Download Key Concepts, continued

1.8.1: Proving the Vertical Angles Theorem
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Warm-Up #12 Monday, 3/7/2016
Advanced Integrated 2
1. When two dice are tossed, what is the probability that the sum is
NOT 11?
2. What is the chance of getting exactly 10 heads in 35 coin flips?
3. A family has five children, what is the probability that exactly one
of them is a girl?
1.8.1: Proving the Vertical Angles Theorem
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Warm-Up #12 Monday, 3/7/2016
Integrated 2
1. A company has to select 3 officers from a pool of 6 candidates.
How many different ways can this be done if the officer are in
different positions?
2. A coach must choose how to line up his five starters from a team
of 12 players. How many different ways can the coach choose the
starters?
3. In a class of 33 students, 12 have blue eyes, and 14 have brown
hair. This includes 5 who have both. Create a Venn diagram.
1.8.1: Proving the Vertical Angles Theorem
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Homework Monday, 3/7/2016
Vertical Angles
1.8.1: Proving the Vertical Angles Theorem
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Vertical Angles
1.8.1: Proving the Vertical Angles Theorem
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•Theorem: a statement or conjecture
that can be proven true by given,
definitions, postulates, or already
proven theorems
•Postulate: a statement that describes a
fundamental relationship between basic
terms of geometry. Postulates are
accepted as true without proof.
•Conjecture: an educated guess based on
known information
1.8.1: Proving the Vertical Angles Theorem
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Key Concepts
• Angles can be labeled with one point at the vertex,
three points with the vertex point in the middle, or with
numbers. See the examples that follow.
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1.8.1: Proving the Vertical Angles Theorem
Key Concepts, continued
• Be careful when using one vertex point to name the
angle, as this can lead to confusion.
• 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.
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1.8.1: Proving the Vertical Angles Theorem
Key Concepts, continued
• Straight angles are angles with rays in opposite
directions—in other words, 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.
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1.8.1: Proving the Vertical Angles Theorem
Key Concepts, continued
• 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.
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1.8.1: Proving the Vertical Angles Theorem
Key Concepts, continued
Adjacent angles
∠ABC is adjacent to
∠CBD. They share vertex
B and
.
∠ABC and ∠CBD have no
common interior points.
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1.8.1: Proving the Vertical Angles Theorem
Key Concepts, continued
Nonadjacent angles
∠ABE is not adjacent to ∠FCD.
They do not have a common vertex.
(continued)
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1.8.1: Proving the Vertical Angles Theorem
Key Concepts, continued
Nonadjacent angles
∠PQS is not adjacent to ∠PQR. They share common
interior points within ∠PQS.
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1.8.1: Proving the Vertical Angles Theorem
Key Concepts, continued
• Linear pairs are pairs of adjacent angles whose
non-shared sides form a straight angle.
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1.8.1: Proving the Vertical Angles Theorem
Key Concepts, continued
Linear pair
∠ABC and ∠CBD are a
linear pair. They are
adjacent angles with
non-shared sides,
creating a straight angle.
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1.8.1: Proving the Vertical Angles Theorem
Key Concepts, continued
Not a linear pair
∠ABE and ∠FCD are not
a linear pair. They are not
adjacent angles.
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1.8.1: Proving the Vertical Angles Theorem
Key Concepts, continued
• Vertical angles are nonadjacent angles formed by
two pairs of opposite rays.
Theorem
Vertical Angles Theorem
Vertical angles are congruent.
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1.8.1: Proving the Vertical Angles Theorem
Key Concepts, continued
Vertical angles
∠ABC and ∠EBD are
vertical angles. ÐABC @ ÐEBD
∠ABE and ∠CBD are
vertical angles. ÐABE @ ÐCBD
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1.8.1: Proving the Vertical Angles Theorem
Key Concepts, continued
Not vertical angles
∠ABC and ∠EBD are not vertical angles.
and
are
not opposite rays. They do not form one straight line.
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1.8.1: Proving the Vertical Angles Theorem
Key Concepts, continued
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.
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1.8.1: Proving the Vertical Angles Theorem
Key Concepts, continued
• Informally, 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.
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1.8.1: Proving the Vertical Angles Theorem
Key Concepts, continued
• In the diagram below, the angles form a linear pair.
m∠ABD + m∠DBC = 180
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1.8.1: Proving the Vertical Angles Theorem
Key Concepts, continued
• The next diagram shows a pair of supplementary
angles that are nonadjacent.
m∠PQR + m∠TUV = 180
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1.8.1: Proving the Vertical Angles Theorem
Key Concepts, continued
Theorem
Supplement Theorem
If two angles form a linear pair, then they are
supplementary.
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1.8.1: Proving the Vertical Angles Theorem
Key Concepts, continued
• Angles have the same congruence properties that
segments do.
