Download Proving the Vertical Angles Theorem - 3

PROVING THE VERTICAL
ANGLES THEOREM
Adapted from Walch Education
Key Concepts
2
Angles can be labeled with one point at the vertex,
three points with the vertex point in the middle, or with
numbers.
If the vertex point serves as the vertex for more than
one angle, three points or a number must be used to
name the angle. 1.8.1: Proving the Vertical Angles Theorem
Straight Angle
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•
Straight angles are straight lines.
Straight angle
Not a straight angle
∠BCD is a straight angle. Points ∠PQR is not a straight angle.
B, C, and D lie on the same line. Points P, Q, and R do not lie on
the same line.
1.8.1: Proving the Vertical Angles Theorem
Adjacent Angles
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•
•
Adjacent angles are angles that lie in the same
plane and share a vertex and a common side. They
have no common interior points.
Nonadjacent angles have no common vertex or
common side, or have shared interior points.
1.8.1: Proving the Vertical Angles Theorem
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Adjacent angles
∠ABC is adjacent to
∠CBD. They share vertex
B and
.
∠ABC and ∠CBD have no
common interior points.
1.8.1: Proving the Vertical Angles Theorem
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Nonadjacent angles
∠ABE is not adjacent to ∠FCD.
They do not have a common vertex.
1.8.1: Proving the Vertical Angles Theorem
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Nonadjacent angles
∠PQS is not adjacent to ∠PQR. They share common
interior points within ∠PQS.
1.8.1: Proving the Vertical Angles Theorem
Linear pair
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
Linear pairs are pairs of adjacent angles whose nonshared sides form a straight angle
Linear pair
∠ABC and ∠CBD are a
linear pair. They are
adjacent angles with
non-shared sides,
creating a straight
angle.
1.8.1: Proving the Vertical Angles Theorem
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Not a linear pair
∠ABE and ∠FCD are not
a linear pair. They are not
adjacent angles.
1.8.1: Proving the Vertical Angles Theorem
Vertical Angles
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•
Vertical angles are nonadjacent angles formed by
two pairs of opposite rays.
Theorem
Vertical Angles Theorem
Vertical angles are congruent.
1.8.1: Proving the Vertical Angles Theorem
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Vertical angles
∠ABC and ∠EBD are ÐABC @ ÐEBD
vertical angles.
∠ABE and ∠CBD are ÐABE @ ÐCBD
vertical angles.
1.8.1: Proving the Vertical Angles Theorem
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Not vertical angles
∠ABC and ∠EBD are not vertical angles.
and
are
not opposite rays. They do not form one straight line.
1.8.1: Proving the Vertical Angles Theorem
Angle Addition Postulate
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Postulate
Angle Addition Postulate
If D is in the interior of ∠ABC,
then m∠ABD + m∠DBC = m∠ABC.
If m∠ABD + m∠DBC =
m∠ABC, then D is in
the interior of ∠ABC.
1.8.1: Proving the Vertical Angles Theorem
Key Concepts
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•
•
•
The Angle Addition Postulate means that the
measure of the larger angle is made up of the sum
of the two smaller angles inside it.
Supplementary angles are two angles whose sum
is 180º.
Supplementary angles can form a linear pair or be
nonadjacent.
1.8.1: Proving the Vertical Angles Theorem
Supplementary Angles
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•

In the diagram below, the angles form a linear pair.
m∠ABD + m∠DBC = 180
1.8.1: Proving the Vertical Angles Theorem
Supplementary Angles
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•

