parallel lines cut by a transversal definitions parallel transversal angle vertical angle...
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DEFINITIONS
• PARALLEL• TRANSVERSAL• ANGLE• VERTICAL ANGLE• CORRESPONDING ANGLE• ALTERNATE INTERIOR ANGLE• ALTERNATE EXTERIOR ANGLE
Parallel lines cut by a transversal
123 4
567 8
< 1 and < 2 are called SUPPLEMENTARY ANGLES
DEFINITION:They form a straight angle measuring 180 degrees.
Parallel lines cut by a transversal
123 4
567 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
123 4
567 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
123 4
567 8
Name other vertical pairs:
< 2 and < 4
< 6 and < 8
< 5 and < 7
Parallel lines cut by a transversal
123 4
567 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
123 4
567 8
Name other corresponding pairs:
< 2 and < 6
< 3 and < 7
< 4 and < 8
Parallel lines cut by a transversal
123 4
567 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
123 4
567 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
123 4
567 8
Name other alternate exterior pairs:
< 2 and < 8
< 1 and < 7
Parallel lines cut by a transversal
123 4
567 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
x + 102x + 20
What do you know about the angles?
Write the equation.
Solve for x.
SUPPLEMENTARY
2x + 20 + x + 10 = 180
3x + 30 = 1803x = 150
x = 50
TRY IT OUT
2x - 60
3x - 120
What do you know about the angles?
Write the equation.
Solve for x.
ALTERNATE INTERIOR
3x - 120 = 2x - 60
x = 60Subtract 2x from both sides
Add 120 to both sides
Holt Geometry
3-2 Angles Formed by Parallel Lines and Transversals
Warm UpIdentify each angle pair.
1. 1 and 3
2. 3 and 6
3. 4 and 5
4. 6 and 7 same-side int s
corr. s
alt. int. s
alt. ext. s
Holt Geometry
3-2 Angles Formed by Parallel Lines and Transversals
Find each angle measure.
Example 1: Using the Corresponding Angles Postulate
A. mECF
x = 70
B. mDCE
mECF = 70°
Corr. s Post.
5x = 4x + 22 Corr. s Post.
x = 22 Subtract 4x from both sides.
mDCE = 5x
= 5(22) Substitute 22 for x.
= 110°
Holt Geometry
3-2 Angles Formed by Parallel Lines and Transversals
Check It Out! Example 1
Find mQRS.
mQRS = 180° – x
x = 118
mQRS + x = 180°
Corr. s Post.
= 180° – 118°
= 62°
Subtract x from both sides.
Substitute 118° for x.
Def. of Linear Pair
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°
32
1
m<1 + m<2 + m<3 = 180°
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
3
2
1 4
exterior angle
remote interior
angles
m<4 = m<1 + m<2
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 Exterior Angle Theorem
1
2 3 4
P
Q R
In the triangle below, recall that 1, 2, and 3 are _______ angles ofΔPQR.
interior
Angle 4 is called an _______ angle of ΔPQR.exterior
An exterior angle of a triangle is an angle that forms a _________ with one ofthe angles of the triangle.
linear pair
In ΔPQR, 4 is an exterior angle at R because it forms a linear pair with 3.
____________________ of a triangle are the two angles that do not forma linear pair with the exterior angle.Remote interior angles
In ΔPQR, 1, and 2 are the remote interior angles with respect to 4.
Exterior Angle Theorem Exterior Angle Theorem
1
2
3 4 5
In the figure below, 2 and 3 are remote interior angles with respect towhat angle? 5
an example with numbers
x
82°
30° y
find x & y
x = 68°
y = 112°
40x 10x2
30x find all the angle measures
80°, 60°, 40°
Do you hear the sirens?????
• Determine the measure of <4,
• If <3 = 50, <2 = 70
Holt Geometry
4-2 Angle Relationships in Triangles
Find mB.
Example 3: Applying the Exterior Angle Theorem
mA + mB = mBCD Ext. Thm.
15 + 2x + 3 = 5x – 60 Substitute 15 for mA, 2x + 3 for mB, and 5x – 60 for mBCD.
2x + 18 = 5x – 60 Simplify.
78 = 3xSubtract 2x and add 60 to both sides.
26 = x Divide by 3.
mB = 2x + 3 = 2(26) + 3 = 55°
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~.