Download Alternate exterior angles

KELOMPOK 5
Anggota Kelompok:
1. Fiqih Dheni Kartika (4101414010)
2. Maitsaa Kaamiliaa (4101414011)
3. Siti Aminah Silviani (4101414024)
4. Cahyaningrum Uswati(4101414151)
Kesejajaran Garis
Definition 5-1
Skew Lines are two nonintersecting lines that
do not lie in the same plane.
Definition 5-2
 A line and a plane are parallel if they have no
points in common.
Definition 5-3
Parallel planes are planes that have no points
in common.
A
B
Definition 5-4
 A transversal is a line that intersect two coplanar lines in two
different points.
Line l intersects lines m and n in two different points to form
interior angles and exterior angles. l is called a transversal.
l
1 2
l
m
3 4
7
n
Interior Angles
5 6
8
Exterior Angles
m
n
Alternate interior angles are two interior
angles with different vertices on opposite
sides of the transversal.
2
1
4
3
and are called alternate interior angles.
and are called alternate interior angles.
 Alternate exterior angles are two exterior angles with
different vertices on opposite sides of the transversal.
5
8
6
7
and are called alternate exterior angles.
and are called alternate exterior angles.
Corresponding angles are on the same side of
the transversal. One of the angles is an
exterior angle, one is an interior angle.
1
7
4
5
6
2
3
8
There are four pairs of corresponding angles :
and and and and 
Theorem 5-1
If two lines are cut by a transversal and
a pair of
corresponding angles are congruent, then the lines are
parallel.
A
p
B
q
2
1
r
Given : Lines p, q, and r with  1 ≅  2
Proof : p || q
Plan : Assume that p || q (not parallel). Then consider
the triangle that would be formed, and find a
contradiction.
A
2
B 1
Statements :
1. Suppose p || q
2. Then p and q intersect at a point,
say C, and ∆ ABC is formed
3.  2 is an exterior angle of ∆
ABC
4.  1 is a remote interior angle of
2
5. m  2 > m  1
6. m  1 = m  2 (contradiction
to m  2 > m  1)
7. therefore, p||q
C
Reason :
1. Indirect proof assumption
2. Restatement of 1
3. Definition of exterior angle
4. Definition of remote interior angle
5. Exterior Angle Theorem
6. Given
7. Logic of indirect proof
Theorem 5-2
If two lines are cut by a transversal and a pair of alternate
interior angles are congruent then the lines are parallel.
Statements



4.
≅ 
≅ 
≅ 
p || q
Reasons
1. Given
2. Congruent
3. Transitive Property of Congruence
4. Logic of indirect proof
Theorem 5-3
If two lines are cut by a transversal and a pair of alternate
exterior angles are congruent then the lines are parallel.
Given : ≅ 
Prove : p || q
Statements
Reasons
 ≅ 
 ≅ 
3. p || q
1. Congruent
2. Theorem 5.2
3. Logic of indirect proof
Theorem 5-4
If two lines are cut by a transversal and a pair of interior
angles on the same side of the transversal are supplementary,
then the lines are parallel.
Given : + 
Prove : p || q
Statements
Reasons



4.
1.
2.
3.
4.
+ 
+ 
≅ 
p || q
Given
Suplementary
Theori 1
Logic of indirect proof
Soal dan Pembahasan
HALAMAN 178 NOMOR 17
Given :
ABC ≅ BCD
𝐵𝐹 bisects ABC
𝐶𝐺 bisects BCD
Prove :
𝐵𝐹 || 𝐶𝐺
Answer :
B
G
A
2
3
D
Answer :
F
C
ABC ≅ BCD
m  =
1
m
2
1
m
2
ABC
(Given)
(Bisect)
m  =
BCD
(Bisect)
m  m  (Restatement 2 and 3)
Therefore : 𝐵𝐹 || 𝐶𝐺 (Theorem 5-2)
HALAMAN 176 NOMOR 6
Which pairs of angles could you prove
congruent to show that 𝐶𝐷 || 𝐴𝐵 in the
figure?
2
1
3
4O 5 6
C
7
A
D
8
B
Answer :
 ≅ (corresponding angles) theorem 5.1
 ≅ 
(corresponding angles) theorem 5.1
 ≅ 
(alternate interior angles) theorem 5.2
 ≅ 
(alternate interior angles) theorem 5.2
HALAMAN 177 NOMOR 8
Complete the following two-column proof of Theorem 5-2
Given :  1 ≅  2
Prove : p || q
3
2
1
Statements
Reasons
1.  1 ≅  2
2.  2 ≅  3
  1 ≅  3
4. p || q
1. .....
2. ......
3. Transitive Property of Congruence
4. .....
Answer:
Statements



4.
≅ 
≅ 
≅ 
p || q
Reasons
1. Given
2. Congruent
3. Transitive Property of
Congruence
4. Logic of indirect proof
HALAMAN 173 NOMOR 10
In a periscope a pair of mirrors are mounted parallel to each other
as shown. The path of light becomes a transversal. Which pair of
angles is an alternate interior pair?  1 and 3, 1 and 4, 2
and 3, or 2 and 4?
mirror
1
2
3
Answer : 2 and 3
4
mirror
HALAMAN 176 NOMOR 5
What pairs of angles could you prove
congruent to show that 𝐴𝐵 || 𝐷𝐶 in the figure?
C
D
A 1
3
2
4
B
Answer :
≅ 
Menurut theorema 5-2
HALAMAN 178 NOMOR 19
Given :
m2 + m3 + m5 = 180
4 ≅ 5
Prove :
𝐴𝐵|| 𝐶𝐷
C
A
4
D
1
3 2
5
B
Answer :
D
C 1
3 2
4
5
D
B
1) m m m 
(Given)
2) m m m 
(sudut berpelurus)
3) m = m 
(Restatement 1 and 5)
4) m = m 
(Given)
5) m = m = m 
(Restatement 3 and 4)
Therefore : 𝐴𝐵 || 𝐶𝐷
(Theorema 1)