ii) theorem - If a pair of alternate angles formed by a transversal of two lines exe is congruent then the two lines are parallel Given ! - line is transversal of line landline m Lo and 2b is a congruents pair of an a ternate angles. that is La ane = =lb To prove proof: Lat = 180° angles in lineat pair. La & = Lb & . L btcc = Butcb and Lc are interior angies on the same side of transversal . .: line illinem test given
Answers
- Answer:-
What Are Alternate Interior Angles?
Alternate angles are of two types:
Alternate interior angles
Alternate exterior angles
Alternate interior angles: Alternate interior angles are the angles formed when a transversal intersects two coplanar lines. They are on the inner side of the coplanar lines but are on the alternate opposite sides of the transversal.
Alternate interior angles are:
1.
∠
3
and
∠
6
2.
∠
4
and
∠
5
alternate interior angles
Alternate exterior angles: Alternate exterior angles are formed on the exterior of the coplanar lines but on the alternate opposite sides of the transversal.
Alternate exterior angles are:
1.
∠
1
and
∠
8
2.
∠
2
and
∠
7
Alternate interior an alternate exterior angle
Alternate interior angles are:
1.
∠
3
and
∠
6
2.
∠
4
and
∠
5
Alternate exterior angles are:
1.
∠
1
and
∠
8
2.
∠
2
and
∠
7
What Does the Alternate Interior Angles Theorem States?
The alternate interior angles theorem states that “if a transversal crosses the set of parallel lines, the alternate interior angles are congruent.”
In the figure given below, a set of parallel lines
m
and
n
are intersected by the transversal and the following pairs of alternate interior angles are formed:
∠
1
and
∠
2
∠
3
and
∠
4
alternate interior angles
Since the given lines
m
and
n
are parallel, therefore the alternate interior angles will be congruent.
∠
1
=
∠
2
∠
3
=
∠
4
Interior angles on the same side of the transversal are called 'consecutive interior angles' or co-interior angles in short. Co-interior angles are supplementary when the lines are parallel.
∠
2
+
∠
3
=
180
∘
∠
6
+
∠
7
=
180
∘
Proof alternate interior angles theorem
Corresponding angles are a pair of angles on the similar corners of each of two lines on the same side of the transversal line. Corresponding angles formed by two parallel lines and a transversal are equal.
Corresponding angles are:
1.
∠
2
and
∠
4
2.
∠
1
and
∠
3
3.
∠
5
and
∠
7
4.
∠
6
and
∠
8
What Is the Alternate Interior Angles Theorem?
Theorem And Proof
Statement: The theorem states that “if a transversal intersects parallel lines, the alternate interior angles are congruent."
Alternate Interior Angles Theorem Converse
The converse of alternate interior angles theorem states that if two lines are intersected by a transversal forming congruent alternate interior angles, then the lines are parallel.
Thus according to the converse of alternate interior angles theorem, the below-given lines will be parallel if
∠
D
is
40
∘
and
∠
B
is
140
∘
. This is because their corresponding alternate interior angles are of the measure
40
∘
and
140
Important Note
Co-interior angles are supplementary when the lines are parallel.
Solved Examples
Example 1
Cathy has been asked to find the pairs of alternate interior angles from the given diagram. Can you help her?
Solution
The alternate interior angles lie on "alternate" sides of the transversal, and they must be in the "interior" of the two parallel lines.
The alternate interior angles are:
∠
q
and
∠
z
∠
r
and
∠
a
Example 2
Julia asks her students if they can find the relationship between
∠
q
and
∠
d
if the given lines
m
and
n
are parallel. Can you tell how are
∠
q
and
∠
d
related to each other?
parallel lines
Solution
The given lines are parallel and are intersected by a transversal.
According to the alternate interior angles theorem, the two congruent alternate interior angles are produced. They are:
∠
q
and
∠
d
and
∠
c
and
∠
p
∴
∠
q
and
∠
d
are congruent alternate interior angles.
∴
∠
q
=
∠
d
Example 3
If the given two lines are parallel and are intersected by a transversal, what will be the measure of
∠
X
and
∠
Y
?
parallel lines
Solution
The given lines are parallel and according to the alternate interior angles theorem, the given angle of measure
85
∘
and
∠
X
are congruent alternate interior angles.
Therefore,
85
∘
=
∠
X
Similarly,
95
∘
and
Y
are congruent alternate interior angles.Therefore,
95
∘
=
∠
Y
∴
∠
X
=
85
∘
∠
Y
=
95
∘
Interactive Questions on Alternate Interior Angles Theorem
Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.
Q.1. Dolly is asked to determine whether the following statement is True or False. Let's help her.
The angles that lie on the outer side of the parallel lines but on the opposite sides of the transversal are alternate interior angles
True
True
False
False
Q2. Alternate Interior Angles are _______________.
Congruent
Congruent
Supplementary
Supplementary
Complementary
Complementary
Q.3
Rachel knows the definition of alternate-interior angles and she is trying to identify an alternate-interior angle pair.
Can we help her with this?
Interior angles example problem
∠
Y
C
D
=
∠
Q.4 a
Ryan's teacher explained the concept of same side interior angles and asked him to identify and write the same side interior angles.
Given that
←→
A
B
∥
←→
C
D