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 .
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1.8.1: Proving the Vertical Angles Theorem
Key Concepts, continued
Theorem
Angles supplementary to the same angle or to congruent
angles are congruent.
If mÐ1+ mÐ2 = 180 and mÐ2 + mÐ3 = 180, then Ð1@ Ð3.
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1.8.1: Proving the Vertical Angles Theorem
Key Concepts, continued
• 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.
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1.8.1: Proving the Vertical Angles Theorem
Key Concepts, continued
• The symbol for indicating perpendicular lines in a
diagram is a box at one of the right angles, as shown
below.
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1.8.1: Proving the Vertical Angles Theorem
Key Concepts, continued
• The symbol for writing perpendicular lines is ^, and is
read as “is perpendicular to.”
• In the diagram,
.
• 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.
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1.8.1: Proving the Vertical Angles Theorem
Key Concepts, continued
• Remember that 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.
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1.8.1: Proving the Vertical Angles Theorem
Key Concepts, continued
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.
(continued)
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1.8.1: Proving the Vertical Angles Theorem
Key Concepts, continued
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
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Key Concepts, continued
• Complementary angles are two angles whose sum
is 90º.
• Complementary angles can form a right angle or be
nonadjacent.
• The following diagram shows a pair of nonadjacent
complementary angles.
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1.8.1: Proving the Vertical Angles Theorem
Key Concepts, continued
mÐB + mÐE = 90
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1.8.1: Proving the Vertical Angles Theorem
Key Concepts, continued
• The diagram at right
shows a pair of adjacent
complementary angles
labeled with numbers.
mÐ1+ mÐ2 = 90
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1.8.1: Proving the Vertical Angles Theorem
Key Concepts, continued
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.
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1.8.1: Proving the Vertical Angles Theorem
Common Errors/Misconceptions
• not recognizing the theorem that is being used or that
needs to be used
• setting expressions equal to each other rather than
using the Complement or Supplement Theorems
• mislabeling angles with a single letter when that letter is
the vertex for adjacent angles
• not recognizing adjacent and nonadjacent angles
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1.8.1: Proving the Vertical Angles Theorem
Guided Practice
Example 4
Prove that vertical angles are congruent given a pair of
intersecting lines,
and
.
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1.8.1: Proving the Vertical Angles Theorem
Guided Practice: Example 4, continued
1. Draw a diagram and label three adjacent
angles.
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1.8.1: Proving the Vertical Angles Theorem
Guided Practice: Example 4, continued
2. Start with the Supplement Theorem.
Supplementary angles add up to 180º.
mÐ1+ mÐ2 = 180
mÐ2 + mÐ3 = 180
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1.8.1: Proving the Vertical Angles Theorem
Guided Practice: Example 4, continued
3. Use substitution.
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
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1.8.1: Proving the Vertical Angles Theorem
Guided Practice: Example 4, continued
4. Use the Reflexive Property.
m∠2 = m∠2
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1.8.1: Proving the Vertical Angles Theorem
Guided Practice: Example 4, continued
5. Use the Subtraction Property.
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.
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1.8.1: Proving the Vertical Angles Theorem
Guided Practice: Example 4, continued
6. Use the definition of congruence.
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.
✔
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1.8.1: Proving the Vertical Angles Theorem
Guided Practice: Example 4, continued
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1.8.1: Proving the Vertical Angles Theorem
Guided Practice
Example 5
In the diagram at right,
perpendicular bisector of
If AD = 4x – 1 and
DC = x + 11, what are
the values of
AD and DC?
is the
.
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1.8.1: Proving the Vertical Angles Theorem
Guided Practice: Example 5, continued
1. Use the Perpendicular Bisector Theorem
to determine the values of AD and DC.
If a point is on the perpendicular bisector of a
segment, then that point is equidistant from the
endpoints of the segment being bisected. That
means AD = DC.
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1.8.1: Proving the Vertical Angles Theorem
Guided Practice: Example 5, continued
2. Use substitution to solve for x.
AD = 4x – 1 and
DC = x + 11
Given equations
AD = DC
Perpendicular Bisector Theorem
4x – 1 = x + 11
Substitute 4x – 1 for AD and
x + 11 for DC.
3x = 12
Combine like terms.
x=4
Divide both sides of the
equation by 3.
1.8.1: Proving the Vertical Angles Theorem
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Guided Practice: Example 5, continued
3. Substitute the value of x into the given
equations to determine the values of AD
and DC.
AD = 4x – 1
DC = x + 11
AD = 4(4) – 1
DC = (4) + 11
AD = 15
DC = 15
AD and DC are each 15 units long.
✔
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1.8.1: Proving the Vertical Angles Theorem