# Theorems on Solid Geometry

Some specific theorems on solid geometry are discussed here under this section.

### Axioms:

The following two fundamental propositions may be considered as axioms:

Proposition 1: One and only one plane can be drawn through any two intersecting straight lines.

Proposition 2: Two intersecting planes cut one another in a straight line and in no other point outside the line of intersection.

The above two propositions lead to the following conclusions.

(a) A straight line intersects a plane at one point only or lies wholly in the plane or is parallel to the plane.

(b) Infinite number of planes can be drawn through a given straight line.

(c) The straight line joining two given points on a plane lies wholly in the plane if it be produced indefinitely in either direction.

(d) The position of a plane is determined if it passes through

(i) two intersecting straight lines;

(ii) a given straight line and a given point outside the line;

(iii) two parallel straight lines;

(iv) three non-collinear points.

Example: Show that two parallel lines and any of its transversal lie in the same plane.

Let LM and NO be two parallel lines and XY, a transversal intersects LM at R and NO at S. We are to prove that the lines LM, NO and XY lie in the same plane (i.e., they are co-planar).

Proof: Since two parallel straight lines are co-planar, let us assume that the parallel tines LM and NO lie in the plane g. Now, the point R lies on the line LM and the point S on the line NO. Hence, it is evident that both the points R and S lie in the plane g. Therefore, the straight line joining the points R and S (i.e. the straight line XY) lies in the plane g.

Therefore, the straight lines LM, NO and XY lie in the same plane g.

Therefore, the straight lines LM, NO and XY are co-planar

Geometry

Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.

## Recent Articles

1. ### Adding 1-Digit Number | Understand the Concept one Digit Number

Sep 17, 24 02:25 AM

Understand the concept of adding 1-digit number with the help of objects as well as numbers.

2. ### Counting Before, After and Between Numbers up to 10 | Number Counting

Sep 17, 24 01:47 AM

Counting before, after and between numbers up to 10 improves the child’s counting skills.

3. ### Worksheet on Three-digit Numbers | Write the Missing Numbers | Pattern

Sep 17, 24 12:10 AM

Practice the questions given in worksheet on three-digit numbers. The questions are based on writing the missing number in the correct order, patterns, 3-digit number in words, number names in figures…

4. ### Arranging Numbers | Ascending Order | Descending Order |Compare Digits

Sep 16, 24 11:24 PM

We know, while arranging numbers from the smallest number to the largest number, then the numbers are arranged in ascending order. Vice-versa while arranging numbers from the largest number to the sma…