Download Math 2201 Unit 2: Properties of Angles and Triangles Read

Math 2201
Unit 2: Properties of Angles and Triangles
Read Learning Goals, p. 67 text.
Ch. 2 Notes
§2.1 Exploring the Angles Formed by Intersecting Lines (0.5 class)
Read Goal p. 70 text.
Outcomes:
1. Define a transversal. pp. 70, 518
2. Define interior angles and identify them in a diagram. pp. 71, 516
3. Define exterior angles and identify them in a diagram. pp. 71, 515
4. Define corresponding angles and identify them in a diagram. pp. 71, 514
5. Define same side interior angles and identify them in a diagram. See notes
6. Define alternate interior angles and identify them in a diagram. pp. 75, 514
7. Define alternate exterior angles and identify them in a diagram. pp. 76, 514
8. Define vertically opposite angles (vertical angles) and identify them in a diagram. See
notes
This chapter is about the geometry of angles and triangles. You will use inductive reasoning to make
conjectures about the properties of angles and triangles and then use deductive reasoning to prove, or
counterexamples to disprove, these conjectures. Your proofs will often be in the two-column format.
Def n : A transversal is a line that intersects (cuts
across) two or more other lines.
Transversal
A transversal creates two general types of angles, interior and exterior angles.
Def n : Interior angles are angles that lie inside the
lines that the transversal intersects. In the diagram
below, a, b, c, and d are interior angles.
Transversal
a
c
1
d
b
Def n : Exterior angles are angles that lie outside the lines that the transversal intersects. In the diagram
below, e, f , g , and h are exterior angles.
Transversal
e
g
f
h
Def n : Corresponding angles are two angles, one exterior and one interior, that are not adjacent but lie
on the same side of the transversal. In the diagram below,
a and e, b and f , c and g, and d and h
are corresponding angles. Notice that the lines that
Transversal
form these angles make an “F” shape
a
E.g.:
c
a
e
e
g
b
d
f
h
Def n : Same-side interior angles are two interior angles that lie on the same side of the transversal. In
the diagram below, c and e, and d and f are same-side interior angles. Notice that the lines that
form these angles make a square-ish “C” shape
Transversal
E.g.:
d
c
a
c
e
e
f
g
b
d
f
h
Def n : Alternate interior angles are two interior angles that lie on opposite sides of the transversal. In
the diagram below, c and f , and d and e are alternate interior angles. Notice that the lines that
form these angles make a “Z” shape
Transversal
E.g.:
d
c
a
c
f
e
\
e
e
g
2
f
h
b
d
3
Def n : Alternate exterior angles are two exterior angles that are not adjacent and that lie on opposite
sides of the transversal. In the diagram below, a and h, and b and g are alternate interior angles.
E.g.:
Transversal
a
c
e
g
b
d
f
h
The last pair of angles we need are vertically opposite angles. They are not formed by three lines but by
two lines.
Def n : Vertically opposite angles (vertical angles) are two angles that are formed when two lines
intersect. In the diagram below, a and d , and b and c are vertically opposite angles.
a
c
4
b
d
§2.2 Exploring the Angles Formed by Parallel Lines (2 classes)
Read Goal p. 73 text.
Outcomes:
1. Define parallel lines and use the symbol for parallel lines. p. 75 and notes
2. On a diagram, indicate that lines are parallel using the correct symbol(s). p. 75
3. Generalize, using inductive reasoning, the relationships between pairs of angles
(corresponding angles, same-side interior angles, alternate interior angles, and alternate
exterior angles ) formed by transversals and parallel lines, with or without technology. pp.
71,78
4. Define the term converse. pp. 71, 514
5. Given a statement, write the converse of the statement. p. 71
6. Determine which angle properties do NOT apply if lines are not parallel. p. 71
7. Define a postulate. See notes
8. State the 4 general types of reasons used in two-column proofs. See notes
9. State the Angle Addition Postulate. See notes
Def n : Parallel lines are lines in the same plane that do not intersect. The symbol
to”. In the diagram to the right, lines l and m are parallel, so we can write l m .
On diagrams that we draw, we place arrow heads (or double or triple arrow
heads if necessary) between the ends of the lines to indicate that the lines are
parallel.
means “is parallel
l
m
l
The Relationships between the Angles Formed by Two Parallel Lines and a Transversal
Corresponding Angles
Draw a transversal that intersects the two parallel lines below.
5
Measure any two corresponding angles. What do you notice about the measures? Measure a different
pair of corresponding angles. What do you notice about those measures?
Use inductive reasoning to make a conjecture about the measures of corresponding angles when two
lines are cut by a transversal.
Conjecture: When two parallel lines are cut by a transversal, the measures of corresponding angles are
___________________.
Should you make this conjecture if the lines are not parallel? (Y or N) _____________
Alternate Interior Angles
Draw a transversal that intersects the two parallel lines below.
Measure any two alternate interior angles. What do you notice about the measures? Measure a
different pair of alternate interior angles. What do you notice about those measures?
Use inductive reasoning to make a conjecture about the measures of alternate interior angles when two
lines are cut by a transversal.
Conjecture: When two parallel lines are cut by a transversal, the measures of alternate interior angles
are ___________________.
Should you make this conjecture if the lines are not parallel? (Y or N) _____________
Alternate Exterior Angles
Draw a transversal that intersects the two parallel lines below.
6
Measure any two alternate exterior angles. What do you notice about the measures? Measure a
different pair of alternate exterior angles. What do you notice about those measures?
Use inductive reasoning to make a conjecture about the measures of alternate exterior angles when two
lines are cut by a transversal.
Conjecture: When two parallel lines are cut by a transversal, the measures of alternate exterior angles
are ___________________.
Should you make this conjecture if the lines are not parallel? (Y or N) _____________
Def n : If the sum of the measures of two angles equals 180 then the angles are supplementary.
130
50
Since 130  50  180 , the two angles above are supplementary.
Same-Side Interior Angles
Draw a transversal that intersects the two parallel lines below.
Measure any two same-side interior angles. What do you notice about the measures? Measure a
different pair of same-side interior angles. What do you notice about those measures?
Use inductive reasoning to make a conjecture about the measures of same-side interior angles when two
parallel lines are cut by a transversal.
Conjecture: When two parallel lines are cut by a transversal, the same-side interior angles are
___________________.
Should you make this conjecture if the lines are not parallel? (Y or N) _____________
Def n : A theorem is a formula or statement that can be proved from other formulas or statements.
The last four conjectures can be grouped into one statement called the Parallel Lines Theorem (PLT).
7
Parallel Lines Theorem (PLT) (See In Summary, p. 78 text)
If two parallel lines are cut by a transversal then,




