the sum of the measures of the interior
angles of a hexagon.
b. Explain how to determine the sum of the measures of the interior
angles for any polygon.
8. Use the method described in your answer to Item 7 to complete the
table below.
Number of Sum of the Measures
Polygon Calculations
d. Sides of the Interior Angles
e
v
er
s
e Triangle 3
s r
ht
g
All ri Quadrilateral 4
d.
ar
Bo Pentagon 5
e
g
e
Coll Hexagon
0
1
0
2
© Heptagon
Octagon
Nonagon
Decagon
Dodecagon
n-gon n
Unit 2 • Congruence, Triangles, and Quadrilaterals 95
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Interior Angles of Polygons
ACTIVITY 2.1
continued PPlleennttyy ooff PPoollyyggoonnss
SUGGESTED LEARNING STRATEGIES: Think/Pair/Share,
Create Representations
My Notes
9. Observe in Item 8 that as the number of sides increases by one, the
sum of the angle measures also increases by a constant amount. What
type of function models this behavior?
10. For the fi rst six polygons in Item 8, plot the ordered pair (number of
sides, sum of angle measures) on the axes below. Carefully choose and
label your scale on each axis.
11. T he data points you graphed above should appear collinear.
Write an equation for the line determined by these points.
Write the equation in slope-
d.
intercept form: ve
er
s
e
y = mx + b s r
ht
g
All ri
12. State the numerical value of the slope of the line in Item 11 and ard.
o
describe what the slope value tells about the relationship between the e B
g
number of sides and the sum of the measures of the interior angles of olle
C
a polygon. Use units in your description. 0
1
0
2
©
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Interior Angles of Polygons
ACTIVITY 2.1
PPlleennttyy ooff PPoollyyggoonnss continued
SUGGESTED LEARNING STRATEGIES: Think/Pair/Share,
Create Representations
My Notes
13. If the function S represents the sum of the measures of the interior
angles as a function of the number of sides n, write an algebraic rule
for S(n).
14. Use S(n) to determine the value of S(7.5). For this value, explain the
signifi cance, if any, given that S(n) represents the sum of the angle
measures of an n-sided polygon.
Linear functions have continuous domains consisting of all real numbers.
However, some contexts restrict the domain of linear functions. If the
graph of a linear model consists of discrete points, the linear function is
said to have a discrete domain.
15. What is the domain of S(n)?
16. In a regular polygon, each interior angle has the same angle measure.
ACADEMIC VOCABULARY
Based upon your results from the table in Item 8, determine the angle
A regular polygon is a polygon
measure in degrees of each interior angle for each regular polygon below.
that is both equiangular, with
all angles congruent, and
Sum of the Measure of
equilateral, with all sides
Number Measures of the Each Interior
ed. Polygon congruent.
v of Sides Interior Angles Angle
er
s
s re (degrees) (degrees)
ht
g
All ri Triangle
d.
ar Quadrilateral
o
B
e
eg Pentagon
oll
C
10 Hexagon
0
2
©
Heptagon
Octagon
Nonagon
Decagon
Dodecagon
n-gon
Unit 2 • Congruence, Triangles, and Quadrilaterals 97
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Interior Angles of Polygons
ACTIVITY 2.1
continued PPlleennttyy ooff PPoollyyggoonnss
SUGGESTED LEARNING STRATEGIES: Create
Representations, Look for a Pattern, Quickwrite
My Notes
17. For each regular polygon listed in the table in Item 16, plot the
ordered pair (number of sides, measure of each interior angle) on the
axes below. Carefully choose and label your scale on each axis.
e
r
u
s
a
Mees)
or Angle (in degre
ri
e
nt
I
Number of Sides
18. T he points plotted in Item 17 should not appear collinear. Explain
how that conclusion could have been drawn from the data alone.
19. If the function E represents the measure of each interior angle as a
function of the number of sides n, write an algebraic rule for E(n).
TECHNOLOGY d.
e
v
er
s
e
s r
Use the TABLE or GRAPHING ht
g
ccoalmcuploantoern.t E onft ea r g trhaep ha lignegb raic 20. As n gets very large, what appears to be happening to the measure of d. All ri
each angle? What causes this behavior? oar
function for E(n) in y to B
1 e
g
explore the measure of e
oll
individual angle measures 0 C
1
of a regular polygon as the 20
©
number of sides increase.
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Interior Angles of Polygons
ACTIVITY 2.1
PPlleennttyy ooff PPoollyyggoonnss continued
SUGGESTED LEARNING STRATEGIES: Think/Pair/Share,
Look for a Pattern
My Notes
21. Determine the angle measures in the interior of the concave decagon.
Th e fi ve acute angles are congruent, and the fi ve refl ex angles are MATH TERMS
congruent. A concave polygon is a
polygon that has one or more
interior angles measuring more
than 180°.
A convex polygon is a polygon,
all of whose angles measure less
than 180°.
22. Does your function S(n) in Item 13 apply to this concave decagon?
Explain.
A concave polygon has sides that
appear to cave in. In a concave
polygon, like pentagon ABCDE
below, it is possible to draw a
line segment with endpoints in
the interior of the polygon such
23. Does your function E(n) in Item 19 apply to this concave decagon?
that the segment contains points
Explain.
in the exterior of the polygon.
A C
B
d.
e
v
ser E D
e
s r
ht
g
All ri
d.
ar
o
B
ge MATH TERMS
e
Coll A refl ex angle is an angle with a
0
01 measure greater than 180° and
2
© less than 360°. To determine
the measure of a refl ex angle,
use the fact that there are 360°
about a point in the plane.
You know how to measure the
related angle whose measure is
less than 180°.
Unit 2 • Congruence, Triangles, and Quadrilaterals 99
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Interior Angles of Polygons
ACTIVITY 2.1
continued PPlleennttyy ooff PPoollyyggoonnss
CHECK YOUR UNDERSTANDING
WWrriittee y yoouur ra nanswsweresr osn o nno nteobteobooko pka ppearp. eSrh.o Swh ow your wor k5.. If the sum of the measures of the interior angles of
your work. a polygon is 2700°, how many sides does it have?
Determine the missing angle measure for each a. 9 b. 15
polygon.
c. 17 d. Not enough information
1.
6. If the measure of each interior angle of a
87°
polygon is 150°, how many sides does it have?
108° 140°
a. 10 b. 12
100° k
c. 15 d. Not enough information
7. Given (cid:2)APT with angle measures as labeled,
2. solve for x and calculate the three angle measures.
b
94° P
18°
(6x+1)°
3. (5x-17)° (9x-24)°
A T
Draw polygons to satisfy the given conditions.
8. An equiangular quadrilateral that is not
equilateral.
4.
88° 9. An equilateral hexagon that is not equiangular. ed.
v
er
s
(13x −3)° 10. A concave pentagon with a refl ex angle s re
measuring 200°. ght
(7x+15)° 80° All ri
11. A regular polygon with an angle measuring 135°. d.
ar
o
12. MATHEMATICAL As the number of sides of a e B
g
REFLECTION regular polygon increases, olle
C
what happens to the shape of the polygon? 10
0
2
©
100 SpringBoard®Mathematics with Meaning™ Geometry
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