Download Unit 3

Unit 3 Angles and Transversals
Unit 3: This unit Introduces Transversals, and angles
based on transversals, including corresponding
angles, Same Side Interior Angles, Alternate Exterior
and Alternate Interior Angles, and the unique
properties of Perpendicular Transversals.
125 2
3 4
5 X+ 15
7 8
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Standards
SPI’s taught in Unit 3:
SPI 3108.1.1 Give precise mathematical descriptions or definitions of geometric shapes in
the plane and space.
SPI 3108.1.4 Use definitions, basic postulates, and theorems about points, lines, angles, and
planes to write/complete proofs and/or to solve problems.
SPI 3108.2.1 Analyze, apply, or interpret the relationships between basic number concepts
and geometry (e.g. rounding and pattern identification in measurement, the relationship of
pi to other rational and irrational numbers)
SPI 3108.4.2 Define, identify, describe, and/or model plane figures using appropriate
mathematical symbols (including collinear and non-collinear points, lines, segments, rays,
angles, triangles, quadrilaterals, and other polygons).
CLE (Course Level Expectations) found in Unit 3:
CLE 3108.3.1 Use analytic geometry tools to explore geometric problems involving parallel
and perpendicular lines, circles, and special points of polygons.
CLE 3108.4.1 Develop the structures of geometry, such as lines, angles, planes, and planar
figures, and explore their properties and relationships.
CFU (Checks for Understanding) applied to Unit 3:
3108.4.8 Apply properties and theorems about angles associated with parallel and
perpendicular lines to solve problems.
3108.4.21 Use properties of and theorems about parallel lines, perpendicular lines, and
angles to prove basic theorems in Euclidean geometry (e.g., two lines parallel to a third line
are parallel to each other, the perpendicular bisectors of line segments are the set of all
points equidistant from the endpoints, and two lines are parallel when the alternate interior
angles they make with a transversal are congruent).
Transversal
• A transversal is a line which cuts across two or
more coplanar lines –they may or may not be
parallel
• Transversals create special angle pairs which
have properties that we can use to solve their
measure of degree
Transversal
Parallel Lines are indicated by arrows
Corresponding Angles
• Corresponding Angles:
These are angles which
occupy corresponding (or
matching) positions on two
different lines• Angle 1 corresponds with
Angle 5
• Which angle corresponds
with angle 4?
• Which angle corresponds
with angle 2?
1
2
3 4
5 6
7 8
Corresponding Angles on Parallel Lines
• Corresponding Angles Postulate
– If a transversal intersects 2 parallel
lines, then corresponding angles are
congruent
• If the measure of angle 3 is 75 degrees,
what is the measure of angle 2? (Vertical
Angles)
• What is the measure of Angle 7?
(Corresponding Angles)
• What is the measure of Angle 6?
• What is the measure of Angle 4
(Supplementary Angles)?
• What is the measure of Angle 8
(corresponding angles)?
1 2
3 4
5 6
7 8
Alternate Interior Angles
• These angles lie between the two lines,
on opposite sides of the transversal
• Alternate Interior Angles Theorem
– If a transversal intersects 2 parallel lines, then
alternate interior angles are congruent
• Here, angle 3 and angle 6 are Alternate
angles.
• What is the alternate interior angle
that matches up with angle 5?
• If angle 5 is 130 degrees, what is the
measure of angle 4?
• What is the measure of Angle 6?
• What is the measure of Angle 3?
1 2
3 4
5 6
7 8
Consecutive Interior Angles, or Same
Side Interior Angles
• If two angles lie between the two lines on
the same side of the transversal they are
consecutive interior angles, or same-side
interior angles
• Same-Side Interior Angles Theorem
– If a transversal intersects 2 parallel
lines, then same-side interior angles
are supplementary
• Angle 3 and Angle 5 are Same Side
Interior Angles
• If angle 5 is 120 degrees, what is the
measure of angle 3?
• What is the measure of angle 4?
• What is the measure of angle 6?
1 2
3 4
5 6
7 8
Alternate Exterior Angles
• If two angles lie outside the two
lines, on opposite sides of the
transversal, then they are
Alternate Exterior Angles
• Alternate Exterior Angles Theorem
– If a transversal intersects 2 parallel lines,
then Alternate Exterior Angles are
congruent
• Angle 1 and Angle 8 are
alternate exterior angles
• What is the alternate exterior
angle for angle 7?
• If angle 8 measures 140 degrees,
what is the measure of angle 1?
1 2
3 4
5 6
7 8
Check for Learning
• Do problems 21-24 on page 144
Perpendicular Transversals
• If a transversal is
perpendicular to one of two
parallel lines, it is
perpendicular to the other
parallel line
• Here, we are given enough
information to conclude that
line c is perpendicular to line
a, because it forms a right
(90 degree) angle
• Because line a is parallel to
line b, we can conclude that
line c is also perpendicular to
line b
a
b
c
Example
• Find the value of X
• These are like puzzles:
– Here, we see that angle 4 is vertical
with an angle that is 125 degrees.
Therefore, angle 4 is 125 degrees
– We see that angle 4, and the angle
(x+15) are same-side interior angles
– We know that Same-Side interior
angles are supplementary (they add
up to 180 degrees)
– Therefore 125 + X + 15 = 180
– Or: 140 + X = 180
– Or: X = 40
125 2
3 4
5 X+ 15
7 8
Assignment
• Pages 153-154 Problems 7-20 (Guided
practice)
• Worksheets:
– 3-1
– Angle Pairs on two transversals
– Transversal on Parallel Lines
Unit 3 Quiz 1
Use the picture at the right to answer the following questions
1.
2.
3.
4.
5.
6.
7.
Name a pair of Corresponding Angles
Name a pair of Alternate Interior Angles
Name a pair of Same Side Interior Angles
Name a pair of Alternate Exterior Angles
(Yes/No) Are Corresponding Angles congruent?
(Yesn/No) are Same Side Interior Angles congruent?
If angle 1 measures 130 degrees, what is the
measure of angle 8?
8. If angle 2 measures 65 degrees, and angle 6 is (2x15), what is x?
9. If angle 7 measures 48 degrees, what is the measure
of angle 4?
10. If angle 3 measures (4x+8) and angle 6 measures
(8x), what is x?
1 2
3 4
5 6
7 8
Unit 3 Final Extra Credit
• The measure of Angle 1 is 6x
• The measure of Angle 2 is 2x-20
1. What is the value of x?
2. How many degrees is the
measure of angle 6?
3. What is the measure of angle 5?
4. What type of angle pair are
angle 3 and angle 6?
5. What can you conclude about
line a and line b?
2 points each, show all work
(equations as required)
a
b
1 2
3 4
5 6
7 8