Transcript Transversals

Geometry
First
Quarter
G.CO.9
G.CO.10
G.CO.11
Block 28: Daily Activities
Complementary Angles
Add to 90 Degrees
Remember: Complementary Angles
Do not Have to Be Adjacent
Supplementary Angles
Add to 180 Degrees
Remember: Supplementary Angles
Do not Have to Be Adjacent
Vertical Angles
are Congruent
Linear Pair
Angles are
Supplementary
Parallel Lines
Obj: Be able to prove and use results
about parallel lines and transversals.
Definitions
1. Parallel lines – Two lines are parallel lines if
they are coplanar and do not intersect.
2. Skew lines—Lines that do not intersect and
are not coplanar.
3. Parallel planes—two planes that do not
intersect.
1) Think of each segment in the
diagram. Which appear to fit
the description?
a. Parallel to AB and contains D CD
b. Perpendicular to AB and
A
contains D. AD
c. Skew to AB and contains D.
B
C
D
DG, DH , DE
a.
Name the plane(s) that
contains D and appear to be
parallel to plane ABE
DCH
F
E
G
H
Postulate 13: Parallel Postulate
• If there is a line and a point not on the line,
then there is exactly one line through the
point parallel to the given line.
P
l
Exterior Angles
Interior Angles
1, 2, 7, 8
3, 4, 5, 6
Consecutive Interior
Angles or Same Side
Interior
3 and 5
4 and 6
Alternate Exterior
Angles
1 and 8
2 and 7
Alternate Interior
Angles
Corresponding
Angles
3 and 6
4 and 5
1 and 5, 2 and 6
3 and 7, 4 and 8
1
3
6
5
7
8
2
4
Parallel Lines Construction Activity
1. Using a straightedge, draw two nonparallel intersecting
n
lines m and n.
A
m
2. At point A, construct a line parallel to line m, by copying the
angle formed by the intersection of lines m and n.
n
A
m
3. Measure all eight angles formed by the parallel lines and
transversal.
If 2 parallel lines are cut by a transversal, then the pairs of
corresponding angles are congruent.
l
m
1
2
If corr 's  , then ||.
1  2
If ||, then corr 's  .
If 2 || lines are cut by a transversal, then the pairs
of alternate interior angles are congruent.
l
1
2
m
If alt int 's  , then ||.
1  2
If ||, then alt int 's  .
If 2 || lines are cut by a transversal, then the pairs of
consecutive interior angles are supplementary.
l
12
m
m1  m2  180
If ||, then con int 's supp.
or
If ||, then ss int 's supp.
If con int 's supp, then ||.
or
If ss int 's supp, then ||.
If 2 || lines are cut by a transversal, then the pairs of
alternate interior angles are congruent.
l
m
1
2
1  2
If ||, then alt ext 's  .
If alt ext 's  , then ||.
Parallel Lines Theorem
If two lines are parallel to the same line, then
they are parallel to each other.
p
q
r
If p║q and q║r,
then p║r.
Examples
1. Find the measure of angle a and b if t // m.
Justify your reasoning using transformations.
b
t
125°
a
m
a = 125° by translating the given angle along the transversal
b = 125° by rotating the given angle 180°
Examples
1. Find the measure of angle a and b if t // m.
Justify your reasoning using transformations.
65°
b
t
a
m
The given angle and b form a linear pair, therefore b = 115°.
Rotate b 180° and translate along the transversal onto a.
Therefore a = 115°
Identify the type of angle pair that is given.
Find the value of x.
3.
4.
Block 29: Daily Activities
Correctly match the following and include the transformation
that maps one angle onto its pair.
E
_________
1. Alternate Interior Angles
A. ∠4 and ∠6
Translate ∠3 along the transversal onto ∠7.
Then rotate 180° onto ∠6
C
_________
2. Alternate Exterior Angles
B. ∠1 and ∠ 5
B
________
3. Corresponding Angles
C. ∠2 and ∠7
Rotate ∠2 180° and then translate along the
transversal onto ∠7
Translate ∠1 along the transversal onto ∠5
A
_________
4. Consecutive Interior Angles
D. ∠1 and ∠4
Not a transformation
D
_________
5. Vertical Angles
Rotate ∠1 180°
l
m
1 2
3 4
5 6
7 8
E. ∠3 and ∠6
Alternate: Transversal
Alternate: Transversal
Interior: Parallel Lines
Exterior: Parallel Lines
Same Position with Respects to
Transversal and Parallel Lines
Alternate Interior
Angles are Congruent
Alternate Exterior
Angles are Congruent
Corresponding Angles
are Congruent
Same Side: Transversal
Interior: Parallel Lines
Same Side Interior
Angles are
Supplementary
Block 30: Daily Activities
Parallel Lines and Transversals Warmup
Special Segments in Triangles
Objective:
1) Be able to identify the median of a triangle.
