GMAT / GRE Mathematics Tutorial Series: Triangles

Triangle properties and types

In this series, we cover the properties and types of triangles.

Definition: A triangle is a three-sided polygon, having three edges and three vertices.

A triangle has three line segments and three angles.

As per the figure above, AB, BC, CA are the three line segments, and ∠A, ∠B, ∠C are the three angles.

Triangles are of three types based on their sides – The equilateral triangle, Isosceles triangle, and the Scalene triangle.

Triangles are of three types based on their angles – Acute-angled triangle, Obtuse-angled triangle, Right-angled triangle.

Also Check GMAT / GRE Tutorial Series Quadratic Equation

Types of triangles based on sides

Equilateral triangle: A triangle in which all three sides are of equal lengths.

All the angles in an equilateral triangle are also equal as all the sides are equal. 

Isosceles triangle: A triangle that has two sides to be of the same length is an isosceles triangle.

In an Isosceles triangle, the two angles opposite to the equal sides are equal.

Scalene triangle: A triangle in which all three sides are of different lengths.

Types of triangles based on angles

Acute-angled triangle: A triangle in which all the angles are acute or less than 90°.

Obtuse-angled triangle: A triangle in which one of the angles is obtuse or (greater than 90° but less than 180°). 

Right-angled triangle: A triangle in which one of the angles is a right-angle or 90°.

In the figure above, the side opposite to the right angle, AC, is called the hypotenuse of the triangle.

For a Right triangle ABC,

AC^² = AB^² + BC^²

This is called the Pythagorean Theorem.

Based on this theory, from the above GMAT exam triangle Questions, we can say that, 10² = 8² + 6²

Only triangles that satisfy this condition can be a right-angled triangle.

Therefore, using the Pythagorean Theorem we can find out if a triangle is Right-angled or not.

Also Check GMAT / GRE Tutorial Series Probability

Types of triangles

Right-angled triangles are of different types as well. Let’s focus on a special pair of right triangles that are:

1. 45-45-90 triangle
2. 30-60-90 triangle

45-45-90 triangle:

A 45-45-90 triangle, is a right triangle, with the other two angles at 45°.

This is also called an isosceles right triangle.

In the given triangle, ∆ DEF, sides, DE = DF and ∠D = 90°.

The sides are in the ratio 1 : 1 : √2, in a 45-45-90 triangle.

30-60-90 triangle:

A 30-60-90 triangle, is a right triangle with the other two angles as 30° and 60°.

As none of the sides or angles are equal, this is a scalene triangle.

The sides are in the ratio 1 : √3 : 2, in a 30-60-90 triangle

Like any other right triangle, these two triangles also satisfy the Pythagorean Theorem.

Basic properties of triangles

• The total sum of the three angles in a triangle is 180°. This is the angle-sum property.
• Secondly, the lengths of two sides, when added together are greater than the third side of the triangle. Also, the difference in lengths of any two sides will always be lesser than the third side of the triangle.
• Thirdly, the side that faces the largest angle, is the longest side. The side that faces the smallest angle is then the shortest side of the triangle.

As per the triangle shown above, ∠B is the largest angle and the side opposite to it (also called hypotenuse), is the longest side of the triangle.

As per the triangle shown above, ∠A is the largest angle and BC is the largest side of the triangle, as it is opposite ∠A.

• The exterior angle property of a triangle states that the exterior angle of a triangle will be equal to the sum of its interior opposite angles

Here, in ∆ABC, the exterior angle is ∠ACD.

Therefore, ∠ACD = ∠CAB + ∠ABC, as per exterior angle property.

Also Check GMAT / GRE Tutorial Series Ratio Analysis

Similarity and Congruence in Triangles

Congruent figures are exactly equal to each other in terms of shape and size.

If two shapes are congruent to each other then,

• they remain congruent even if they are moved or rotated
• their mirror images are also congruent

Two geometrical shapes are said to be congruent if they can exactly map each other.

Figures that have the same shape but proportional sizes are also said to be similar figures and will remain so even if the figures are rotated or moved. 

Similarity of triangles

Two triangles are said to be similar if the corresponding angles of two triangles are congruent and the lengths of the corresponding sides are proportional.

The similarity is written as ∆ ABC ∼ ∆ XYZ and we can say that ∆ ABC ‘is similar to’ ∆ XYZ.