A pair of supplementary angles that are
nonadjacent.
m∠PQR + m∠TUV = 180
1.8.1: Proving the Vertical Angles Theorem
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Theorem
Supplement Theorem
If two angles form a linear pair, then they are
supplementary.
Theorem
Angles supplementary to the same angle or to congruent
angles are congruent.
If
and
, then
.
1.8.1: Proving the Vertical Angles Theorem
Properties of Congruence
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Theorem
Congruence of angles is reflexive, symmetric, and
transitive.
• Reflexive Property: Ð1@ Ð1
• Symmetric Property: If Ð1@ Ð2, then Ð2 @ Ð1.
• Transitive Property: If Ð1@ Ð2 and Ð2 @ Ð3,
then Ð1@ Ð3 .
1.8.1: Proving the Vertical Angles Theorem
Perpendicular Lines
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•
Perpendicular lines form four adjacent and
congruent right angles.
Theorem
If two congruent angles form a linear pair, then they are
right angles.
If two angles are congruent and supplementary, then
each angle is a right angle.
1.8.1: Proving the Vertical Angles Theorem
Perpendicular Lines
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•
•
•
The symbol for writing perpendicular lines is
is read as “is perpendicular to.”
^
, and
Rays and segments can also be perpendicular.
In a pair of perpendicular lines, rays, or segments,
only one right angle box is needed to indicate
perpendicular lines.
1.8.1: Proving the Vertical Angles Theorem
Concepts, continued
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•
•
Perpendicular bisectors are lines that intersect a
segment at its midpoint at a right angle; they are
perpendicular to the segment.
Any point along the perpendicular bisector is
equidistant from the endpoints of the segment that
it bisects.
1.8.1: Proving the Vertical Angles Theorem
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Theorem
Perpendicular Bisector Theorem
If a point lies on the perpendicular bisector of a
segment, then that point is equidistant from the
endpoints of the segment.
If a point is equidistant from the endpoints of a
segment, then the point lies on the perpendicular
bisector of the segment.
1.8.1: Proving the Vertical Angles Theorem
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Theorem
If
is the perpendicular bisector of AC, then DA = DC.
If DA = DC, then
is the perpendicular bisector of AC .
1.8.1: Proving the Vertical Angles Theorem
Complementary Angles
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•
•
Complementary angles are two angles whose sum
is 90º.
Complementary angles can form a right angle or
be nonadjacent.
1.8.1: Proving the Vertical Angles Theorem
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Complementary Angles
mÐB + mÐE = 90
1.8.1: Proving the Vertical Angles Theorem
Complementary Angles
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•
The diagram at right
shows a pair of adjacent
complementary angles
labeled with numbers.
mÐ1+ mÐ2 = 90
1.8.1: Proving the Vertical Angles Theorem
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Theorem
Complement Theorem
If the non-shared sides of two adjacent angles form a
right angle, then the angles are complementary.
Angles complementary to the same angle or to
congruent angles are congruent.
1.8.1: Proving the Vertical Angles Theorem
Practice
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
Prove that vertical angles are congruent given
a pair of intersecting lines,
and
.
1.8.1: Proving the Vertical Angles Theorem
Draw a Diagram
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1.8.1: Proving the Vertical Angles Theorem
Supplement Theorem
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
Supplementary angles add up to 180º.
mÐ1+ mÐ2 = 180
mÐ2 + mÐ3 = 180
1.8.1: Proving the Vertical Angles Theorem
Use Substitution
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
Both expressions are equal to 180, so they are
equal to each other. Rewrite the first equation,
substituting m∠2 + m∠3 in for 180.
m∠1 + m∠2 = m∠2 + m∠3
1.8.1: Proving the Vertical Angles Theorem
Reflexive Property
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
m∠2 = m∠2
1.8.1: Proving the Vertical Angles Theorem
Subtraction Property
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Since m∠2 = m∠2, these measures can be subtracted
out of the equation m∠1 + m∠2 = m∠2 + m∠3.
This leaves m∠1 = m∠3.
1.8.1: Proving the Vertical Angles Theorem
Definition of Congruence
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
Since m∠1 = m∠3, by the definition of congruence,
Ð1@ Ð3.

∠1 and ∠3 are vertical angles and they are
congruent. This proof also shows that angles
supplementary to the same angle are congruent.
1.8.1: Proving the Vertical Angles Theorem
Try this one…
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

In the diagram at
right,
is the
perpendicular
bisector of
.
If AD = 4x – 1 and
DC = x + 11, what
are the values of AD
and DC?
1.8.1: Proving the Vertical Angles Theorem
Thanks For Watching!
Dr. Dambreville