Understanding definition of vertical angles:
Mathematicians define vertical angles as the angles having same vertex and that are not adjacent to each other, i.e. they do not have a common side. This definition of vertical angles is universally accepted.
Proof of Vertical angles theorem:
The vertical angles theorem states that if two angle are vertical, than they are always equal. The proof of vertical angle theorem is given below.
As shown in above figure, ∠a and ∠c are vertical angles according to the vertical angles definition. Now we know that ∠a and ∠b are supplementary to each other. Also ∠b and ∠c are supplementary, this gives ∠a + ∠b = 180° and ∠b + ∠c = 180°. Comparing both the equations & subtracting ∠b from both sides, ∠a = ∠c i.e. they are equal.
Significance of vertical angles in real life:
There is lot of significance of vertical angles in real life. If you look around, you can find various examples of vertical angles. If you see the tiles on your floor, you can find this concept easily. Similar concepts can be found in many human as well as natural designs.
Lets see vertical angles examples:
Lets have some examples of vertical angles. Consider the above diagram for reference.
(i) Find the value of ∠c and ∠a if ∠b is 88°.
Since ∠b and ∠c are supplementary, we have ∠b + ∠c = 180 => ∠c = 180 – 88 = 92°. Now ∠a and ∠c are vertical angles and the theorem says that they should be equal. Hence ∠a = 92°.
(ii) Find value of ∠d in above example:
Since ∠a and ∠c are vertical, ∠b and ∠d are also vertical to each other and hence equal. Thus we get value of ∠d as 92°.
This vertical angles examples can be applied to many geometrical problems with ease.
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