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TEKS Focus:
(6)(A) Verify theorems about angles formed by
the intersection of lines and line segments,
including vertical angles, and angle formed by
parallel lines cut by a transversal and prove
equidistance between the endpoints of a
segment and points on its perpendicular bisector
and apply these relationships to solve problems.
(1)(D) Communicate mathematical ideas,
reasoning, and their implications using multiple
representations, including symbols, diagrams,
graphs, and language as appropriate.
(1)(F) Analyze mathematical relationships to
connect and communicate mathematical ideas.
This means that the angles do not overlap.
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What are Vertical Angles?
Another angle pair relationship exists between two
angles whose sides form two pairs of opposite rays.
Vertical angles are two nonadjacent angles whose sides
are opposite rays. Remember that opposite rays will
form a line.
1 and 3 are vertical angles, as are 2 and 4.
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An angle bisector is a ray that divides an
angle into two congruent ( ) angles.
JK bisects LJM; thus LJK  KJM.
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What are Complementary Angles?
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What are Supplementary Angles?
Complementary and Supplementary angles
do not have to be next to each other (see
above), but they can be adjacent.
Answer always, sometimes, or never
for the following statements:
1.Complementary angles are adjacent.
2.Supplementary angles form linear pairs.
3.Linear pairs form supplementary angles.
Answers: 1 is sometimes; 2 is sometimes; 3 is always.
Example: 1
Tell whether the angles are only adjacent, adjacent and
form a linear pair, or not adjacent.
AEB and BED
Adjacent and Linear Pair
AEB and BEC
Adjacent Only
DEC and AEB
Not Adjacent
Example: 2
Name the pairs of vertical angles.
HML and JMK are vertical angles.
HMJ and LMK are vertical angles.
If mHML = 60°, then mLMK = 120°,
then mKMJ = 60°, and mJMK = 120°.
What do you notice about the measures of vertical angles?
They are equal, therefore the vertical angles are congruent.
Example: 3
Find the measure of each of the following.
A. complement of F
(90 – x)
90 – 59 = 31
B. supplement of G
(180 – x)
180 – (7x+10) = 180 – 7x – 10
= (170 – 7x)
Example: 4
Find the measure of each of the following.
A. complement of E
(90 – x)°
90° – (7x – 12)° = 90° – 7x° + 12°
= (102 – 7x)°
B. supplement of F
(180 – x)
180 – 116.5° =
Example: 5
Find the mEDG and mFDH. Are they a pair of vertical angles?
EDG and FDH are vertical angles
and they have the same measure.
(4x - 36) 
(x + 24) 
4x - 36 = x + 24
Vertical angles are congruent.
3x - 36 = 24
Subtraction Property of Equality.
3x = 60
Addition Property of Equality.
Division Property of Equality.
x = 20
mEDG = 4(20) – 36 = 44
mFDH = 20 + 24 = 44
Substitution Property
of Equality
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Light passing through a fiber
optic cable reflects off the
walls of the cable in such a
way that 1 ≅ 2, 1 and 3
are complementary, and 2
and 4 are complementary.
If m1 = (3x + 4)° and
m3 = (2x + 21) °, find m1,
m2, m3, and m4.
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If 1  2, then m 1 = m 2.
If 3 and 1 are
complementary, then
m1 + m3 = 90°.
(3x + 4) + (2x + 21) = 90
5x + 25 = 90
5x = 65
x = 13
m1 = 3(13) + 4 = 43°
m3 = 2(13) + 21 = 47°
m2 = m1 = 43°
m4 = m3 = 47°
 Discuss
the following question
within your group:
 Would this example be worked
the same way if the directions had
said that KM was an angle
bisector for JKL?
 Write on KM that it is called an
angle bisector for this example.
Example: 7
Tell whether the angles are only adjacent, adjacent and
form a linear pair, or not adjacent.
5 and 6
Adjacent and Linear Pair
7 and SPU
Not Adjacent
7 and 8
Not Adjacent
Example: 8
What if...? Suppose m3 = 27.6°. Find m1,
m2, and m4.
If 1  2, then m1 = m2.
If 3 and 1 are complementary,
then m1 = (90 – 27.6)°.
If 4 and 2 are complementary,
then m4 = (90 – 27.6)°.