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Alternate Angles

Alternate angles are one of the special kinds of angles in geometry. These are the set of non adjacent angles for either sides of the transversal. Alternate angles are shaped by two parallel lines crossed by the transversal.

In geometry, the intersection of a straight line on two or more parallel lines is called transversal. In the figure shown below, EF and GH are two parallel lines. RS is the transversal that cuts EF at L and GH at M.

Alternate Angles

Measure of alternate angles is always same.

$\therefore$ $\angle$3 = $\angle$ 5 and $\angle$ 4 = $\angle$ 6

$\angle$2 = $\angle$ 8 and $\angle$ 1 = $\angle$ 7.

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Alternate Angles Definition

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Angles that are formed between the coplanar lines that are cut by a tranversal are known as interior angles. In the plane, transversal intersects two given lines $l$ and $m$ in two distinct points E and F.

Definition of Alternate Interior Angles

In the figure:

  • $\angle$3, $\angle$4, $\angle$5 and $\angle$6 are alternate interior angles.
  • $\angle$1, $\angle$2, $\angle$7 and $\angle$8 are alternate exterior angles.

Types of Alternate Angles

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Alternate Angles are classified in to two different types based on the position of the angles. The two types of alternate angles are:

  • Alternate Interior Angles
  • Alternate Exterior Angles


Alternate Angles Theorem

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Prove that if two parallel lines are cut by a transversal, the alternate interior angles are equal.

Theorem of Alternate Interior Angles

Proof: Suppose that alternate exterior angles are equal, $\angle$b = $\angle$c and $\angle$a = $\angle$d.

Step 1: $\angle$ a + $\angle$1 = 180 Step 1: $\angle$ b + $\angle$4 = 180
Step 2: $\angle$ 3 + $\angle$d = 180 Step 2: $\angle$ 2 + $\angle$c = 180
From Step 1 and Step 2
$\angle$ a + $\angle$1 = $\angle$ 3 + $\angle$d
From Step 1 and Step 2
$\angle$ b + $\angle$4 = $\angle$ 2 + $\angle$c
Given: $\angle$a = $\angle$d Given: $\angle$b = $\angle$c
$\angle$1 = $\angle$ 3 $\angle$4 = $\angle$ 2

and

These are two pairs of alternate angles.

Converse:

A transversal intersects two lines. If the alternate angles are equal, then the lines are parallel.

Alternate Angle

If or then AB is parallel to CD.

If two parallel lines are cut by a transversal, the alternate angles are equal.

Alternate Corresponding Angles

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Alternate angles are formed on opposite sides of a line which crosses two or more parallel lines whereas corresponding angles can be formed on the same side of a line, which crosses two or more parallel lines. Alternate angle are spotted by letter Z or reverse of it and corresponding angles are spotted by letter F or reverse of F.
Alternate and Corresponding Angles

Alternate Angles Examples

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Given below are some of the examples on alternate angles.

Solved Example

Question: Find the measure of the set of alternate angles in the figure given below

Alternate angles Examples


Solution:

The given angle 3 is equal to $55^{0}$.

Angle 3 is the interior angle of the transversal and its alternate angle is 6.

Therefore, $\angle$3 = $\angle$6 = $55^{0}$

Next pair of alternate interior angles is angle 4 and 5

Angle 3 and 4 lies in the same line.

Therefore, 55 + $\angle$ 4 = $180^{0}$

Then, $\angle$4 =$125^{0}$
$\therefore$ $\angle$
4 = $\angle$5 = $125^{0}$

Angle 1 and 4 are vertical angles. They are congruent in measure.

Therefore, $\angle$1 = $\angle$4 = $125^{0}$

Then, the alternate pair of the exterior angle 1 is angle 8.

Therefore, $\angle$1 = $\angle$ 8 = $125^{0}$

Angle 2 and 3 are vertical angles. They are congruent in their measure.

So, $\angle$2 = $\angle$3 = $55^{0}$
alternate pair of exterior angle of 2 is the angle 7.

Therefore, angle $\angle$2 = $\angle$ 7 = $55^{0}$.



More topics in Alternate Angles
Alternate Interior Angles Alternate Exterior Angles
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