Vertically Opposite Angles
Last Updated : 30 May, 2024
Vertically Opposite Angles, also called Vertical Angles; are angles that stand across from each other when two lines intersect. Imagine two straight lines crossing like a giant letter 'X.' The angles formed on opposite sides of this intersection are called vertically opposite angles. These angles share a unique relationship – they are equal to each other. So, if you know the measurement of one vertically opposite angle, you automatically know the size of the other.
In this article, we have covered the concept of vertical angles. such as definition, theorem and proof, properties, formation, and applications of vertical angles.
What are Vertically Opposite Angles?
The angles opposite to each other when two lines intersect each other are called Vertical Angles. When two lines cross each other, they create four angles. Among these, two pairs of angles are not next to each other, and these pairs are known as vertical angles. In the provided illustration, (∠1, ∠3) and (∠2, ∠4) represent the two sets of vertical angles formed by the intersection of the lines.
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Vertically Opposite Angle Theorem
The Vertical Angles Theorem, also known as the Vertically Opposite Angles Theorem, states that when two lines intersect, the pairs of opposite vertical angles formed are always equal or congruent.
Theorem Statement
When two lines intersect, the pairs of opposite vertical angles formed are congruent.
Proof of Vertically Opposite Angles Theorem
The proof is based on the concept of straight angles. Knowing that angles on a straight line add up to 180 degrees, we can illustrate the proof with the following steps:
Consider the angles ∠1 and ∠2 in the figure.
∠1 + ∠2 = 180° (since they form a linear pair of angles) . . . (1)
Similarly, consider the angles ∠1 and ∠4.
∠1 + ∠4 = 180° (since they form a linear pair of angles) . . . (2)
By comparing equations (1) and (2), we find ∠1 + ∠2 = 180° = ∠1 + ∠4.
According to the transitive property (if a = b and b = c, then a = c), we can rewrite the statement as
∠1 + ∠2 = ∠1 + ∠4 . . .(3)
By eliminating ∠1 on both sides of equation (3), we get ∠2 = ∠4.
Similarly, we can use the same set of statements to prove that ∠1 = ∠3.
When two lines intersect, then pairs of opposite angles known as vertical angles are formed. Vertical angles are formed by a pair of non-adjacent angles that are positioned across from each other. This occurs at the point of intersection, and the angles share a common vertex.
Applications of Vertically Opposite Angles
Applications of Vertically Opposite Angles are mentioned below:
Surveying and Navigation: In navigation and surveying, understanding angles is crucial. Vertically opposite angles play a role in determining directions and establishing precise measurements.
Structural Engineering: Engineers use vertically opposite angles when designing and analyzing structures. The angles help assess the stability and balance of various architectural elements.
Traffic Control Systems: In traffic control systems, the arrangement of roads often involves intersections. Vertical angles contribute to designing efficient traffic flow and ensuring safety at intersections.
Satellite Communication: When positioning satellite dishes, the correct alignment is crucial for optimal communication. Vertical angles are considered to ensure the accurate pointing of satellite antennas.
Physics Experiments: Physics experiments involving optics and light often require the consideration of angles. Vertically opposite angles may come into play in the setup and analysis of these experiments.
Robotics and Automation: In robotics, particularly robotic arms and automated systems, understanding angles is vital. Vertically opposite angles may be involved in the programming and control of these systems.
Architectural Design: Architects use geometric principles, including vertically opposite angles, when planning structures. This ensures aesthetic appeal and functional stability in architectural designs.
Photography and Videography: Professionals in photography and videography consider angles for framing shots. Vertically opposite angles may influence the composition and visual impact of images and videos.
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Solved Examples on Vertically Opposite Angles
Following are some solved examples on Vertically Opposite Angles:
Example 1: If ∠A and ∠B are vertical angles, and the measure of ∠A is 120 degrees, what is the measure of ∠B?
Vertical angles are always congruent, meaning they have the same measure. Therefore, if ∠A measures 120 degrees, ∠B will also have the same measure.
So, the measure of ∠B is 120 degrees.
Example 2: If the sum of the measures of two vertical angles is 120 degrees, what is the measure of each individual angle?
Denote the measures of the two vertical angles as ∠X and ∠Y.
Given that ∠X + ∠Y = 120 degrees.
We need to find the individual measures of ∠X and ∠Y.
Since vertical angles are congruent, ∠X = ∠Y.
Let ∠X = ∠Y = x (for simplicity).
Now, the equation becomes:
x + x = 120
⇒ 2x = 120
⇒ x = 60
So, each angle measures 60 degrees. Therefore, ∠X = 60 degrees and ∠Y = 60 degrees.
Practice Questions on Vertically Opposite Angles
Q1: In the given figure, if ∠A = 60°, find the measure of its vertically opposite angle.
Q2: Two lines intersect, forming vertically opposite angles. If ∠1 = 120°, what is the measure of ∠3?
Q3: If ∠X and ∠Y are vertically opposite angles, and ∠X measures 75°, what is the measure of ∠Y?
Q4: In a quadrilateral, two intersecting lines create vertically opposite angles. If ∠P = 110°, find the measure of ∠Q.
Q5: In a parallelogram, ∠A and ∠C are vertically opposite angles. If ∠A measures 80°, what is the measure of ∠C?
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