Download UNIT 5 • SIMILARITY, RIGHT TRIANGLE TRIGONOMETRY, AND

UNIT 5 • SIMILARITY, RIGHT TRIANGLE TRIGONOMETRY, AND PROOF
Lesson 5: Proving Theorems About Lines and Angles
Instruction
Prerequisite Skills
This lesson requires the use of the following skills:
•
identifying and labeling points, lines, and angles
•
using the addition and subtraction properties of angles
Introduction
Think about crossing a pair of chopsticks and the angles that are created when they are opened at
various positions. How many angles are formed? What are the relationships among those angles?
This lesson explores angle relationships. We will be examining the relationships of angles that lie
in the same plane. A plane is a two-dimensional figure, meaning it is a flat surface, and it extends
infinitely in all directions. Planes require at least three non-collinear points. Planes are named using
those points or a capital script letter. Since they are flat, planes have no depth.
Key Concepts
•
ngles can be labeled with one point at the vertex, three points with the vertex point in the
A
middle, or with numbers. See the examples that follow.
A
C
1
B
B
∠ABC
∠B
∠1
•
Be careful when using one vertex point to name the angle, as this can lead to confusion.
•
I f the vertex point serves as the vertex for more than one angle, three points or a number must
be used to name the angle.
U5-230
CCSS IP Math II Teacher Resource
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UNIT 5 • SIMILARITY, RIGHT TRIANGLE TRIGONOMETRY, AND PROOF
Lesson 5: Proving Theorems About Lines and Angles
Instruction
•
S traight angles are angles with rays in opposite directions—in other words, straight angles
are straight lines.
Straight angle
Not a straight angle
D
C
P
R
B
Q
∠PQR is not a straight angle. Points P,
Q, and R do not lie on the same line.
∠BCD is a straight angle. Points B, C,
and D lie on the same line.
•
djacent angles are angles that lie in the same plane and share a vertex and a common side.
A
They have no common interior points.
•
onadjacent angles have no common vertex or common side, or have shared interior
N
points.
Adjacent angles
Nonadjacent angles
E
A
A
P
S
C
B
D
Q
C
R
F
B
D
∠ABC is adjacent to ∠CBD .
They share vertex B and BC .
∠ABE is not adjacent to ∠FCD .
They do not have a common vertex.
∠ABC and ∠CBD have no
common interior points.
∠PQS is not adjacent to ∠PQR . They share
common interior points within ∠PQS .
U5-231
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CCSS IP Math II Teacher Resource
5.5.1
UNIT 5 • SIMILARITY, RIGHT TRIANGLE TRIGONOMETRY, AND PROOF
Lesson 5: Proving Theorems About Lines and Angles
Instruction
• Linear pairs are pairs of adjacent angles whose non-shared sides form a straight angle.
Linear pair
Not a linear pair
C
A
E
A
B
B
C
F
D
D
∠ABE and ∠FCD are not a linear pair.
∠ABC and ∠CBD are a linear pair.
They are adjacent angles with non-shared They are not adjacent angles.
sides, creating a straight angle.
•
Vertical angles are nonadjacent angles formed by two pairs of opposite rays.
Theorem
Vertical Angles Theorem
Vertical angles are congruent.
Vertical angles
Not vertical angles
C
A
A
B
E
C
D
B
E
D
∠ABC and ∠EBD are vertical angles.
∠ABC ≅ ∠EBD
∠ABE and ∠CBD are vertical angles.
∠ABE ≅ ∠CBD
∠ABC and ∠EBD are not vertical angles.
BC and BD are not opposite rays.
They do not form one straight line.
U5-232
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UNIT 5 • SIMILARITY, RIGHT TRIANGLE TRIGONOMETRY, AND PROOF
Lesson 5: Proving Theorems About Lines and Angles
Instruction
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 .
A
D
C
B
•
I nformally, 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. Postulates are true statements that don’t
need proofs.
•
Supplementary angles are two angles whose sum is 180º.
•
Supplementary angles can form a linear pair or be nonadjacent.
•
In the following diagram, the angles form a linear pair.
m∠ABD + m∠DBC = 180
DD
CC
BB
AA
•
The next diagram shows a pair of supplementary angles that are nonadjacent.
m∠PQR + m∠TUV = 180
PP
25º
25º
QQ
TT
RR
UU
155º
155º
VV
Theorem
Supplement Theorem
If two angles form a linear pair, then they are supplementary.
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CCSS IP Math II Teacher Resource
5.5.1
UNIT 5 • SIMILARITY, RIGHT TRIANGLE TRIGONOMETRY, AND PROOF
Lesson 5: Proving Theorems About Lines and Angles
Instruction
•
Angles have the same congruence properties that segments do.
Theorem
Congruence of angles is reflexive, symmetric, and transitive.
•
Reflexive Property: ∠1 ≅ ∠1
•
Symmetric Property: If ∠1 ≅ ∠2 , then ∠2 ≅ ∠1 .
•
Transitive Property: If ∠1 ≅ ∠2 and ∠2 ≅ ∠3 , then ∠1 ≅ ∠3 .
Theorem
Angles supplementary to the same angle or to congruent angles are
congruent.
If m∠1 + m∠2 = 180 and m∠2 + m∠3 = 180 , then ∠1 ≅ ∠3 .
• Perpendicular lines form four adjacent and congruent right angles, or 90º 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.
•
he symbol for indicating perpendicular lines in a diagram is a box at one of the right angles,
T
as shown below.
Q
R
P
S
•
The symbol for writing perpendicular lines is ⊥ , and is read as “is perpendicular to.”
U5-234
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UNIT 5 • SIMILARITY, RIGHT TRIANGLE TRIGONOMETRY, AND PROOF
Lesson 5: Proving Theorems About Lines and Angles
Instruction
•
In the diagram, SQ ⊥ PR .
•
Rays and segments can also be perpendicular.
•
I n a pair of perpendicular lines, rays, or segments, only one right angle box is needed to
indicate perpendicular lines.
•
erpendicular bisectors are lines that intersect a segment at its midpoint at a right angle;
P
they are perpendicular to the segment.
•
ny point along the perpendicular bisector is equidistant, or the same distance, from the
A
endpoints of the segment that it bisects.
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.
D
C
A
B
E
If DE is the perpendicular bisector of AC , then DA = DC.
If DA = DC, then DE is the perpendicular bisector of AC .
•
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.
U5-235
© Walch Education
CCSS IP Math II Teacher Resource
5.5.1
UNIT 5 • SIMILARITY, RIGHT TRIANGLE TRIGONOMETRY, AND PROOF
Lesson 5: Proving Theorems About Lines and Angles A
P
m∠B + m∠E = 90
C
E
A
S
F
55º
P
35º
B
Instruction
35º
B
1
D
C
2
Q
R
E
F
55º of adjacent
The next diagram shows a pair
complementary anglesS labeled with numbers.
•
1
m∠1 + m∠2 = 90
D
A
2
R
P
35º
B
E
Q
C
55º
S
F
1
D
Q
2
R
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.
Common Errors/Misconceptions
•
not recognizing the theorem that is being used or that needs to be used
•
s etting 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
U5-236
CCSS IP Math II Teacher Resource
5.5.1
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