the measures of corresponding angles are equal.
the measures of alternate interior angles are equal.
the measures of alternate exterior angles are equal.
same-side interior angles are supplementary.
Note that the PLT consists of a premise (If two parallel lines are cut by a transversal) and four
conclusions (each of the bulleted statements).
Also note that there are simple two-column proofs for some of the statements that make up the PLT on
pp. 75. However, we will be focusing on using the PLT to complete other proofs and to find the missing
angles in diagrams involving parallel lines.
Def n : The converse of a given statement is a statement formed by switching the premise and the
conclusion of the given statement.
Let’s form the four converses that arise from the PLT. They can be used to prove that two lines are
parallel. (See In Summary, p. 78 text)
Converse of PLT #1: If two lines are cut by a transversal and the measures of corresponding angles are
equal then the lines are parallel.
Converse of PLT #2: If two lines are cut by a transversal and the measures of alternate interior angles
are equal then the lines are parallel.
Converse of PLT #3: If two lines are cut by a transversal and the measures of alternate exterior angles
are equal then the lines are parallel.
Converse of PLT #4: If two lines are cut by a transversal and the same-side interior angles are
supplementary then the lines are parallel.
Do # 5, p. 72 text in your homework booklet.
The Relationships between the Angles Formed by Two Intersecting Lines
Vertically Opposite (Vertical) Angles
Draw two intersecting lines below.
8
Measure one pair of vertical angles. What do you notice about the measures? Measure a different pair of
vertical angles. What do you notice about those measures?
Use inductive reasoning to make a conjecture about the measures of vertical angles when two lines
intersect.
Conjecture: When two lines intersect, the measures of the vertical angles are ___________________.
E.g.: Find the value of alpha  α  , beta β  , and gamma  γ  in the diagram below.
Since AC BM , we can use the PLT to find two of the missing
angles.
Since CAB and MBE are corresponding angles, they have
the same measures so m MBE    65 .
Since ACB and MBC are alternate interior angles, they have
the same measures so m MBC    80 .
Since ABE is a straight angle, we know that       180 . Substituting values for  and  gives
65  80    180
145    180
145  145    180  145
  35
E.g.: Use the PLT to find the values of x and y and the measures of the marked angles.
6 x  65
5 y  15
3y  5
2x  5
Since the angles marked 3 y  5 and 5 y  15 are same-side interior angles, they are supplementary.
Therefore,
9
3 y  5  5 y  15  180
8 y  20  180
8 y  20  20  180  20
8 y  160
8 y 160

