Download Slide 1

3-3
3-3 Proving
ProvingLines
LinesParallel
Parallel
Warm Up
Lesson Presentation
Lesson Quiz
Holt
Geometry
Holt
Geometry
3-3 Proving Lines Parallel
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 =
complementary.
If A and  B are
then mA + mB
3. If AB + BC = AC,
90°, then A and B are
complementary,
=90°.
then A, B, and C are collinear.
If A, B, and C are collinear, then AB + BC = AC.
Holt Geometry
3-3 Proving Lines Parallel
Objective
Use the angles formed by a transversal
to prove two lines are parallel.
Holt Geometry
3-3 Proving Lines Parallel
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 Geometry
3-3 Proving Lines Parallel
Holt Geometry
3-3 Proving Lines Parallel
#1
Use the Converse of the Corresponding Angles
Postulate and the given information to show
that ℓ || m.
1  5
1  5
ℓ || m
Holt Geometry
1 and 5 are corresponding angles.
Conv. of Corr. s Post.
3-3 Proving Lines Parallel
#2
Use the Converse of the Corresponding Angles
Postulate and the given information to show
that ℓ || m.
m4 = (2x + 10)°,
m8 = (3x – 55)°, x = 65
m4 = 2(65) + 10 = 140
m8 = 3(65) – 55 = 140
m4 = m8
4  8
ℓ || m
Holt Geometry
Conv. of Corr. s Post.
3-3 Proving Lines Parallel
#3
Use the Converse of the Corresponding Angles
Postulate and the given information to show
that ℓ || m.
m 1 = m  3
1  3
ℓ || m
Holt Geometry
1 and 3 are
corresponding angles.
Conv. of Corr. s Post.
3-3 Proving Lines Parallel
#4
Use the Converse of the Corresponding Angles
Postulate and the given information to show
that ℓ || m.
m7 = (4x + 25)°,
m5 = (5x + 12)°, x = 13
m7 = 4(13) + 25 = 77
m5 = 5(13) + 12 = 77
m7 = m5
7  5
ℓ || m
Holt Geometry
Conv. of Corr. s Post.
3-3 Proving Lines Parallel
Holt Geometry
3-3 Proving Lines Parallel
#1
Use the given information and the theorems you
have learned to show that r || s.
2  6
2  6
2 and 6 are alternate interior angles.
r || s
Conv. Of Alt. Int. s Thm.
Holt Geometry
3-3 Proving Lines Parallel
#2
Use the given information and the theorems you
have learned to show that r || s.
m6 = (6x + 18)°,
m7 = (9x + 12)°, x = 10
m6 = 6x + 18
= 6(10) + 18 = 78
m7 = 9x + 12
= 9(10) + 12 = 102
Holt Geometry
3-3 Proving Lines Parallel
#2 (Continued)
Use the given information and the theorems you
have learned to show that r || s.
m6 = (6x + 18)°,
m7 = (9x + 12)°, x = 10
m6 + m7 = 78° + 102°
= 180°
r || s
Holt Geometry
6 and 7 are same-side
interior angles.
Conv. of Same-Side Int. s Thm.
3-3 Proving Lines Parallel
#3
Refer to the diagram. Use the given information
and the theorems you have learned to show
that r || s.
m4 = m8
4  8 Congruent angles
4  8
4 and 8 are alternate exterior angles.
r || s
Conv. of Alt. Ext. s Thm.
Holt Geometry
3-3 Proving Lines Parallel
#4
Refer to the diagram. Use the given information
and the theorems you have learned to show
that r || s.
m3 = 2x, m7 = (x + 50),
x = 50
m3 = 2x
= 2(50) = 100°
m7 = x + 50
= 50 + 50 = 100°
m3 = 100 and m7 = 100
3  7
r||s Conv. of the Alt. Int. s Thm.
Holt Geometry
3-3 Proving Lines Parallel
Proof # 1
Given: l || m , 1  3
Prove: p || r
Holt Geometry
3-3 Proving Lines Parallel
Proof 1 Continued
Statements
Reasons
1. l || m
1. Given
2. 1  3
2. Given
3. 1  2
3. Corresponding < Post.
4. 2  3
4. Transitive POC
5. r ||p
5. Conv. of Alt. Ext s Thm.
Holt Geometry
3-3 Proving Lines Parallel
Proof # 2
Given: 1  4, 3 and 4 are supplementary.
Prove: ℓ || m
Holt Geometry
3-3 Proving Lines Parallel
Proof 2 Continued
Statements
1.
2.
3.
4.
5.
6.
1  4
< 3 and < 4 are supp
m<1 = m<4
m3 + m4 = 180
m3 + m1 = 180
2  3
7. m<2 = m<3
8. m2 + m1 = 180
9. ℓ || m
Holt Geometry
Reasons
1. Given
2. Given
3. Def of  <s
4. Def. of Supp <s
5. Substitution
6. Vert.s Thm.
7. Def of  <s
8. Substitution
9. Conv. of Same-Side
Interior s Thm.
3-3 Proving Lines Parallel
Proof # 3
Given: 1 and 3 are supplementary.
Prove: m || n
Holt Geometry
3-3 Proving Lines Parallel
Proof 3 Continued
Statements
Reasons
1. <1 and <3 are supp
2. 2 and 3 are supp
1. Given
3. 1  2
3.  Supplements Thm.
4. m ||n
4. Conv. Of Corr < Post.
Holt Geometry
2. Linear Pair Thm
3-3 Proving Lines Parallel
Ms. Anderson’s Rowing Example # 1
During a race, all
members of a rowing
team should keep the
oars parallel on each
side. If on the port side
m < 1 = (3x + 13)°,
m<2 = (5x - 5)°, & x = 9.
Show that the oars are
parallel.
3x+13 = 3(9) + 13 = 40°
5x - 5 = 5(9) - 5 = 40°
The angles are congruent, so the oars are || by the
Conv. of the Corr. s Post.
Holt Geometry
3-3 Proving Lines Parallel
Ms. Anderson’s Rowing Example # 2
Suppose the
corresponding angles on
the starboard side of the
boat measure (4y – 2)°
and (3y + 6)°, & y = 8.
Show that the oars are
parallel.
4y – 2 = 4(8) – 2 = 30°
3y + 6 = 3(8) + 6 = 30°
The angles are congruent, so the oars are || by the
Conv. of the Corr. s Post.
Holt Geometry
3-3 Proving Lines Parallel
Exit Question
• Complete Exit Questions to be turned in
– This is not graded
Holt Geometry
3-3 Proving Lines Parallel
Homework
• Page 166-167: # 1-10 & # 22
• Chapter 3 Test on Friday December 21st
(C-level)
Holt Geometry