Transcript PARALLEL LINES CUT BY A TRANSVERSAL

PARALLEL LINES CUT BY A
TRANSVERSAL
DEFINITIONS
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PARALLEL
TRANSVERSAL
ANGLE
VERTICAL ANGLE
CORRESPONDING ANGLE
ALTERNATE INTERIOR ANGLE
ALTERNATE EXTERIOR ANGLE
DEFINITIONS
• SUPPLEMENTARY ANGLE
• COMPLEMENTARY ANGLE
• CONGRUENT
Parallel lines cut by a transversal
2
3
6
7 8
5
1
4
Parallel lines cut by a transversal
2
3
6
1
4
5
7 8
< 1 and < 2 are called SUPPLEMENTARY ANGLES
DEFINITION:They form a straight angle measuring 180
degrees.
Parallel lines cut by a transversal
2
3
6
1
4
5
7 8
Name other supplementary pairs:
< 2 and < 3
< 3 and < 4
< 4 and < 1
< 5 and < 6
< 6 and < 7
< 7 and < 8
< 8 and < 5
Parallel lines cut by a transversal
2
3
6
1
4
5
7 8
< 1 and < 3 are called VERTICAL
ANGLES
They are congruent m<1 = m<3
DEFINITION: The angles formed
from two lines are crossing.
Parallel lines cut by a transversal
2
3
6
7 8
1
4
5
< 2 and < 4
< 6 and < 8
Name other vertical pairs:
< 5 and < 7
Parallel lines cut by a transversal
2
3
6
1
4
5
7 8
< 1 and < 5 are called CORRESPONDING ANGLES
They are congruent
m<1 = m<5
DEFINITION: Corresponding angles occupy the same position
on the top and bottom parallel lines.
Parallel lines cut by a transversal
2
3
6
1
4
5
7 8
< 2 and < 6
< 3 and < 7
Name other corresponding pairs:
< 4 and < 8
Parallel lines cut by a transversal
2
3
6
1
4
5
7 8
< 4 and < 6 are called ALTERNATE INTERIOR ANGLES
They are congruent
m<4 = m<6
DEFINITION:Alternate Interior on the inside of the two parallel
lines and on opposite sides of the transversal.
Parallel lines cut by a transversal
2
3
6
1
4
5
7 8
Name other alternate interior pairs:
< 3 and < 5
Parallel lines cut by a transversal
2
3
6
1
4
5
7 8
< 1 and < 7 are called ALTERNATE EXTERIOR ANGLES
They are congruent
m<1 = m<7
Alternate Exterior on the outside of the two parallel lines and on
opposite sides of the transversal.
Parallel lines cut by a transversal
2
3
6
1
4
5
7 8
< 2 and < 8
Name other alternate exterior pairs:
< 1 and < 7
Parallel lines cut by a transversal
2
3
6
1
4
5
7 8
< 4 and < 5 are called CONSECUTIVE INTERIOR ANGLES
The sum is 180.
m<4 = m<5
DEFINITION: Consecutive Interior on the inside of the two
parallel lines and on same side of the transversal. Sum = 180
TRY IT OUT
2
3
6
1
4
5
7 8
The m < 1 is
60 degrees.
What is the m<2 ?
120 degrees
TRY IT OUT
2
3
6
1
4
5
7 8
The m < 1 is
60 degrees.
What is the m<5 ?
60 degrees
TRY IT OUT
2
3
6
1
4
5
7 8
The m < 1 is
60 degrees.
What is the m<3 ?
60 degrees
TRY IT OUT
120 60
60 120
120
60
60 120
TRY IT OUT
2x + 20
What do you know about the angles?
Write the equation.
Solve for x.
x + 10
SUPPLEMENTARY
2x + 20 + x + 10 = 180
3x + 30 = 180
3x = 150
x = 50
TRY IT OUT
3x - 120
2x - 60
What do you know about the angles? ALTERNATE INTERIOR
Write the equation.
Solve for x.
Subtract 2x from both sides
Add 120 to both sides
3x - 120 = 2x - 60
x
=
60
3-2
Angles Formed by Parallel Lines
and Transversals
Warm Up
Identify each angle pair.
1. 1 and 3
corr. s
2. 3 and 6
alt. int. s
3. 4 and 5
alt. ext. s
4. 6 and 7
same-side int s
Holt Geometry
3-2
Angles Formed by Parallel Lines
and Transversals
Example 1: Using the Corresponding Angles
Postulate
Find each angle measure.