8
8
y  20
So these two angles have measures 3  20   5  65 and 5  20  15  115 .
Since the angles marked 6 x  65 and 5 y  15 are vertically opposite angles, we know they have the same
measure. Therefore
6 x  65  115
6 x  65  65  115  65
6 x  180
6 x 180

6
6
x  30
Finally, the angle marked 2 x  5 must be 65 since it is vertically opposite to the angle marked 3 y  5 .
Do #’s 4 a, b, 15, 20 pp. 79-82 text in your homework booklet.
Def n : A postulate is a statement that is accepted without proof.
E.g.: Angle Addition Postulate (AAP): If point B lies in the interior of
m AOB  m BOC  m AOC .
A
AOC , then
B
O
C
So now you have four different types of reasons that you can use in a two-column proof.
 Given information
 Definitions
 Postulates (including properties of algebra)
 Theorems that you have already proved.
Now let’s use the PLT and its converses to complete some two-column proofs.
10
E.g.: #16, p. 81 text.
Statements
AB DE; DE FG
m BAC  m ACF
m CDE  m DCF
m ACD  m ACF  m DCF
m ACD  m BAC  m CDE
Reasons
Given
PLT (alternate interior angles)
PLT (alternate interior angles)
Angle Addition Postulate
Substitution
E.g.: #12, p. 80 text.
Statements
m FOX  m FRS
SR XO
m FXO  m FPQ
Reasons
Given
Converse of PLT (corresponding angles)
PQ XO
Given
Converse of PLT (corresponding angles)
PQ SR
Transitive Property
11
§2.3 Angle Properties in Triangles (1 class)
Read Goal p. 86 text.
Outcomes:
1. Define an auxiliary line and use it to complete proofs. See notes
2. Prove the angle sum of a triangle theorem (ASTT). pp. 86-87
3. Define and identify an exterior angle of a polygon. pp. 87, 95, 515
4. Define and identify non-adjacent interior angles (remote interior angles). pp. 88, 516
5. Determine and prove the relationship between an exterior angle of a triangle and its nonadjacent interior angles (remote interior angles). p. 88
Def n : An auxiliary line is a line (ray, segment) added to a diagram to help complete a proof.
We are going to use an auxiliary line to prove that the sum of the measures of the angles of any triangle
is 180 . This statement is known as the angle sum of a triangle theorem (ASTT).
E
B
D
A
Statements
Draw EF parallel to AD
m EBA  m ABD  m DBF  180
m EBA  m BAD; m DBF  m BDA
m BAD  m ABD  m BDA  180
F
Reasons
Auxiliary Line
Definition of straight angle.
PLT (Alternate interior angles).
Substitution
Def n : An exterior angle of a polygon is the angle formed by a side of a polygon and the extension of
an adjacent side. In the triangle below, ACD is an exterior angle.
12
Carefully extend any side of the triangle drawn below to form an exterior angle. Find the measure of this
exterior angle. Measure the two angles that are inside the triangle and as far away from the exterior
angle as possible. These are called the remote interior angles.
B
A
C
Make a conjecture about the measure of the exterior angle and the measures of the two remote interior
angles.
Conjecture: The measure of the exterior angle is equal to the __________ of the two remote interior
angles.
E.g.: Find m S in the diagram to the right.
S
x
According to the last conjecture,
3 x  x  80
3 x  x  x  x  80
2 x  80
2 x 80

