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Corresponding Angles

In geometry, we study about lines and angles. Few angles are defined in reference with a set of parallel lines and a transversal. Just to recall that parallel lines are the lines which are at equal distance from one another. A straight line is said to be transversal, if the line cuts two or more parallel lines at different points. There are following four types of angles :
(1) Alternate angles
(2) Corresponding angles
(3) Vertically opposite angles
(4) Interior angles on the same side of the transversal

Here, we shall understand about corresponding angles in this page. Corresponding angles are formed when a transversal cuts a pair of parallel lines. Let us have a look at the following figure :

Corresponding Angles


In the figure shown above, transversal cuts the parallel lines m and n. When a transversal cuts two parallel lines, eight angles are formed. The pair of angles which are either above or below the parallel lines are known as corresponding angles. The corresponding angles are equal to each other. In the above figure, the pair of corresponding angles are -
$\angle$1 = $\angle$5
$\angle$2 = $\angle$6
$\angle$4 = $\angle$8
$\angle$3 = $\angle$6

To identify these angles it is easy by writing the letter F and its reverse along the two parallel lines.

Identify Corresponding angles

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Corresponding Angles Definition

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When two parallel lines are intersected by transversal, the angles in matching corners are called corresponding angles.
Definition of Corresponding Angles

In the figure, m || n (m is parallel to n)

Pairs of corresponding angles are:

($\angle$ a, $\angle$e), ($\angle$d, $\angle$h), ($\angle$b, $\angle$f), and ($\angle$c, $\angle$h)

The measure of the alternate corresponding angles is equal.
  • $\angle$ a = $\angle$e
  • $\angle$d = $\angle$h
  • $\angle$b = $\angle$f
  • $\angle$c = $\angle$g

Corresponding Angles Postulate

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The corresponding angles postulate states that, two lines are cut by a transversal are parallel iff corresponding angles are congruent.
Postulate for Corresponding Angles
If two parallel lines are cut by a transversal then corresponding angles are congruent. The pairs of angles which lie on the same side of transversal and parallel lines are congruent.

Congruent corresponding angles are:
  • $\angle$ a $\cong$ $\angle$e
  • $\angle$d $\cong$ $\angle$h
  • $\angle$b $\cong$ $\angle$f
  • $\angle$c $\cong$ $\angle$g

Corresponding Angles Theorem

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Statement: If two parallel lines cut by a transversal then corresponding angles are congruent.

Converse of Corresponding Angles

Given: m and n are parallel lines.

To Prove: Corresponding angles are equal.

Proof:
We are given with two parallel lines.

Statement Reason
Step 1
m || n Given
Step 2
$\angle$2 + $\angle$3 = 180$^o$
$\angle$2 is supplementary to $\angle$3
(Straight Angle Theorem)
Step 3 $\angle$5 + $\angle$6 = 180$^o$ $\angle$5 is supplementary to $\angle$6
(Straight Angle Theorem)
Step 4 $\angle$2 + $\angle$3 = $\angle$5 + $\angle$6 From Step 2 and Step 3
Step 5 $\angle$3 = $\angle$5 Alternate Interior Angle Theorem
Step 6 $\angle$2 = $\angle$6 Using Step 5 in Step 4

$\therefore$ $\angle$2 = $\angle$6 (Corresponding angles). Hence proved.

Similarly, this can be proven for every pair of corresponding angles.

Corresponding Angles Converse

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If corresponding angles are congruent then two lines cut by a transversal are parallel.

Theorem of Corresponding Angles
Given: Corresponding angles are congruent

To Prove:
m || n

Proof:
By the straight angle theorem, $\angle$ 2 + $\angle$ 3 = 180$^o$

We know that, the sum of all interior angles on either side of transversal = 180$^o$.

So, by interior angle theorem, both the lines are parallel.

=> m || n. Hence Proved

Corresponding Angles in Real Life

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Key and window pane are good examples to understand different pairs of corresponding angles.
Pictures of Corresponding Angles

Corresponding Angles in Triangles

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Corresponding angles and sides are a pair of same angles or sides that are in the same spot in two different shapes. If two triangles are congruent, then the corresponding angles are equal and corresponding sides are proportional.

Consider two congruent triangles $\triangle$ ABC and $\triangle$ PQR. There are 3 pairs of corresponding congruent angles.

In Triangle Corresponding Angles
In $\triangle$ ABC and $\triangle$ PQR, $\angle$ A is corresponding to $\angle$P, $\angle$B is corresponding to $\angle$Q and $\angle$C is corresponding to $\angle$ R.

Since triangles are congruent, $\angle$ A $\cong$ $\angle$P, $\angle$B $\cong$ $\angle$Q and $\angle$C $\cong$ $\angle$ R.

How to Solve Corresponding Angles?

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Solving Corresponding Angles

The corresponding angles formed by a transversal are equal. In the above figure, angle d and angle f are corresponding angles. Another pair of corresponding angles are (e and c), (h and b) and (a and g).

Therefore, we have angle f = angle d

angle e = angle c

angle h = angle b

angel g = angle a

Using the above formulas, we can easily solve the corresponding angles problems.

Corresponding Angles Examples

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Given below are some of the examples on corresponding angles.

Solved Example

Question:

In the given figure, if the angle a is 52 degree, then find the other seven angles.

Corresponding Angles Examples


Solution:

Given that $\angle$a = 52 degree and $\angle$g and $\angle$a are corresponding angles. So, $\angle$a and $\angle$g are equal. Therefore $\angle$g = 52 degree.

We know that a straight angle has a measure of 180 degree. So, $\angle$a + $\angle$d = 180 degree.

52 + $\angle$d = 180

$\angle$d = 180 - 52

$\angle$d = 128

Here, $\angle$d and $\angle$f are corresponding angles. Therefore, $\angle$d and $\angle$f are equal. Therefore, $\angle$f is also 128 degree.

We know that, a straight angle has a measure of 180 degree. So, $\angle$a + $\angle$b = 180 degree.

52 + $\angle$b = 180

$\angle$b = 180 - 52

$\angle$b = 128

Here, $\angle$b and $\angle$h are corresponding angles. Therefore, $\angle$b and $\angle$h are equal. Therefore, $\angle$h is also 128 degree.

We know that a straight angle has a measure of 180 degree. So, $\angle$b + $\angle$c = 180 degree.

128 + $\angle$c = 180

$\angle$c = 180 - 128

$\angle$c = 52

Here, $\angle$c and $\angle$e are corresponding angles. Therefore, $\angle$c and $\angle$e are equal. Therefore, $\angle$c is also 52 degree.

Therefore, $\angle$a = $\angle$g = $\angle$c = $\angle$e = 52 degree.

$\angle$b = $\angle$h = $\angle$d = $\angle$f = 128 degree.



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