Download The Triangle-Sum Property The Triangle-Sum Property

Chapter 14
Lesson
The Triangle-Sum
Property
14-7
Vocabulary
revolution
interior angles of a polygon
exterior angles of a polygon
BIG IDEA
In all triangles, the sum of the measures of the
three angles is the same number.
Ancient Babylonians realized that each night at the same time, stars are
in a slightly different position than the night before. They have rotated a
small amount. In a year, they rotate all the way to their original position,
1
because Earth goes around the sun in a year. So, they rotate _
of the
365
way around a circle each day. The number 365 is not an easy number
to use in making calculations. Because 360 is divisible by 2, 3, 4, 5, 6, 8,
9, 10, and 12, it is an easier number to use. In fact, 360 is the smallest
number divisible by all these numbers. So the Babylonians thought of
1
the stars as moving _
of the way around the sky each day. From this
360
we get that one full revolution is a rotation of magnitude 360º.
Activity
MATERIALS thin paper such as tracing paper, a pencil, and a ruler
Step 1 Draw any triangle ABC and trace it onto the thin paper.
Label the angles 1, 2, and 3 at points A, B, and C, as shown.
A
1
B
3
2
C
Step 2 Measure to find the midpoint M of BC. Rotate ABC 180º about M.
(Move the triangle you originally drew and trace it again.)
The preimage and image are shown. A' is the image of A.
When you perform this rotation, where are the images of B and C?
A
SMP10_A_SE_C14_L07A_T_002
1
B
2
3
M
3
2
C
1
A'
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Some Important Geometry Ideas
SMP10_A_SE_C14_L07A_T_003
Lesson 14-7
Step 3 Measure to find the midpoint N of A'C. Rotate A'BC 180º about N.
B' is the image of B.
A
1
2
3
B
M
3 C
2 1
N
3
1 2
B'
A'
Step 4 Measure to find the midpoint O of B'C.
Rotate A'B'C 180º about O. A* is the image of A'.
SMP10_A_SE_C14_L07A_T_004
A
1
2
3
B
3 C
2 13
M
N
O
3
1 2
A*
2
1
B'
A'
Step 5 Measure to find the midpoint P of A*C.
Rotate A*B'C 180º about P. B* is the image of B'.
B*
A
1
1
2
3
B
C
3 2
2 13
M
N
2
A*
O
3
1 2
3
P
1
B'
A'
Step 6 Measure to find the midpoint Q of B*C.
Rotate A*B*C 180º about Q. Where is the image of A*?
B*
A
2 1
SMP10_A_SE_C14_L07A_T_006
3
Q
1
B
2
3
P
C3 1 2
M
2 1
N
1 2
3
3
2
A*
O
3
1
B'
A'
You should find that the triangles completely cover the region around point C.
Is this result the same for your classmates who started with different triangles?
SMP10_A_SE_C14_L07A_T_007
The Triangle-Sum Property
37
Chapter 14
In the Activity, notice that at point C, ∠1, ∠2, and ∠3 each appear
twice. Together, these six angles form a full revolution around
point C. One full revolution measures 360º. The Activity illustrates
that 2(m∠1 + m∠2 + m∠3) = 360º, so m∠1 + m∠2 + m∠3 = 180º.
This is a property of any triangle.
Triangle-Sum Property
The sum of the measures of the three angles of any triangle
is 180º.
QY
QY
What is the sum of the
measures of the angles of
an isosceles triangle?
Example 1
Suppose two angles of a triangle measure 36.4º and 64.6º. What is the
measure of the third angle of the triangle?
Solution Let x be the measure of the third angle. The sum of the measures of
the three angles is 180º, so
xº + 36.4º + 64.6º = 180º.
x + 101 = 180
x = 79
The third angle measures 79º.
Applying the Triangle-Sum Property
Recall that complementary angles are two angles whose measures
add to 90º. In the diagram of Example 2, ∠TOP and ∠TOS are
complementary angles, because m∠POS = 90º.
P
Example 2
In the diagram at the right, find x.
19º
Solution x = m∠OTS, and ∠OTS is in OTS.
So x + y + m∠TOS = 180°. m∠TOS = 43°.
To find y, use POS.
m∠P + m∠POS + y = 180°
19° + 90° + y = 180°
T
y = 71°
x
So x + 71° + 43° = 180°
x = 66°
38
O
43º
y
S
Some Important Geometry Ideas
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Lesson 14-7
Angles formed by the sides of a polygon are its interior angles.
Angles that form linear pairs with interior angles of a polygon are
exterior angles of the polygon. In the diagram in Example 3,
∠ ABC, ∠BCA, and ∠CAB are interior angles. ∠DAC, ∠CBE, and
∠ACF are exterior angles.
Example 3
Find the measure of ∠DAC in the drawing at the right.
Solution Let x be the measure of ∠DAC. Let y be the
measure of ∠BAC.
Use the Triangle-Sum Property to find y.
D A
xº
B
yº
E
23º
120º
F
C
23° + 120° + y° = 180°
143 + y = 180
y = 37
Arithmetic
Add –143 to both sides.