2
2
x  40
R
80
3x
T
So m S  40 .
Let’s prove our last conjecture and make it into a theorem.
2
1
m
m
m
m
1 m
3 m
1 m
1 m
Statements
2  m 3  180
4  180
2  m 3  m 3 m 4
2m 4
3 4
Reasons
ASTT
Definition of a straight angle.
Substitution
Subtraction Property
Do #’s 3, 5, 7, 10, 14, 15, pp.90-92 text in your homework booklet.
13
§2.4 Angle Properties in Polygons (1 class)
Read Goal p. 94 text.
Outcomes:
1. Define and give an example of a convex polygon. pp. 96, 514
2. Determine how the number of sides of a convex polygon is related to the sum of its interior
angles. p. 94
3. Determine the sum of the exterior angles of a convex polygon. p. 95
4. Define and give an example of a regular polygon. p. 97
5. Determine how the number of sides of a regular polygon is related to the measure of each
interior angle. p. 96
Def n : A convex polygon is a polygon in which each interior angle measures less than 180 .
In the figure below, some of the polygons are convex and some are non-convex (concave). Label the
figure correctly.
________________________
______________________________
Complete the table below for the sum of the measures of the interior angles of a convex polygon. (See
pp. 94, 96 text)
Polygon
Triangle
Quadrilateral
Pentagon
Hexagon
Heptagon
Octagon
n-gon
# Sides (n)
3
4
5
6
7
8
# Triangles
1
2
3
4
5
6
n
n2
Sum of Measures of Interior Angles (S)
180
360
Make a conjecture about the relationship between the number of sides of a convex polygon, n, and the
sum of the measures of the interior angles, S.
Conjecture: The sum of the measures of the interior angles of a convex polygon is ________________
___________________________________________________________________________________.
14
Below, write the expression for the sum of the measures of the interior angle of a regular polygon with n
sides. (See p. 99)
E.g.: The sum of the measures of the interior angles of a convex polygon is known to be between 2500
and 2600 . How many sides does the polygon have?
The sum of the measures of the interior angles of a convex polygon is  n  2 180  . Let
 n  2 180   2600
180n  360  2600
180n  360  360  2600  360
180n  2960
180 n 2960

180
180
n  16.4
Since the largest the sum of the interior angles can be is 2600 , the number of sides must be less than
16.4 . So the polygon has 16 sides.
Complete the table below for the sum of the measures of the exterior angles of a convex polygon. (See
p. 95 text)
Polygon
Triangle
Quadrilateral
Pentagon
Hexagon
Heptagon
Octagon
n-gon
# Sides
3
4
5
6
7
8
# Exterior Angles
3
4
5
6
7
8
n
n
Sum of Measures of Exterior Angles
360
360
Make a conjecture about the sum of the measures of the exterior angles of a convex polygon.
Conjecture: The sum of the measures of the exterior angles of a convex polygon is _______________.
E.g.: The sum of the measures of the interior angles of a polygon is four times the sum of the measures
of its exterior angles, one at each vertex. How many sides does the polygon have?
15
Let n be the number of sides of the polygon, then
***The sum of the measures of the interior angles of a polygon is  n  2 180  .
The sum of the measures of its exterior angles is 360 .
***Four times the sum of the measures of its exterior angles is 4  360  1440 .
Therefore,
 n  2 180   1440
 n  2  180 

180
n28
n2 2  8 2
1440
180
n  10
So the polygon has 10 sides.
Def n : A regular polygon is a polygon in which side is the same length and each interior angle has the
same measure.
E.g.:
16
Complete the table below for the measure of each interior angle of a regular polygon. (See p. 97 text)
Polygon
# Sides (n)
Sum of Measures of
Interior Angles (S)
3
180
4
360
Regular (Equilateral) Triangle
Regular Quadrilateral (Square)
Regular Pentagon
5
Regular Hexagon
6
Regular Heptagon
7
Regular Octagon
8
n-gon
n
Measure of Each Interior
Angle
180
 60
3
360
 90
4
 n  2180
Make a conjecture that relates the measure of each interior angle to the number of sides of a regular
polygon.
Conjecture: The measure of each interior angle of a regular polygon is _________________________
___________________________________________________________________________________.
Below write the expression for the measure of each interior angle of a regular polygon with n sides. (See
p. 99)
E.g.: The measure of each interior angle of a regular polygon is eight times that of an exterior angle.
How many sides does the polygon have?
Let n be the number of sides of the polygon, then
The sum of the measures of the interior angles of a polygon is  n  2 180  .
***The measure of each interior angle is
 n  2 180 
n
The sum of the measures of its exterior angles is 360 .
The measure of an exterior angle is
360
n
 360 
***Eight times the measure of an exterior angle is 8 
.
 n 
17
Therefore,
 n  2 180   8  360 


n
 n 
180n  360 2880

n
n
Since the two fractions above are equal and their denominators are equal, then their numerators must
also be equal. Therefore,
180n  360  2880
180n  360  360  2880  360
180n  3240
180 n 3240

