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Plane Geometry

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Last updated date: 28th May 2024
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Plane Geometric Figures

Are you someone who is finding it difficult to study the topic of Plane Geometry - Explanation, Types, Examples, and FAQs on your own and is unable to find correct guidance? Then this is your destination. Vedantu has brought a very detailed and well-analyzed article to explain this topic to you. But how reliable is the information, you ask. Well to answer these questions, Vedantu has ensured that the best subject matter experts were assigned to complete this task.


The best way to go about this topic is to start by understanding the meaning of plane geometric figures first. 


In-plane geometry, plane geometric figures including 2-dimensional shapes such as squares, rectangles, triangles, and circles are also called flat shapes. On the other hand, In solid geometry, 3-dimensional geometric shapes such as a cone, cube, cuboid, cylinder, etc. are also called solids. The fundamental concept of geometry is based on points, planes, and lines, defined in coordinate geometry. With the help of geometric concepts, we do not only understand the shapes we see in real life but also can calculate the volume, area, and perimeter of shapes.


Examples of Plane Geometry

As already mentioned, plane Geometry deals with flat shapes that can also be drawn on a piece of paper. These plane geometric figures include triangles, squares, lines, and circles of two dimensions. That being said, plane geometry is also referred to as two-dimensional geometry. All the 2D figures consist of only two measures such as length and breadth. These shapes do not deal with the depth of the shapes. Some examples of plane figures are triangles, rectangles, squares, circles, and so on.


Important Terminologies in Plane Geometry

Below are some of the important terminologies in plane geometry:

  1. Point: A point is known to be a precise position or place on a plane. A dot generally denotes them. It is however crucial to know that a point is not a thing, but a place or location. Also, remember that a point contains no dimension; rather, it has the only position.

  2. Line: A line is straight and has no curves, consisting of no thickness and stretches out in both directions without end (boundlessly). It is crucial to mark a point that is the combination of infinite points together to make a line. In geometry, we consist of horizontal lines and vertical lines which are termed as x-axis and y-axis respectively. Lines can also be classified into the 2 parts as follows:

  • Line Segment – If a line consists of a starting and an endpoint then it is referred to as a Line Segment. For example, a ruler

  • Ray – If a line consists of a starting point and has no endpoint it is known as a Ray. An example of a ray includes Sun Rays.


Plane Angle in Geometry

Under the domain of planar geometry, an angle is a figure created by two rays, known as the sides of the angle, sharing a common endpoint, known as the vertex of the angle. The dimension of a plane angle is two. 


Types of Plane Angle

  • Acute Angle – An acute angle also called a Sharp angle is an angle smaller than a right angle. This implies that the measurement of an acute angle can range between 0 – 90 degrees.

  • Obtuse Angle – An obtuse angle is an angle that measures more than 90 degrees but is less than 180 degrees.

  • Right Angle – An angle exactly at 90 degrees is a right angle.

  • Straight Angle – An angle that measures precisely 180 degrees is straight, i.e. the angle is formed by a straight line.


Plane Angle Formula

The angle between planes is equivalent to the angle between their normal vectors. That implies, the angle between planes is equivalent to an angle between lines l1 and l2, which is perpendicular to lines of planes crossing and lying on planes themselves.


Angle formulas between two planes are as below:


\[ Cos \alpha  = \frac{\left | A_{1}.A_{2} + B_{1}.B_{2} + C_{1}.C_{2} \right |}{\sqrt{A{_{1}}^{2}.{_{1}}^{2}.C{_{1}}^{2}} \sqrt{A{_{2}}^{2}B{_{2}}^{2}.C{_{2}}^{2}}}\]


Solved Examples 

Example:

In the figure given below, AB is parallel to CD. Find out the value of a+b?

  

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Solution: 

We are aware that angle b needs to be equal to its vertical angle (the angle directly "across" the bisection of the line). Thus, it is 20°. 

In addition, given the properties of parallel lines, we know that the supplementary angle must be 40°.  Based on the principle of supplements, we know that a + 40° = 180°. 

Now, Solving for angle a, we obtain a = 140°.

Hence, a + b = 140° + 20° 

= 160°


Example:

In a rectangle PQRS, both diagonals are constructed and bisect at point O. 

Let the measure of angle POQ equal a degree.

Let the measure of angle QOR equal b degrees.

Let the measure of angle ROS equal c degrees.

Find the measure of angle POS concerning a, b, and/or c.