2) Be able to apply the Mid-segment Theorem.
3) Be able to use triangle measurements to find the
longest and shortest side.
1. Using a straight edge, draw a triangle. Label the vertices
A, B, and C.
2. Using a compass, construct the midpoint of AB and CB.
Label the midpoints D and E, respectively.
3. What do you notice about the relationship between DE and AC ?
Figure
Picture
B
D
A
E
C
2DE  AC
Definition
The midsegment of a
triangle is parallel to
the side it does not
touch and is half as
long.
Example
1) Given: JK and KL are midsegments.
Find JK and AB.
B
K
J
6
A
L
10
C
JK  5
AB  12
Example
2) Find x.
2  3x  7   7 x  6
6 x  14  7 x  6
8 x
3x  7
33xx77
7x  6
Figure
Picture
Definition
Intersection
A segment whose
endpoints are a vertex
of the triangle and the
midpoint of the
opposite side.
The concurrence of the
medians is called the
centroid.
1. Using a straight edge, draw a triangle. Label the vertices
A, B, and C.
2. Using a compass, construct the midpoint of all 3 sides.
3. Using a straightedge, draw a segment from each midpoint to
its opposite vertice.
3. What do you notice about the three segments ?
Concurrency of Medians of a Triangle
The medians of a triangle
intersect at a point that is
two thirds of the distance
from each vertex to the
midpoint of the opposite
side.
If P is the centroid of ∆ABC,
then
AP = 2/3 AD,
BP = 2/3 BF, and
CP = 2/3 CE
B
D
E
C
P
F
A
28
The longest side of a triangle is
always opposite the largest
angle and the smallest side is
always opposite the smallest
angle.
_____5. QT
_____6. QR
_____7.
Block 31: Daily Activities
Block 32: Daily Activities
Quadrilateral Activity
• Students will first cut out their set of triangles.
• Mark the triangle on both sides if there are congruent
sides or angles.
• Using 2 or more triangles, they must transform the
original triangles to form quadrilaterals
• Then glue these quadrilaterals onto the butcher paper.
• Next to the quadrilateral write down any
characteristics that are displayed on the diagrams.
• Present your findings.
Block 33: Daily Activities
Quadrilaterals
Objectives:
 Be able to discover properties of
quadrilaterals.
Parallelograms
A parallelogram is a quadrilateral with both pairs of opposite sides
parallel.
P
Q
S
R
When you mark diagrams of quadrilaterals, use matching
arrowheads to indicate which sides are parallel. For
example, in the diagram above, PQ║RS and QR║SP. The
symbol
PQRS is read “parallelogram PQRS.”
Thm
6.2
If a quadrilateral is a
parallelogram, then its
opposite sides are
congruent.
Thm
6.3
If a quadrilateral is a
parallelogram, then the
opposite angles are
congruent.
Thm
6.4
If a quadrilateral is a
parallelogram, then its
consecutive angles are
supplementary.
Thm
6.5
If a quadrilateral is a
parallelogram, then its
diagonals bisect each
other.
P
Q
S
PQ  RS
R
SP  QR
S
P
R
Q
P
S
R
Q
P  R
Q   S
mP  mQ  180
mQ  mR  180
mR  mS  180
mS  mP  180
P
Q
M
S
QM  SM
R
PM  RM
1) Find the value of each variable in
the parallelogram below.
y  11
2) WXYZ is a parallelogram.
Find the value of x.
3x  18
 4x  9
Example:
3) PQRS is a parallelogram.
Find the value of x.
Q
3x
120
Theorems about Parallelograms
Thm 6.6
Thm 6.7
Thm 6.8
Thm 6.9
If both pairs of opposite sides are
congruent, then the quadrilateral
is a parallelogram.
If both pairs of opposite angles
are congruent, then the
quadrilateral is a parallelogram.
If an angle of a quadrilateral is
supplementary to both of its
consecutive angles, then the
quadrilateral is a parallelogram.
P
Q
R
P
P
S
R
Q
P
Q
S
R
Q
If the diagonals of a quadrilateral
bisect each other, then the
quadrilateral is a parallelogram.