180
180
n  18
So the regular polygon has 18 sides.
Do #’s 1, 3, 6-8, 11, pp. 99-101 text in your homework booklet.
18
§2.5 Exploring Congruent Triangles (One-half class)
Read Goal p. 104 text.
Outcomes:
1. Explain what is meant by congruent triangles. See notes.
2. Define and give an example of an included (contained) angle. See notes
3. Define and give an example of a non-included (non-contained) angle. See notes
4. Define and give an example of an included (contained) side. See notes
5. Determine five ways to show that two triangles are congruent. p. 105
Def n : Congruent triangles    ' s  are triangles in which corresponding sides have the same length
and corresponding angles have the same measure.
In the diagram to the right, JMK  PRQ . This means
that:
Sides
Angles
JM  PR
m J m P
JK  PQ
m K m Q
m M m R
MK  QR
If we want to work backwards and show that two triangles are congruent, do we have to show that three
sets of corresponding sides are equal in length and that three sets of corresponding angles have the same
measure? Luckily, the answer is NO!
Proving (showing) that Two Triangles are Congruent
Method 1: SSS (side-side-side)
If three pairs of corresponding sides are equal, then the two triangles are congruent.
Using the diagram below, if you can show that AB  DE , BC  EF , and AC  DF then you have
proved that ABC  DEF .
19
Def n : An included (contained) angle is an angle that is formed by two sides of the triangle.
Method 2: SAS (side-angle-side)
If two pairs of corresponding sides and the included (contained) angles are equal, then the two
triangles are congruent.
Using the diagram below, if you can show that AB  DE , BC  EF , and m B  m E then you have
proved that ABC  DEF .
Def n : An included (contained) side is a side that is formed by two angles of the triangle.
Method 3: ASA (angle-side-angle)
If two pairs of corresponding angles and the included (contained) sides are equal, then the two
triangles are congruent.
Using the diagram below, if you can show that m B  m E , BC  EF , and m C  m F then you
have proved that ABC  DEF .
20
Method 4: AAS (angle-angle-side)
If two pairs of corresponding angles and the non-included (non-contained) sides are equal, then
the two triangles are congruent.
Using the diagram below, if you can show that m B  m E , m C  m F , and AC  DF then you
have proved that ABC  DEF .
Method 5: HL (hypotenuse-leg)
If the hypotenuse and a leg of one right triangle are equal to the hypotenuse and a leg of another
right triangle, then the two triangles are congruent.
Using the diagram below, if you can show that, AB  DE and AC  DF then you have proved that
ABC  DEF .
Do #’s 1-3, p. 106 text in your homework booklet.
21
§2.6 Proving Congruent Triangles (2 classes)
Read Goal p. 107 text.
Outcomes:
1. Prove that triangles, or that corresponding parts of triangles, are congruent.
To show that parts of triangles are congruent, it is often necessary to first prove that the triangles
themselves are congruent using deductive reasoning (often SSS, SAS, ASA, AAS, or HL). These proofs
will often be in the two-column format.
Proof using SSS
Prove that m NQY  m PQY
Statements
NQ  PQ, NY  PY
YQ  YQ
NQY  PQY
m NQY  m PQY
Reasons
Given
Common side
SSS
Corresponding parts of congruent triangles are
equal. (CPCTE)
Proof using SAS
Prove that AB  ED
Statements
AC  EC, BC  DC
m ACB  m ECD
ABC  EDC
AB  ED
Reasons
Given
Vertically opposite angles are equal.
SAS
Corresponding parts of congruent triangles are
equal. (CPCTE)
22
Proof using ASA
Prove that BP  CP
Statements
AP  5, DP  5
AP  DP
m BPA  m CPD
BAP is a right angle
CDP is a right angle
m BAP  90
m CDP  90
m BAP  m CDP
BPA  CPD
BP  CP
Reasons
Given
Substitution
Vertically opposite angles are
equal.
Given
Definition of a right angle.
Substitution
ASA
CPCTE
Proof using AAS
Prove that PS  QS
m SPR  m SQR
m SRP  m SRQ
RS  RS
SPR  SQR
PS  QS
Statements
Reasons
Given
Common side
AAS
CPCTE
23
Proof using HL
Prove that m QPS  m RPS
Statements
QP  RP
SP  SP
QSP is a right angle
RSP is a right angle
QSP is a right triangle
RSP is a right triangle
QSP  RSP
m QPS  m RPS
Reasons
Given
Common side
Given
Definition of a right triangle.
HL
CPCTE
Do #’s 1-2, 4-6, 8, 9, 11, pp. 112-114 text in your homework booklet.
Do #’s 2-5, 7, 8, 10, 12, 13, 15-17, pp. 119-120 text in your homework booklet.
24