Proof by mathematical induction

Introduction Geometry is a very organized and logical subject. The guiding light for solving Geometric problems is Definitions, Geometry Postulates, and Geometry Theorems. So before moving onto the geometry theorems list, let us discuss these to aid in geometry postulates and theorems list. Definitions are what we use for explaining things. E.g.: - You know that a circle is a round figure but did you know that a circle is defined as lines whose points are all equidistant from one point at the center. Or did you know that an angle is framed by two non-parallel rays that meet at a point? This is what is called an explanation of Geometry. Geometry. Image Postulates Geometry Postulates are something that can not be argued. It’s like set in stone. Example: - For 2 points only 1 line may exist. It is the postulate as it the only way it can happen. Or when 2 lines intersect a point is formed. Example 1 We can also say Postulate is a common-sense answer to a simple question. Theorems Unlike Postulates, Geometry Theorems must be proven. To prove a Geometry Theorem we may use Definitions, Postulates, and even other Geometry theorems. For example: If I say two lines intersect to form a 90° angle, then all four angles in the intersection are 90° each. Geometry Theorems Key components in Geometry theorems are Point, Line, Ray, and Line Segment. Let us go through all of them to fully understand the geometry theorems list. Point In maths, the smallest figure which can be drawn having no area is called a point. Line A straight figure that can be extended infinitely in both the directions Ray A line having one endpoint but can be extended infinitely in other directions. Line Segment A line having two endpoints is called a line segment. Now let’s discuss the Pair of lines and what figures can we get in different conditions. Pair of lines image 1. Angles Two rays emerging from a single point makes an angle. Angles image 1 2. Linear Pair When two or more than two rays emerge from a single point. Then the angles made by such rays are called linear pairs. Linear pair image 3. Parallel Lines When the perpendicular distance between the two lines is the same then we say the lines are parallel to each other. Parallel lines image 4. Vertically opposite angles If there are two lines crossing from one particular point then the opposite angles made in such a condition are equals. Vertically opposite angles Now that we are familiar with these basic terms, we can move onto the various geometry theorems. To make it easier to connect and hence apply, we have categorized them according to the shape the geometry theorems apply to. Angle Theorems The relation between the angles that are formed by two lines is illustrated by the geometry theorems called “Angle theorems”. Some of the important angle theorems involved in angles are as follows: 1. Alternate Exterior Angles Theorem When two parallel lines are cut by a transversal then resulting alternate exterior angles are congruent. Alternate Exterior Angles Theorem Image The alternate exterior angles have the same degree measures because the lines are parallel to each other. 2. Alternate Interior Angles Theorem When two parallel lines are cut by a transversal then resulting alternate interior angles are congruent. Alternate Interior Angles Theorem Image The alternate interior angles have the same degree measures because the lines are parallel to each other. One way to find the alternate interior angles is to draw a zig-zag line on the diagram. zig-zag line on the diagram image zig-zag line on the diagram Image 2 3. Congruent Complements Theorem If two angles are complementary to the same angle or of congruent angles, then the two angles are congruent. Congruent Complements Theorem Image 4. Congruent Supplements Theorem If two angles are supplements to the same angle or of congruent angles, then the two angles are congruent. Congruent Supplements Theorem Image 5. Right Angles Theorem If two angles are both supplement and congruent then they are right angles. Right Angles Theorem Image 6. Same-Side Interior Angles Theorem If two parallel lines are cut by a transversal, then the interior angles on the same side of the transversal are supplementary. Same-Side Interior Angles Theorem Image 7. Vertical Angles Theorem Angles that are opposite to each other and are formed by two intersecting lines are congruent. Vertical Angles Theorem Image Now let us move onto geometry theorems which apply on triangles. Triangle Theorems We know that there are different types of triangles based on the length of the sides like a scalene triangle, isosceles triangle, equilateral triangle and we also have triangles based on the degree of the angles like the acute angle triangle, right-angled triangle, obtuse angle triangle. Though there are many Geometry Theorems on Triangles but Let us see some basic geometry theorems. Theorem 1 In any triangle, the sum of the three interior angles is 180°. Example Suppose XYZ are three sides of a Triangle, then as per this theorem; ∠X + ∠Y + ∠Z = 180° Theorem 2 If a side of the triangle is produced, the exterior angle so formed is equal to the sum of corresponding interior opposite angles. Example For a triangle, XYZ, ∠1, ∠2, and ∠3 are interior angles. And ∠4, ∠5, and ∠6 are the three exterior angles. Theorem 3 The base angles of an isosceles triangle are congruent. Suppose a triangle XYZ is an isosceles triangle, such that; XY = XZ [Two sides of the triangle are equal] Hence, ∠Y = ∠Z Where ∠Y and ∠Z are the base angles. Now Let’s learn some advanced level Triangle Theorems. Theorem 3: If a line is drawn parallel to one side of a triangle to intersect the midpoints of the other two sides, then the two sides are divided in the same ratio.