Solution:

Intersecting lines create 2 pairs of vertical angles that are congruent. Thus, we can conclude that b = measure of angle POS.

Moreover, intersecting lines form adjacent angles which are supplementary (summate to 180 degrees). Thus, we can deduce that a + b + c + (measure of angle POS) = 360 degrees

Substituting the 1st equation into the 2nd equation, we obtain

a + (measure of angle POS) + c + (measure of angle POS) = 360 degrees

2(measure of angle POS) + a + c = 360 degrees

2(measure of angle POS) = 360 – (a + c)

Divide by two and obtain:

measure of angle POS = 180 – 1/2(a + c)


Conclusion

So this completes one of the most important topics in the syllabus of geometry. If you go through the previous year's question papers of any exam that has mathematics as a subject you will find that questions from this topic are always asked.


Vedantu understands the significance of Plane Geometry from the exam as well as from the long-term perspective as well therefore we have bought these materials free for everyone to access. Vedantu can help you with many other maths topics similarly.

FAQs on Plane Geometry

1. What is Geometry and its Applications?

Geometry is a very important branch of mathematics that significantly deals with the lines, angles, dimensions, shapes, and sizes of different things we observe in everyday life. In Euclidean geometry, there are 2D shapes and 3D shapes. That being said, geometry has wide usage not only in academic maths but in daily life in the following ways:

  • Used in the field of construction, such as constructing roads, buildings, dams, bridges, etc.

  • Helps in surveying, mapping, graphing, and navigation.

  • In software industries, it is commonly used in gaming, graphics designing, animations, etc.

  • In the medical field, is used for CT scanning and MRI’s

2. What is the Difference Between Similarity and Congruence in Geometry?

Many of us believe that both the terms are synonyms to each other; however, it is not the case. There mean a bit different in a way that:

  • Similarity- Two shapes or objects are said to be similar given that they have the same shape or contain an equal angle but do not have the same size.

  • Congruence- Two shapes or objects are congruent if they have the completely same shape and size. Therefore, they are equal.

3. What is the Unit of Plane Angle?

The radian is the SI-derived unit of a plane angle.

4. What are the applications of Plane Geometry?

The topic of Plane Geometry has a lot of practical applications, thus it holds a special place in the syllabus. Whenever we travel across the highways and wonder how are these built by engineers, the answer lies in geometry. Civil engineers use geometry in designing and constructing such architectural marvels as dams, bridges, etc. Interior designers who ensure that our houses look stunningly beautiful and presentable use plane geometry to set new items in the open spaces. Artists and painters use geometry to express their ideas and ideologies and also to conceptualize their paintings. You can read Geometry in Daily life by Vedantu to gain further insights.

5. How do I study Plane Geometry?

Vedantu believes that every student has their unique of studying any topic in every subject. The same applies to Plane Geometry. you can start the topic by reading Vedantu’s free study materials on the website followed by covering the topic from the NCERT books of mathematics. Also, remember to solve all the questions in the chapter because many times questions are asked directly front these in the board exams. After this, proceed towards solving the RD Sharma’s maths book which is available on Vedantu’s website for free. This offers advanced-level questions required for entrances exams.

6. Is Vedantu reliable for studying Plane Geometry?

Vedantu is one of the most well-known names in the ed-tech industry in India. We have constantly delivered toppers in various examinations. Vedantu ensures that you are supplied only with the best quality of free study materials and web content designed by our subject matter experts. The same dedication is shown while preparing Plane Geometry. The entire gamut of topics has been covered fully including all the important dimensions you need to know. You can completely rely on Vedantu for this and other topics as it is a very trustworthy brand.

7. How do I download Plane Geometry for free?

On Vedantu’s website, the topic of Plane Geometry is covered in the form of web content and the form of a pdf. Thus you cannot download it. But you need not get disheartened, as this content is freely available for all the students. You can simply reach the website and access it. Not no need to sign in either. It is more hassle-free than downloading a pdf. Because does not create a burden on the storage of your devices such as pc and smartphones. You can access it from Vedantu’s mobile application as well.

8. How useful is this topic in engineering?

The topic is very useful in various branches of engineering. If you are pursuing civil and mechanical engineering, it is inevitable for you to be a master in this subject. This topic has wide applications in construction and building activities. Also if you are someone who is planning to make a career in architecture then also this topic will be very helpful in designing. We suggest not to underestimate this topic as it will be very useful in the future.