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Now use the fact the ∠DAC and ∠BAC form a linear pair.
x° + y° = 180°
x + 37 = 180
x = 143
Substitution
Add –37 to both sides.
So, m∠DAC = 143°.
Notice ∠DAC = m∠ABC + ∠BCA. This is an example of the Exterior
Angle Theorem for Triangles.
Exterior Angle Theorem for Triangles
In a triangle, the measure of an exterior angle is equal to the
sum of the measures of the interior angles at the other two
vertices of the triangle.
The Triangle-Sum Property
39
Chapter 14
GUIDED
Example 4
Use the information given in the drawing. Explain the steps needed to
find m∠9.
105˚ 2
1 3
5 7
4 6
8
9
Solution The drawing is complicated. Examine it carefully. You may wish
to make an identical drawing and write in angle measures as they are found.
Here is one way to solve the problem.
The arrows indicate that two of the lines are parallel. ∠5 and
the angle that measures 105° are corresponding angles. So
m∠5 = ? . ∠7 forms a linear pair with ∠5, so they are
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supplementary. Thus m∠7 = ? . The triangle has a 90° angle
created by the perpendicular lines, but it also has ∠7, a ?
angle. The Triangle-Sum Property says that the measures of the
three angles add to 180°. Therefore, m∠8 = ? . Finally, ∠9 and
∠8 form a linear pair, so m∠9 = ? .
Questions
COVERING THE IDEAS
A
1
1. At the right is the result of Step 2 in the Activity. Why do the angles
of A'BC have the same measures as the angles of ABC?
2. Multiple Choice Which is true?
B
A In some but not all triangles, the sum of the measures of the
angles is 180º.
B In all triangles, the sum of the measures of the angles is 180º.
2
3
M
3
2
C
1
A'
C The sum of the measures of the angles of a triangle can be any
number from 180º to 360º.
D If two angles of a triangle are complementary, then the measure of SMP10_A_SE_C14_L07A_T_003
the third angle must be greater than 90º.
In 3–6, two angles of a triangle have the given measures. Find the
measure of the third angle.
3. 45º, 45º
4. 2º, 3º
5. 70º, 80º
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Some Important Geometry Ideas
6. xº, 120º - xº
Lesson 14-7
7. In Example 3, find
a. m∠ ACF.
b. m∠FBE.
c. m∠ ACF + m∠FBE + m∠CAD.
8. In Example 4, explain how to find m∠4.
APPLYING THE MATHEMATICS
a. If m∠1 = 40º and m∠2 = 60º, fill in the measures of all the
angles in the six triangles.
b. Give 5 pairs of parallel lines.
c. How do you know the lines are parallel?
1
G
B
2
outlined in the kaleidoscope quilt block at the right, the vertex
angle equals 54º. What is the measure of each base angle?
11. Find m∠DAB below.
12. Find m∠VYZ below.
B
E
3
D
C
10. Many quilt patterns use triangles. In the isosceles triangle
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W
54º
50º
50º
43º
D
F
A
9. Copy the figure at the right.
A
V
C
Y
Z
13. Three lines intersect forming ∠1, ∠2, ∠3, and ∠4 as shown below.
Explain why m∠1 + m∠2 + m∠3 = 180º.
SMP08TM2_SE_C06_T_0141
SMP08TM2_SE_C06_T_0140
3 4
1
2
8
14. Use the information given in the drawing at the right. Find the
measures of angles 1 through 8.
3
15. Explain why a triangleSMP08TM2_SE_C06_T_0139
cannot have two right angles.
16. In an equilateral triangle, all the angles have the same measure.
What is this measure?
123º 2
1
61º
4
5
7 6
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The Triangle-Sum Property
41
Chapter 14
−−−
−−−
17. In the figure below, BA ⊥ AC. Angle BAC is bisected (split into
, and m∠B = 70º. Find the measures of ∠1,
two equal parts) by AD
∠2, and ∠3.
B
D
70º
A
1
2
45º
45º
3
C
18. Multiple Choice Two angles of a triangle have measures a and
b degrees. The third angle must have what measure?
R
P
A 180 - a + b degrees
B 180 + a + b degrees
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58º
C 180 + a - b degrees
D 180 - a - b degrees
−−− −−
19. In the figure at the right, PQ RS = T. Find the measures of
as many angles in the figure as you can.
20º
20. Explain why the measure of an exterior angle must be equal to the
sum of the measures of the interior angles at the other two vertices
of the triangle.
S
55º
T
22º
Q
21. a. Write a theorem about the sum of the measures of the exterior
angles of a triangle.
b. Explain why your theorem is true.
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EXPLORATION
22. You have seen that the sum of the measures of angles in a triangle
on a plane is 180º, but what about a triangle drawn along the
surface of a globe or other sphere? Suppose you draw a triangle by
starting at the North Pole and drawing a line south. Then draw a
line west, then another line north back to the pole. Will the sum of
the angles of a triangle on a sphere equal 180º, be greater than 180º,
or be less than 180º? Experiment using a globe or other sphere and
summarize your results.
QY ANSWER
180°
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Some Important Geometry Ideas