Vertical angles are pairs of opposite angles formed when two lines intersect each other at a point. They are thus also known as vertically opposite angles. Any two intersecting lines form two pairs of vertical angles. In geometry, the word ‘vertical’ means ‘related to a vertex’ or corner.
Vertical Angles
The above figure shows two pairs of vertically opposite angles that are equal.
Vertical Angles Theorem
Vertical Angles Congruence Theorem
Prove Vertical Angles Theorem
To prove:
∠AOE = ∠BOF, ∠AOF = ∠BOE
Proof:
Given that, AB and EF are the two line segments that intersect at point ‘O’ Now, ∠AOF and ∠BOF form a linear pair and thus they are supplementary ∴∠AOF + ∠BOF = 180°……. (1) Similarly, ∠AOF = ∠AOE form another linear pair and are also supplementary ∴ ∠AOF + ∠AOE = 180°……. (2) From (1) and (2) we get, ∠AOF + ∠BOF = ∠AOF + ∠AOE Subtracting ∠AOF from both sides we get, ∠BOF = ∠AOE…… (3) Similarly, ∠BOF and ∠ BOE form a linear pair and are supplementary ∠BOF + ∠ BOE = 180°…….. (4) From (1) and (4) we get, ∠AOF + ∠BOF = ∠BOF + ∠BOE Subtracting ∠BOF from both sides we get, ∠AOF = ∠BOE…….. (5) From (3) and (5) we can say that vertical angles are always congruent or equal to each other. Hence Proved
How to Solve Vertical Angles
Let us find some vertical angles in the given examples to understand the concept better.
Name the pair of vertical angles in the given figure and find the values of the unknown angles.
Solution:
By definition, in the above figure, ∠AOD, ∠BOC and ∠DOB, ∠COA are the two pair of vertical angles Thus, ∠AOD = ∠BOC ∠DOB = ∠COA Given, ∠AOD = 48° Hence, ∠BOC = 48° Now, ∠AOD + ∠DOB = 180° (linear pair of supplementary angles) ∠BOC +∠COA = 180° (linear pair of supplementary angles) Now, ∠AOD + ∠DOB = 180°, here ∠AOD = 48° 48° + ∠DOB = 180° ∠DOB = 180° – 48° ∠DOB = 132° Since, ∠DOB = ∠COA (Vertical angles) ∠COA = 132°
Consider the figure given alongside to identify the pair of vertical angles and find their values.
Solution:
In the given figure, ∠WOX = ∠ZOY (vertical angles) ∠XOZ = ∠WOY (vertical angles) Now, given ∠XOZ = 65° Thus, ∠WOY = y = 65° Again, ∠XOZ + ∠WOX = 180° (linear pair), here ∠XOZ = 65°, ∠WOX = x 65° + x = 180° x = 180°- 65° x = 115° Similarly, ∠ZOY + ∠WOY = 180° (linear pair), here ∠WOY = 65°, ∠ZOY = z z + 65° = 180° z = 180° – 65° z = 115° Alternative method In the given figure, ∠WOX = ∠ZOY (vertical angles) ∠XOZ = ∠WOY (vertical angles) Now, given ∠XOZ = 65° Thus, ∠WOY = y = 65° Again, ∠XOZ + ∠WOX = 180° (linear pair), here ∠XOZ = 65°, ∠WOX = x 65° + x = 180° x = 180°- 65° x = 115° Since, ∠WOX = ∠ZOY (Vertical angles) ∠ZOY = z = 115°
Examples in Real Life
Railroad crossing sign
Opened scissor
Alphabet X
Two wooden sticks meet in a kite
FAQs
Q1. Can Vertical Angles be Adjacent?
Ans. By definition, vertical angels cannot be adjacent or next to each other. This is because the second pair of vertical angles interrupts in their way of becoming adjacent. Adjacent angles take one angle from one pair of vertical angles and another angle from another pair.
Q2. Are Vertical Angles Supplementary?
Ans. As we know, supplementary angles add up to 180°, vertical angles can only be supplementary when the two lines intersect perpendicularly. Thus vertical angles are not always supplementary.
Q3. Are Vertical Angles Complementary?
Ans. Since complementary angles add up to 90°, vertical angles can only be complementary when each of them are acute and measures 45°. Thus acute vertical angles can be complementary when they measure 45°.
Name the pair of vertical angles in the given figure and find the values of the unknown angles.
Consider the figure given alongside to identify the pair of vertical angles and find their values.