Transcript Slide 1

21.1
Angles
Formed
by Parallel
Lines
Angles
Formed
by Parallel
and Transversals
and Transversals
Warm Up
Lesson Presentation
Lesson Quiz
Holt
Geometry
Holt
McDougal
Geometry
Lines
21.1
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 McDougal Geometry
21.1 Angles Formed by Parallel Lines
and Transversals
Objective
Prove and use theorems about the
angles formed by parallel lines and a
transversal.
Holt McDougal Geometry
21.1
Angles Formed by Parallel Lines
and Transversals
Holt McDougal Geometry
21.1
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 McDougal Geometry
Corr. s Post.
Subtract 4x from both sides.
Substitute 22 for x.
21.1
Angles Formed by Parallel Lines
and Transversals
Example 2
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 McDougal Geometry
21.1
Angles Formed by Parallel Lines
and Transversals
Helpful Hint
If a transversal is perpendicular to
two parallel lines, all eight angles are
congruent.
Holt McDougal Geometry
21.1
Angles Formed by Parallel Lines
and Transversals
Remember that postulates are statements
that are accepted without proof.
Since the Corresponding Angles Postulate is
given as a postulate, it can be used to
prove the next three theorems.
Holt McDougal Geometry
21.1
Angles Formed by Parallel Lines
and Transversals
Holt McDougal Geometry
21.1
Angles Formed by Parallel Lines
and Transversals
Example 3
Find each angle measure.
A. mEDG
mEDG = 75° Alt. Ext. s are
Congruent.
B. mBDG
x – 30° = 75° Alt. Ext. s are congruent.
x = 105 Add 30 to both sides.
mBDG = 105°
Holt McDougal Geometry
21.1
Angles Formed by Parallel Lines
and Transversals
Example 4
Find x and y in the diagram.
By the Alternate Interior Angles
Theorem, (5x + 4y)° = 55°.
By the Corresponding Angles
Postulate, (5x + 5y)° = 60°.
5x + 5y = 60
–(5x + 4y = 55)
y=5
Subtract the first equation
from the second equation.
5x + 5(5) = 60
Substitute 5 for y in 5x + 5y =
60. Simplify and solve for x.
x = 7, y = 5
Holt McDougal Geometry
21.1
Angles Formed by Parallel Lines
and Transversals
Lesson Quiz
State the theorem or postulate that is related
to the measures of the angles in each pair.
Then find the unknown angle measures.
1. m1 = 120°, m2 = (60x)°
Alt. Ext. s Thm.; m2 = 120°
2. m2 = (75x – 30)°,
m3 = (30x + 60)°
Corr. s Post.; m2 = 120°,
m3 = 120°
3. m3 = (50x + 20)°, m4= (100x – 80)°
Alt. Int. s Thm.; m3 = 120°, m4 =120°
4. m3 = (45x + 30)°, m5 = (25x + 10)°
Same-Side Int. s Thm.; m3 = 120°, m5 =60°
Holt McDougal Geometry
21.2
Angles Formed by Parallel Lines
Proving
Lines Parallel
and
Transversals
Warm Up
Lesson Presentation
Lesson Quiz
Holt
Geometry
Holt
McDougal
Geometry
21.2
Angles Formed by Parallel Lines
and Transversals
Warm Up
State the converse of each statement.
1. If a = b, then a + c = b + c.
If a + c = b + c, then a = b.
2. If mA + mB = 90°, then A and B are
complementary.
If A and  B are
complementary, then mA +
mAB
B =90°.
3. If
+ BC = AC, then A, B, and C are collinear.
If A, B, and C are collinear, then AB + BC = AC.
Holt McDougal Geometry
21.2
Angles Formed by Parallel Lines
and Transversals
Objective
Use the angles formed by a
transversal to prove two lines are
parallel.
Holt McDougal Geometry
21.2
Angles Formed by Parallel Lines
and Transversals
Recall that the converse of a theorem is found
by exchanging the hypothesis and conclusion.
The converse of a theorem is not automatically
true. If it is true, it must be stated as a
postulate or proved as a separate theorem.
Holt McDougal Geometry
21.2
Angles Formed by Parallel Lines
and Transversals
Holt McDougal Geometry
21.2
Angles Formed by Parallel Lines
and Transversals
The Converse of the Corresponding Angles
Postulate is used to construct parallel lines. The
Parallel Postulate guarantees that for any line ℓ,
you can always construct a parallel line through
a point that is not on ℓ.
Holt McDougal Geometry
21.2
Angles Formed by Parallel Lines
and Transversals
Holt McDougal Geometry
21.2
Angles Formed by Parallel Lines
and Transversals
Lesson Quiz: Part I
Name the postulate or theorem
that proves p || r.
1. 4  5
Conv. of Alt. Int. s Thm.
2. 2  7
Conv. of Alt. Ext. s Thm.
3. 3  7
Conv. of Corr. s Post.
4. 3 and 5 are supplementary.
Conv. of Same-Side Int. s Thm.
Holt McDougal Geometry
21.3
Angles Formed by Parallel Lines
Perpendicular
Lines
and
Transversals
Warm Up
Lesson Presentation
Lesson Quiz
Holt
Geometry
Holt
McDougal
Geometry
21.3
Angles Formed by Parallel Lines
and Transversals
Objective
Prove and apply theorems about
perpendicular lines.
Holt McDougal Geometry
21.3
Angles Formed by Parallel Lines
and Transversals
Vocabulary
perpendicular bisector
distance from a point to a line
Holt McDougal Geometry
21.3
Angles Formed by Parallel Lines
and Transversals
The perpendicular bisector of a segment
is a line perpendicular to a segment at
the segment’s midpoint.
The shortest segment from a point to a
line is perpendicular to the line. This fact
is used to define the distance from a
point to a line as the length of the
perpendicular segment from the point to
the line.
Holt McDougal Geometry
21.3
Angles Formed by Parallel Lines
and Transversals
Example 5: Distance From a Point to a Line
A. Name the shortest segment from point A to BC.
AP
B. Write and solve an inequality for x.
AC > AP AP is the shortest segment.
x – 8 > 12 Substitute x – 8 for AC and 12 for AP.
+ 8 + 8 Add 8 to both sides of the inequality.
x > 20
Holt McDougal Geometry
21.3
Angles Formed by Parallel Lines
and Transversals
HYPOTHESISCONCLUSION
Holt McDougal Geometry
21.3
Angles Formed by Parallel Lines
and Transversals
Example 6
Solve to find x and y in the diagram.
x = 9, y = 4.5
Holt McDougal Geometry