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Holt McDougal Geometry
Angles Formed by Parallel Lines and Transversals
Angles Formed by Parallel Lines and Transversals
Holt Geometry
Warm Up
Lesson Presentation
Lesson Quiz
Holt McDougal Geometry
21.1
Holt McDougal Geometry
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
21.1
Holt McDougal Geometry
Angles Formed by Parallel Lines and Transversals
Prove and use theorems about the angles formed by parallel lines and a transversal.
Objective
21.1
Holt McDougal Geometry
Angles Formed by Parallel Lines and Transversals
21.1
Holt McDougal Geometry
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°
21.1
Holt McDougal Geometry
Angles Formed by Parallel Lines and Transversals
Example 2
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*
21.1
Holt McDougal Geometry
Angles Formed by Parallel Lines and Transversals
If a transversal is perpendicular to two parallel lines, all eight angles are congruent.
Helpful Hint
21.1
Holt McDougal Geometry
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.
21.1
Holt McDougal Geometry
Angles Formed by Parallel Lines and Transversals
21.1
Holt McDougal Geometry
Angles Formed by Parallel Lines and Transversals
Find each angle measure.
Example 3
A. mEDG
B. mBDG
mEDG = 75° Alt. Ext. s are
Congruent.
mBDG = 105°
x – 30° = 75° Alt. Ext. s are congruent.
x = 105 Add 30 to both sides.
21.1
Holt McDougal Geometry
Angles Formed by Parallel Lines and Transversals
Find x and y in the diagram.
Example 4
By the Alternate Interior AnglesTheorem, (5x + 4y)° = 55°.
By the Corresponding Angles Postulate, (5x + 5y)° = 60°.
5x + 5y = 60–(5x + 4y = 55) y = 5
5x + 5(5) = 60
Subtract the first equation from the second equation.
x = 7, y = 5
Substitute 5 for y in 5x + 5y = 60. Simplify and solve for x.
21.1
Holt McDougal Geometry
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. m1 = 120°, m2 = (60x)°
2. m2 = (75x – 30)°, m3 = (30x + 60)°
Corr. s Post.; m2 = 120°, m3 = 120°
Alt. Ext. s Thm.; m2 = 120°
3. m3 = (50x + 20)°, m4= (100x – 80)°
4. m3 = (45x + 30)°, m5 = (25x + 10)°Alt. Int. s Thm.; m3 = 120°, m4 =120°
Same-Side Int. s Thm.; m3 = 120°, m5 =60°
21.1
Holt McDougal Geometry
Angles Formed by Parallel Lines and TransversalsProving Lines Parallel
Holt Geometry
Warm Up
Lesson Presentation
Lesson Quiz
Holt McDougal Geometry
21.2
Holt McDougal Geometry
Angles Formed by Parallel Lines and Transversals
Warm UpState the converse of each statement.
1. If a = b, then a + c = b + c.
2. If mA + mB = 90°, then A and B are complementary.
3. If AB + BC = AC, then A, B, and C are collinear.
If a + c = b + c, then a = b.
If A and B are complementary, then mA + mB =90°.
If A, B, and C are collinear, then AB + BC = AC.
21.2
Holt McDougal Geometry
Angles Formed by Parallel Lines and Transversals
Use the angles formed by a transversal to prove two lines are parallel.
Objective
21.2
Holt McDougal Geometry
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.
21.2
Holt McDougal Geometry
Angles Formed by Parallel Lines and Transversals
21.2
Holt McDougal Geometry
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.
21.2
Holt McDougal Geometry
Angles Formed by Parallel Lines and Transversals
21.2
Holt McDougal Geometry
Angles Formed by Parallel Lines and Transversals
Lesson Quiz: Part I
Name the postulate or theoremthat 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.
21.2
Holt McDougal Geometry
Angles Formed by Parallel Lines and TransversalsPerpendicular Lines
Holt Geometry
Warm Up
Lesson Presentation
Lesson Quiz
Holt McDougal Geometry
21.3
Holt McDougal Geometry
Angles Formed by Parallel Lines and Transversals
Prove and apply theorems about perpendicular lines.
Objective
21.3
Holt McDougal Geometry
Angles Formed by Parallel Lines and Transversals
perpendicular bisectordistance from a point to a line
Vocabulary
21.3
Holt McDougal Geometry
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.
21.3
Holt McDougal Geometry
Angles Formed by Parallel Lines and Transversals
Example 5: Distance From a Point to a Line
AP
B. Write and solve an inequality for x.
AC > AP
x – 8 > 12
x > 20
Substitute x – 8 for AC and 12 for AP.
Add 8 to both sides of the inequality.
A. Name the shortest segment from point A to BC.
AP is the shortest segment.
+ 8 + 8
21.3
Holt McDougal Geometry
Angles Formed by Parallel Lines and Transversals
HYPOTHESISCONCLUSION
21.3
Holt McDougal Geometry
Angles Formed by Parallel Lines and Transversals
Example 6
Solve to find x and y in the diagram.
x = 9, y = 4.5
21.3
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