Plane Geometry Concepts and Problem Solving

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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:

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?

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.

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