Download Third Angle Theorem

Third Angle Theorem
Bill Zahner
Lori Jordan
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Printed: May 8, 2016
AUTHORS
Bill Zahner
Lori Jordan
www.ck12.org
C HAPTER
Chapter 1. Third Angle Theorem
1
Third Angle Theorem
Here you’ll learn the Third Angle Theorem and how to use it to determine information about two triangles with two
pairs of angles that are congruent.
What if you were given 4FGH and 4XY Z and you were told that 6 F ∼
= 6 X and 6 G ∼
= 6 Y ? What conclusion could
you draw about 6 H and 6 Z? After completing this Concept, you’ll be able to make such a conclusion.
Watch This
MEDIA
Click image to the left or use the URL below.
URL: https://www.ck12.org/flx/render/embeddedobject/136991
CK-12 Foundation: Chapter4TheThirdAngleTheoremA
MEDIA
Click image to the left or use the URL below.
URL: https://www.ck12.org/flx/render/embeddedobject/10331
Guidance
Find m6 C and m6 J.
The sum of the angles in each triangle is 180◦ by the Triangle Sum Theorem. So, for 4ABC, 35◦ +88◦ +m6 C = 180◦
and m6 C = 57◦ . For 4HIJ, 35◦ + 88◦ + m6 J = 180◦ and m6 J is also 57◦ .
Notice that we were given that m6 A = m6 H and m6 B = m6 I and we found out that m6 C = m6 J. This can be
generalized into the Third Angle Theorem.
Third Angle Theorem: If two angles in one triangle are congruent to two angles in another triangle, then the third
pair of angles must also congruent.
1
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In other words, for triangles 4ABC and 4DEF, if 6 A ∼
= 6 D and 6 B ∼
= 6 E, then 6 C ∼
= 6 F.
Notice that this theorem does not state that the triangles are congruent. That is because if two sets of angles are
congruent, the sides could be different lengths. See the picture below.
Example A
Determine the measure of the missing angles.
From the markings, we know that 6 A ∼
= D and 6 E ∼
= 6 B. Therefore, the Third Angle Theorem tells us that 6 C ∼
= 6 F.
So,
m6 A + m6 B + m6 C = 180◦
m6 D + m6 B + m6 C = 180◦
42◦ + 83◦ + m6 C = 180◦
m6 C = 55◦ = m6 F
Example B
The Third Angle Theorem states that if two angles in one triangle are congruent to two angles in another triangle,
then the third pair of angles must also congruent. What additional information would you need to know in order to
be able to determine that the triangles are congruent?
In order for the triangles to be congruent, you need some information about the sides. If you know two pairs of
angles are congruent and at least one pair of corresponding sides are congruent, then the triangles will be congruent.
Example C
Determine the measure of all the angles in the triangle:
2
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Chapter 1. Third Angle Theorem
First we can see that m6 DCA = 15◦ . This means that m6 BAC = 15◦ also because they are alternate interior angles.
m6 ABC = 153◦ was given. This means by the Triangle Sum Theorem that m6 BCA = 12◦ . This means that m6 CAD =
12◦ also because they are alternate interior angles. Finally, m6 ADC = 153◦ by the Triangle Sum Theorem.
Watch this video for help with the Examples above.
MEDIA
Click image to the left or use the URL below.
URL: https://www.ck12.org/flx/render/embeddedobject/136992
CK-12 Foundation: Chapter4TheThirdAngleTheoremB
Concept Problem Revisited
For two given triangles 4FGH and 4XY Z, you were told that 6 F ∼
= 6 X and 6 G ∼
= 6 Y.
By the Third Angle Theorem, 6 H ∼
= 6 Z.
Guided Practice
Determine the measure of all the angles in the each triangle.
1.
2.
3
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3.
Answers:
1. m6 A = 86, m6 C = 42 and by the Triangle Sum Theorem m6 B = 52.
m6 Y = 42, m6 X = 86 and by the Triangle Sum Theorem, m6 Z = 52.
2. m6 C = m6 A = m6 Y = m6 Z = 35. By the Triangle Sum Theorem m6 B = m6 X = 110.
3. m6 A = 28, m6 ABE = 90 and by the Triangle Sum Theorem, m6 E = 62. m6 D = m6 E = 62 because they are
alternate interior angles and the lines are parallel. m6 C = m6 A = 28 because they are alternate interior angles and
the lines are parallel. m6 DBC = m6 ABE = 90 because they are vertical angles.
Explore More
Determine the measures of the unknown angles.
1.
2.
3.
4.
4
6
6
6
6
XY Z
ZXY
LNM
MLN
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5.
6.
7.
6
8.
9.
10.
11.
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Chapter 1. Third Angle Theorem
CED
GFH
FHG
6
6
6
6
6
ACB
HIJ
HJI
IHJ
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12.
13.
14.
15.
6
6
6
6
RQS
SRQ
T SU
TUS
Answers for Explore More Problems
To view the Explore More answers, open this PDF file and look for section 4.5.
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