A. mECF
x = 70 Corr. s Post.
mECF = 70°
B. mDCE
5x = 4x + 22
x = 22
mDCE = 5x
= 5(22)
= 110°
Holt Geometry
Corr. s Post.
Subtract 4x from both sides.
Substitute 22 for x.
3-2
Angles Formed by Parallel Lines
and Transversals
Check It Out! Example 1
Find mQRS.
x = 118 Corr. s Post.
mQRS + x = 180°
mQRS = 180° – x
Def. of Linear Pair
Subtract x from both sides.
= 180° – 118° Substitute 118° for x.
= 62°
Holt Geometry
WEBSITES FOR PRACTICE
Ask Dr. Math: Corresponding /Alternate Angles
Project Interactive: Parallel Lines cut by Transversal
Triangle Sum Theorem
The sum of the angle measures in a triangle
equal 180°
1
3
2
m<1 + m<2 + m<3 = 180°
Corollary
•A corollary to a
theorem is a
statement that
follows directly from
that theorem
TRIANGLE ANGLE SUM
THEOREM COROLLARIES
• If 2 angles of 1 triangle are congruent to 2
angles of another triangle, then the 3rd
angles are congruent
• The acute angles of a right triangle are
complementary
• The measure of each angle of an
equiangular triangle is 60o
• A triangle can have at most 1 right or 1
obtuse angle
Exterior Angle Theorem
(your new best friend)
The measure of an exterior angle in a
triangle is the sum of the measures of
the 2 remote interior angles
remote
interior
angles
exterior
angle
2
1
3
m<4 = m<1 + m<2
4
REMOTE INTERIOR ANGLE
• In any polygon, a remote interior angle is an
interior angle that is not adjacent to a given
exterior angle A and B are remote to angle 1
Exterior Angle Theorem
In the triangle below, recall that
ΔPQR.
1,
2, and
interior angles of
3 are _______
Angle 4 is called an exterior
_______ angle of ΔPQR.
linear pair with one of
An exterior angle of a triangle is an angle that forms a _________
the angles of the triangle.
In ΔPQR,
4 is an exterior angle at R because it forms a linear pair with
Remote interior angles of a triangle are the two angles that do not form
____________________
a linear pair with the exterior angle.
In ΔPQR, 1, and 2 are the remote interior angles
with respect to 4.
P
1
Q
2
3 4
R
3.
Exterior Angle Theorem
In the figure below,
what angle? 5
2 and
3 are remote interior angles with respect to
1
2
3
4
5
an example with numbers
find x & y
82°
30°
x
y
x = 68°
y=
112°
• Determine the measure of <4,
30x
find all the angle
• If <3 = 50, <2 = 70
measures
40x
10x2
80°, 60°, 40°
Do you hear the
sirens?????
Exterior Angle Theorem
4-2 Angle Relationships in Triangles
Example 3: Applying the Exterior Angle Theorem
Find mB.
mA + mB = mBCD
Ext.  Thm.
15 + 2x + 3 = 5x – 60
Substitute 15 for mA, 2x + 3 for
mB, and 5x – 60 for mBCD.
2x + 18 = 5x – 60
78 = 3x
Simplify.
Subtract 2x and add 60 to
both sides.
Divide by 3.
26 = x
mB = 2x + 3 = 2(26) + 3 = 55°
Holt Geometry
4-2 Angle Relationships in Triangles
Holt Geometry
There are several ways to prove certain triangles are
similar. The following postulate, as well as the SSS
and SAS Similarity Theorems, will be used in proofs
just as SSS, SAS, ASA, HL, and AAS were used to
prove triangles congruent.
Example 1: Using the AA Similarity Postulate
Explain why the triangles
are similar and write a
similarity statement.
Since
, B  E by the Alternate Interior
Angles Theorem. Also, A  D by the Right Angle
Congruence Theorem. Therefore ∆ABC ~ ∆DEC by
AA~.
Check It Out! Example 1
Explain why the triangles
are similar and write a
similarity statement.
By the Triangle Sum Theorem, mC = 47°, so C  F.
B  E by the Right Angle Congruence Theorem.
Therefore, ∆ABC ~ ∆DEF by AA ~.