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 :
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.
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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 |
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.
Given below are some of the examples on corresponding angles.In the given figure, if the angle a is 52 degree, then find the other seven angles.
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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