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Definition: When a transversal (any line that passes through two other lines) cuts two coplanar lines then the angles that are in between the two coplanar lines and
that lie on the opposite sides of the transversal are called the Alternate Interior angles. In these figures, Line m and line n are coplanar lines. The transversal cuts both the lines. $\angle$a and $\angle$c are angles formed when the transversal cuts m and n and these are one pair of angles that are on opposite sides of transversal l and in between the lines m and n. Thus $\angle$a and $\angle$c are Alternate Interior angles. There is one other such pair of Alternate Interior angles in these figures. $\angle$b and $\angle$d is the other pair of Alternate Interior angles in these figures. In Figure (i), the lines m and n are parallel and in Figure (ii) the lines m and n are not parallel. It is easy to remember that the pair of Alternate Interior angles are on "Alternate" sides of the Transversal, and they are on the "Interior" of the two crossed lines. |
If two parallel lines are cut by a transversal, then the pairs of Alternate Interior angles are always congruent.
Proof:
To prove the theorem we need to prove that, if two lines M and N are cut by the transversal L, and if M ∥ N,
then ∠1≅ ∠5 and ∠4≅ ∠8.
From the above figure, Since M || N, by the Corresponding Angles Postulate,
We have ∠2≅ ∠8
Therefore, by the definition of congruent angles, ∠2= ∠8.
Since ∠1 and ∠2 form a linear pair, they are supplementary and so ∠1 + ∠2 = 180°.
Also, since ∠ 8 and ∠ 5 form a linear pair, they are supplementary and so ∠8 + ∠5 = 180°.
Substituting ∠2 for ∠8 (since ∠2= ∠8),
We get ∠2 + ∠5 = 180°.
Subtracting ∠2 from both sides of above equation, we get ∠5 = 180° - ∠2.
But 180° - ∠2 is ∠1.
Thus, ∠5 = ∠1.
Therefore, ∠1 is congruent to ∠5.
Similarly, it can be proved that ∠4 is congruent to ∠8 using the above same method.
If any two lines are cut be a transversal and the Alternate Interior angles are congruent, then the two lines are parallel.
Proof: Given that alternate interior angles ∠CGH is congruent to ∠FHG and ∠DGH is congruent to ∠EHG. We need to prove that
$\overleftrightarrow{CD}$
and
$\overleftrightarrow{EF}$
are parallel.
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Statements |
Reasons |
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1. ∠CGH = ∠FHG 2. ∠DGH = ∠EHG 3. ∠CGH = ∠AGD 4. ∠AGD = ∠FHG 5. $\overleftrightarrow{CD}$ and $\overleftrightarrow{EF}$ are parallel. |
1. Given 2. Given 3. Vertically Opposite angles. 4. Corresponding Angles. 5. Converse of the Corresponding Angles Postulate. |
Thus we proved that “If any two lines are cut be a transversal and the Alternate Interior angles are congruent, then the two lines are parallel”.
In real life, we come across non coplanar lines in many places. It is easy to understand the alternate interior angles when the lines are parallel in real life. Few such examples are given below.
A window pane is a good example to understand different pairs of angles. In this picture of window pane ∠a and ∠b are corresponding angles.
In this racket the angles marked in red are Alternate interior angles. We can find many such pairs in the getting of a racket.
Solved Examples
Solution:
Given that M and N are parallel lines, we have 105 ° and x congruent (Alternate interior angle theorem).
Thus x = 105 ° ( By definition of congruency).
Solution:
Given that ∠a = 72 °.
By definition of corresponding angles we have, ∠e and ∠a are corresponding angles.
Also, By Corresponding Angles Postulate, ∠a and ∠e are congruent and therefore equal. Thus ∠ e = 72°.
Here, ∠e and ∠c are Alternate interior angles. By Alternate interior angles theorem, we have ∠e=∠c .
Thus we have ∠c = 72°.
In Geometry, We know that a straight angle has a measure of 180 °. So, ∠a + ∠d = 180 °.
72 + ∠d = 180 °
∠ d = 180 - 72
∠ d = 108 °
Here ∠d and ∠f are also alternate interior angles. By Alternate angles theorem, ∠d and ∠f are congruent and therefore equal. Thus, ∠f = 108 °.
Thus the pairs of alternate interior angles are ∠e=∠c=72° and ∠d = ∠f = 108°.
Solution:
From the figure we see that , ∠ DCJ ≅ ∠ AJI (Alternate Interior Angles theorem).
∠ DCJ = ∠ AJI (Definition of Congruent Angles)
But it is given, ∠DCJ = 72 °.
Thus, ∠ AJI = 72 °.
Also from the figure we see that, ∠GFJ≅ ∠HJI (Alternate Interior Angles theorem).
∠GFJ= ∠HJI (Definition of Congruent Angles)
But it is given that ∠ GFJ = 48 °.
Thus we have, ∠HJI= 48 °.
By Angle Addition Postulate we have, ∠ AJH = ∠ AJI + ∠HJI.
Thus, ∠ AJH = 72 ° + 48 °.
Hence, ∠ AJH = 120 °