

Plane Geometric Figures
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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:
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.
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?
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 plane geometry and why is it called two-dimensional?
Plane geometry is the branch of mathematics that studies flat shapes and figures that can be drawn on a piece of paper. It is called two-dimensional (2D) because all its shapes, like squares, circles, and triangles, have only two measurements: length and breadth. They do not have a third dimension like depth or height.
2. What are the three fundamental elements of plane geometry?
The three fundamental, undefined elements that form the basis of plane geometry are:
- Point: A specific location or position on a plane. It has no size or dimension.
- Line: A straight path made up of infinite points that extends endlessly in both directions. It has length but no width.
- Plane: A perfectly flat surface that extends infinitely in all directions. It is the space where all 2D figures are drawn.
3. What is the main difference between a line, a line segment, and a ray?
The main difference lies in their endpoints. A line extends infinitely in both directions and has no endpoints. A line segment is a part of a line with two distinct endpoints, giving it a fixed length. A ray has one starting point (endpoint) and extends infinitely in only one direction.
4. What are the key types of angles based on their measurement in plane geometry?
Angles in plane geometry are classified based on their degree measurement:
- Acute Angle: An angle that measures less than 90°.
- Right Angle: An angle that measures exactly 90°.
- Obtuse Angle: An angle that measures more than 90° but less than 180°.
- Straight Angle: An angle that measures exactly 180°, forming a straight line.
- Reflex Angle: An angle that measures more than 180° but less than 360°.
5. How does plane geometry apply to real-world scenarios like architecture and art?
Plane geometry is crucial in many real-world fields. In architecture, it is used to design floor plans, calculate room areas, and ensure structural stability by using principles of triangles and polygons. In art and design, artists use concepts of symmetry, proportion, and perspective, all rooted in plane geometry, to create visually appealing compositions and realistic drawings.
6. What is the important distinction between congruence and similarity in plane figures?
The distinction is based on size and shape. Congruent figures are identical in both shape and size; you can superimpose one perfectly over the other. Similar figures have the exact same shape (meaning their corresponding angles are equal) but can be of different sizes. All congruent figures are similar, but not all similar figures are congruent.
7. How do axioms and postulates serve as the foundation for all theorems in plane geometry?
Axioms and postulates are statements that are accepted as true without proof. They form the logical starting point for geometric reasoning. For example, the axiom that 'things which are equal to the same thing are equal to one another' allows us to prove relationships between shapes. Every theorem in plane geometry is a logical conclusion derived step-by-step from these fundamental axioms and postulates.
8. Why can't we measure the area of a line in plane geometry?
We cannot measure the area of a line because area is a two-dimensional quantity, requiring both length and width for its calculation. A line, by definition in plane geometry, has only one dimension (length) and possesses no width or thickness. Since it lacks a second dimension, it occupies no surface area on a plane.





