Do you feel overwhelmed while learning a plethora of GMAT Geometry formulas[1] and concepts? Do you feel Geometry is your weak topic? You are not alone. Many test takers avoid solving GMAT Geometry questions, as most of them don’t feel comfortable with this topic.

In this article, you will read-

- Why is GMAT Geometry your weak area?
- How will we help you improve Geometry?
- Basic properties of triangles for GMAT Geometry
- Illustrative examples showing usage of GMAT Geometry Formulas
- Takeaways

- There are
__too many GMAT Geometry formulas to remember__. Even though GMAT tests us on a very limited number of properties, but since these limited properties are not collated in a single place, it is difficult for a test taker to confidently say he knows all the concepts.

- And say, even if someone
__manages to find and learn all the concepts__, most of them are not sure how to apply these concepts to__deduce information from the given question and figure__in the questions tested on the GMAT.

This is the second article in our GMAT Geometry Pitfalls series. Do read the first article click on the link below.NOTE:

Most common mistakes in GMAT Geometry Questions

The next 2 articles in this series are:

## How will we help you with GMAT Geometry Prep?

In this article, we will address both the above issues by:

- Consolidating the
__MOST important GMAT Geometry concepts needed in Triangles__. __Focusing on some GMAT-like and official questions__and illustrating the application of the above concepts in these questions.

## Basic properties of triangles

This section covers the following aspects:

- Finding if a set of given numbers could be the lengths of a triangle or not
- The range of values the side of a triangle can take given the values of the other two sides, and
- The properties of the
__angles of the triangle__

## Concept 1: GMAT Geometry formulas

**Geometry GMAT Concept 1 – Sum of lengths of any two sides of a triangle > Length of the third side**

Suppose you are given three lengths a, b, and c, and asked to find out if we can form a triangle using these three lengths.

*How are you going to approach this problem?*

__Theory:__

To determine if the three lengths will form a triangle or not, we need to know __the most basic property of a triangle:__

Sum of lengths of any two sides of a triangle > Length of the third side

This means that:

**a + b > c****b + c > a****c + a > b**

Please note that __all the three inequalities should satisfy__ for the triangle to exist.

Let us understand this with the help of a very basic GMAT-like question:

## Illustrative Example 1

Q. The three lengths 5, 3, and x are used to form a triangle. Which of the following can be the value of x?

- 6
- 9
- 10
- 12
- 14

Since 5, 3 and x form a triangle, they must satisfy the property –

**Sum of lengths of any two sides of a triangle > Length of the third side**

Therefore, we can write –

**I. 5 + 3 > x II. 3 + x > 5 III. 5 + x > 3 **

** => x < 8 => x > 2 => x > -2**

we can conclude that the range of x is: 2 < x < 8

Out of the given options, the only option that falls within this range is 6. Hence, **option A is our answer.**

Therefore, now we know:

- How to find if the
__given lengths form a triangle or not__ - How to use the above property to
__find the range of values for an unknown side in a triangle__

## Concept 2: GMAT Geometry Formulas

Suppose after finding out that the given lengths form a triangle, you want to find out how the length of the sides can help in **finding a relation between the angles of the triangle.**** **

To understand this, we need to know two more important properties:

**GMAT Geometry Concept 2 – ****The sum of the interior angles of a triangle is constant and is equal to 180°**

Please note that the above property holds true for all triangles irrespective of their size and shape.

__ ____(Notice I am using capital A, B and C to denote the angles of the__

__ ____triangle and a, b, and c for the lengths of the sides of the triangle)__

##### Property 3 – In any triangle, the largest side is always opposite the largest angle and the smallest side is always opposite the smallest angle

Let me explain this a bit more in detail:

If ABC is a triangle, in which we know that the lengths AB, BC, and CA follow the relation AB>BC>AC, then we can conclude that, since __AB is the greatest side the angle opposite to it__ __i.e., angle ACB is the greatest angle__.

And since BC > AC, angles opposite to them also hold the same relation, that means angle BAC > angle ABC

Hence we can conclude that: ∠ ACB > ∠ BAC > ∠ ABC* *

Keep in mind that the **converse is also true**, i.e., if

__the relation between the angles is given, we__

__can use that to find the relation between the sides.__

Let us understand the application of this property through an example:

## Illustrative Example 2

In the triangle given below, all the points denoted are non-overlapping and the angles A, B, and C are distinct. Is AB > AC > BC?

- ∠ y
_{1}= ∠ c_{1} - ∠ BAC < ∠ ABC and neither angle is the greatest in the triangle ABC

Solution:

__Approach__** **

We need to find if the statement AB > AC > BC is true or not. For this, it is enough to get a relation among angles A, B and C since __the sides of a ____triangle follow__ __the same relation as the angles opposite to them.__

__Analyze Statement 1__

** **∠**y _{1} = **

**∠**

**c**

_{1}Therefore, in ΔBYC

BY = BC

But AB = BY + YA

AB > BC

However, __we cannot infer any relationship between BC and AC (or AB and AC)__ from the given information

__Analyze Statement 2__

**Given **∠ **BAC < **∠ **ABC and neither angle is the greatest.**

Also, in the question statement, it is mentioned that angles BAC, ABC, and ACB are distinct, therefore, ∠ACB has to be the greatest angle.

Therefore, we can conclude, ∠ **BAC **< ∠ **ABC **< ∠ **ACB**

Thus, we can conclude that length **BC < AC < AB**

Hence, statement 2 is sufficient to arrive at a unique answer.* *

**Correct Answer: B**

## Takeaways – Article 1 – GMAT Geometry Triangle Concepts

This brings us to the end of the first article in this series. I hope you find it useful and informative. After reading this you should be able to easily:

, to identify if a triangle can be formed from the given three lengths__Apply GMAT Geometry Concept 1__, to also find the range of the unknown side, if the other two lengths of the triangle are given__Apply GMAT Geometry Concept 1__in questions where the relation of angles and sides are required to solve the problem__Apply GMAT Geometry Concept 2__

Keeping these takeaways in mind, can you solve the 2 questions?

**Mastering Important Concepts Tested By GMAT in Triangles – Exercise Question #1****Mastering Important Concepts Tested By GMAT in Triangles – Exercise Question #2**

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**In the next articles on Triangles, read about**

**1. **__Tabular Representation:__ Different types of triangles with their MOST important properties.

**2. ** __Special Properties:__ A few special properties that can be tested by GMAT.

[1] *Formulas* is the plural form of *formula*, the alternate plural of *formula* is *formulae.*

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