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Vertical Angles




 
In geometry, pairs of angles can relate to each other in several ways.
In these lessons, we will learn
  • how to identify vertical angles
  • the vertical angle theorem
  • how to solve problems involving vertical angles
  • how to proof vertical angles are equal
Related Topics: More Geometry Lessons



Vertical Angles

When two lines intersect, the opposite angles form vertical angles or vertically opposite angles. They are called vertical angles because they share the same vertex.

The Vertical Angle Theorem states that

Vertical angles are equal.

Example:
vertical angles

This means that:
 (i)  q and s are vertical angles
 (ii) q = s

Similarly, p is opposite to r
This means that:
 (i)  p and r are vertical angles
 (ii) p = r

Notice also that p and q are supplementary angles i.e. their sum is 180°. Similarly, s and r are supplementary angles.

The following diagram shows another example of vertical angles.

vertical angles




The following video explains more about vertical angles.
How to define and identify vertical angles
A group of examples that identifies vertical angles.


 

Solving Problems using Vertical Angles

Very often math questions will require you to work out the values of angles given in diagrams by applying the relationships between the pairs of angles.

Example:

Given the diagram below, determine the values of the angles x, y and z.

vertical angles

Solution:

Step 1: x is a supplement of 65°.

            Therefore, x + 65° =180° ⇒ x = 180° – 65° = 115°

Step 2: z and 115° are vertical angles.

            Therefore, z = 115°

Step 3: y and 65° are vertical angles.

            Therefore, y = 65°

Answer: x = 115°, y = 65° and z = 115°

Example:

Both AEC and DEB are straight lines.  Find q.

 

Solution:

   ← vertical angles
 q + 45˚= 135˚
    q = 135˚ – 45˚ = 90˚



The following video shows how to use the vertical angle theorem to solve problems.
Identify vertical angles and find the missing angle measures from a diagram.
The following video shows how to find a missing vertical angle in a triangle.


 

Proof of the Vertical Angle Theorem

The following videos will prove that vertical angles are equal.


Rotate to landscape screen format on a mobile phone or small tablet to use the Mathway widget, a free math problem solver that answers your questions with step-by-step explanations.


You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.


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