S
S
R
Theorems about Parallelograms
Thm
6.10
If one pair of opposite sides of a
quadrilateral are congruent and
parallel, then the quadrilateral is
a parallelogram
P
Q
Summary
Proving Quadrilateral are Parallelograms
 Show that both pairs of opposite sides are parallel
 Show that both pairs of opposite sides are congruent
 Show that both pairs of opposite angles are congruent
 Show that one angle is supplementary to both consecutive angles
 Show that the diagonals bisect each other
 Show that one pair of opposite sides are congruent and parallel
S
R
A parallelogram with
four congruent sides.
A parallelogram with
four right angles.
A parallelogram with
four congruent sides,
and four right angles.
Corollaries
– Rhombus Corollary: A quadrilateral is a
rhombus if and only if it has four congruent
sides.
– Rectangle Corollary: A quadrilateral is a
rectangle if and only if it has four right angles.
– Square Corollary: A quadrilateral is a square if
and only if it is a rhombus and a rectangle.
You can use these to prove that a quadrilateral is a
rhombus, rectangle or square without proving first
that the quadrilateral is a parallelogram.
Example:
1) Decide whether the statement is always,
sometimes, or never.
A. A rectangle is a square.
B. A square is a rhombus.
Theorems
Theorem
6.11
Theorem
6.12
Theorem
6.13
A parallelogram is a
rhombus if and only if its
diagonals are perpendicular.
A parallelogram is a rhombus
if and only if each diagonal
bisects a pair of opposite
angles.
A parallelogram is a
rectangle if and only if its
diagonals are congruent.
Examples:
2) Which of the following quadrilaterals
have the given property?
 All sides are congruent.
A.Parallelogram
 All angles are congruent. B.Rectangle
 The diagonals are
congruent.
 Opposite angles are
congruent.
C.Rhombus
D.Square
Example:
P
Q
3) In the diagram at the right,
PQRS is a rhombus. What
2y + 3
is the value of y?
S
5y - 6
R
Trapezoids
A trapezoid is a quadrilateral with exactly one pair
of parallel sides.
Bases: The parallel sides of a trapezoid.
Legs: The nonparallel sides of the trapezoid.
Isosceles Trapezoid: A trapezoid whose legs are congruent.
Midsegment: A segment that connects the midpoints of the legs
and that is parallel to each base. Its length is one half the sum of
the lengths of the bases.
Base
Leg
Midsegment
Base Angles
Base
Leg
Isosceles Trapezoids
A trapezoid that has congruent legs.
A
Theorem
6.14
If a trapezoid is isosceles,
then each pair of base
angles is congruent.
B
D
C
A  B C  D
Theorem
6.15
Theorem
6.16
If a trapezoid has a pair of
congruent base angles,
then it is an isosceles
trapezoid.
A trapezoid is isosceles if
and only if its diagonals
are congruent.
ABCD is isosceles if and only if AC  BD.
A
B
D
C
A
D
B
C
Example
4) CDEF is an isosceles trapezoid with
CE  10 and mE  95 . Find DF , mC ,
mD, and mF .
C
D
95
F
E
Midsegment of a trapezoid
The midsegment of a trapezoid is the segment that connects the
midpoints of its legs.
Theorem 6.17: Midsegment of a trapezoid
The midsegment of a trapezoid
is parallel to each base and its
length is one half the sums of
the lengths of the bases.
MN║AD, MN║BC
MN = ½ (AD + BC)
B
M
A
C
N
D
Example
5) Find the length of the midsegment RT.
Definition
• A kite is a
quadrilateral that
has two pairs of
consecutive
congruent sides, but
opposite sides are
not congruent.
Kite Theorems
Theorem 6.18
• If a quadrilateral is a kite,
then its diagonals are
perpendicular.
• AC  BD
Theorem 6.19
• If a quadrilateral is a kite,
then exactly one pair of
opposite angles is
congruent.
• A ≅ C, B ≅ D
C
B
D
A
C
B
D
A
Example
6) Find the lengths of all
four sides of the kite.
X
12
W
20
U 12
12
Z
Y
Example
7) Find mG and mJ
in the diagram at the
right.
J
H
132°
60°
G
K
Block 34: Daily Activities
Block 35: Daily Activities
Exit Quiz: Quadrilaterals
Block 36: Daily Activities
Quadrilaterals Warmup
1.
3.
2.
4.
Exit Quiz: Quadrilaterals
Block 37: Daily Activities
Block 38: Daily Activities
Block 39: Daily Activities