What are Vertical Angles in Geometry?
Contents
Definition: Vertical angles are a pair of two angles lying on the opposite sides of two intersecting lines.
Vertical angles are also defined as a pair of non-adjacent angles formed by two lines that are intersecting. Vertical angles are also known as vertically opposite angles because the angles are opposite to each other and they share the same vertex.
In Geometry, when two straight lines intersect with each other they form four angles at the intersection. The angles opposite to each other are the vertical angles. In this context, vertical means they share the same vertex.
Vertical Angle Theorem
Characteristics
Vertical angles formed when two lines intersect. They are opposite to each other. They have following characteristics:
- Vertical angles share the same vertex (the common corner point) but they cannot share a side.
- Vertical angles are always equal in degrees to one another and therefore they are called congruent angles.
Uses
Vertical angles are used in many real life scenarios. For example:
- Rail road crossing signs found on roadways near rail roads
- Letter X.
- Open scissors
- In a kite where two wooden sticks hold it together.
- The point where ceiling beams intersect in a X like shape.
- In some floor designs lines intersect to form vertical angles.
- The intersecting lines on the dartboard form 10 pairs of vertical angles. The bull’s eye in the center of the board serves as the vertex.
Importance
Vertical angles are an important part of Geometry.
They are extremely important in triangles, angles formed between two parallel lines and a transversal and sometimes also useful in solving questions related to bearings.
Vertical Angle Formula
If the angles are vertical, they are congruent means they are equal measure in degrees. If the two lines intersect at 90 angles, then the vertical angles are supplementary angles. Vertical angles in perpendicular lines sums to 180.
Vertical angles are not supplementary if the lines are not perpendicular to each other. As we know, when two straight lines intersect with each other they form four angles at the intersection. Both pairs of vertical angles (four angles altogether) always add up to a full angle 360.
Are Vertical Angles Congruent?
Yes, Vertical angles are always congruent that is they are equal in measure. Vertical angles are non-adjacent angles they share same vertex but not same ray, therefore they are equal in measure.
Vertical Angle Examples
Example 1
Two lines are intersecting in the above figure. Four angles are formed by this intersection of two lines.
∠a and ∠d are adjacent angles and ∠c and ∠b are also adjacent angles as they share a common ray. ∠a and ∠b are non-adjacent angles and ∠c and ∠d are also non-adjacent angles as they do not share a common ray.
As we know vertical angles are a pair of non-adjacent angles formed by two lines that are intersecting, therefore ∠a and ∠b are one pair of vertical angles and ∠c and ∠d are the second pair of vertical angles.
Vertical angles are always congruent. Therefore in the above figure:
∠a = ∠b
∠c = ∠d
Example 2
In the above figure, Line AD and Line BC intersect with each other at point O. Four angles are formed at this point.
‘O’ is the vertex. ∠AOB and ∠COD are non-adjacent angles and are vertical angles. There are equal in measure that is of 70°. In the same way, ∠AOC and ∠BOD are non-adjacent angles and are vertical angles.
There are equal in measure that is of 110°. ∠AOB and ∠COD is a one pair of vertical angles. They are congruent.
∠AOB = ∠COD
∠AOC and ∠BOD is another pair of vertical angles. They are also congruent.
∠AOC = ∠BOD Altogether their some is a full angle 360°.
∠AOB + ∠COD + ∠AOC + ∠BOD = 360°
70° + 70° + 110° + 110° = 360°
Calculating Vertical Angles
Example 1
Find ∠a, ∠b and ∠c
Solution: ∠b and 47° are vertical angles, therefore they are congruent.
∠b = 47°
∠a and ∠c are also vertical angles, therefore they are congruent.
∠a = ∠c
Let’s assume they are equal to x°
∠a = ∠c = x°
Altogether their sum is equal to a full angle 360°
∠a + ∠b + ∠c + 47° = 360°
x° + x° + 47° + 47° = 360°
2x° = 266°
x = 133
∠a = 133°, ∠b = 47°, ∠c = 133°
Example 2
Find x, hence find the angle 5x.
Solution: 5x and 2x + 24 are vertical angles, therefore they are congruent.
5x = 2x + 24
Subtract 2x both sides
3x = 24
Divide both sides by 3
x = 8
Therefore the angle 5x = 5(8) = 40°.