Here, the angles, ∠A = ∠X, ∠B =∠Y and ∠C = ∠Z

And sides AB / XY = BC / YZ = CA / ZX

For two triangles to be similar they have to meet the necessary and sufficient conditions:

(1) Side-Side-Side (SSS) criterion for similarity:

According to SSS criterion, when the three sides of a triangle are proportional to the corresponding three sides of another triangle then the triangles are said to be similar.

Here, ∆ PQR ∼ ∆ DEF as

Sides PQ / DE = QR / EF = RP / FD

(2) Side-Angle-Side (SAS) criterion for similarity:

If two corresponding sides of the two triangles are proportional and one included angle is equal to the corresponding included angle then the triangles are similar.

Here, ∆ LMN ∼ ∆ QRS in which

∠L = ∠Q

QS / LN = QR / LM

(3) Angle-Angle-Angle (AAA) criterion for similarity:

If the three corresponding angles of two triangles are equal then the two triangles are said to be similar.

Here ∆ TUV ∼ ∆ PQR ,

As the angles, ∠T = ∠P, ∠U = ∠Q and ∠V = ∠R

Congruence of triangles

For two triangles to be congruent, all the sides and angles of one triangle should be equal to the corresponding sides and angles of the other triangle.

Expressed as ∆ ABC ≅ ∆ XYZ and we can say that ∆ ABC ‘is congruent to’ ∆ XYZ.

 For two triangles to be congruent, the necessary and sufficient conditions to be met are     :
 
(1) Side-Side-Side (SSS) criterion for congruence:

As per this criterion, when the three sides of a triangle are equal to the corresponding three sides of another triangle the triangles are said to be congruent.

Here, ∆ ABC ≅ ∆ XYZ as the sides AB = XY, BC = YZ and AC = XZ.
 
(2) Side-Angle-Side (SAS) criterion for congruence:

As per this criterion, when two sides and the angle between them are equal to the corresponding two sides and the included angle of another triangle, then the triangles are congruent.

Here, ∆ ABC ≅ ∆ XYZ as the sides, AB = XY, and angles ∠A = ∠X and  sides AC = XZ.
 
(3) Angle-Side-Angle (ASA) criterion for congruence: If two angles and the included side of a triangle are equal to the corresponding two angles and the included side of another triangle then the triangles are congruent.

Based on the triangle shown above, ∆ ABD ≅ ∆ CBD in which

Angles ∠ABD = ∠CBD, sides AB = CB and angles ∠ADB = ∠CDB.
 
(4) Right-Angle Hypotenuse criterion of congruence: If the hypotenuse and one side of a right-angled triangle are equal to the corresponding hypotenuse and side of another right-angled triangle, then the triangles are congruent.

Here, angles ∠B = ∠Y = 90° and the sides AB = XY, AC = XZ.

Area of a triangle:

The formula for finding the area of a triangle = (1/2) *Base * Height

To find the area of a triangle, we draw a perpendicular line from the base to the opposite vertex to get the height of the triangle.

Therefore the area of ∆ PQR = (1/2) * (PR * QS) = (1/2) * 6 *4 =12 sq. units.

For a right triangle, it’s easy to find the area, as the height of the triangle is the side that is perpendicular to the base.

XY is the height of the ∆ XYZ  and its area is (1/2) * XZ * XY sq. units.

Now, how do we calculate the area of an obtuse triangle LMN?

For an obtuse triangle, the height of the triangle can be found out by extending the base and drawing a perpendicular line from the opposite vertex.

Therefore, the area of ∆ LMN = (1/2) * LM * NK sq. units.
Practice questions

1)∆ ABC is a right triangle with CD ⊥ AB (⊥ stands for ‘perpendicular’).

Find  i) ∠ACD and  ii) ∠ABC.

A. 25, 35
B. 35, 35
C. 25, 25
D. 35, 25

Answer: C

 Explanation:

Consider ∆ ACD.

∠ADC + ∠DAC + ∠ACD = 180° (as sum of angles in a triangle is 180°)

90 + 65 + ∠ACD = 180° → ∠ACD = 25°

∠ACD + ∠DCB = 90° → 25 + ∠DCB = 90 → ∠DCB = 65°

In ∆ BCD, ∠DCB + ∠CBD + ∠BDC = 180° (angle sum property)

65 + ∠CBD + 90 = 180 → ∠CBD = 25° = ∠ABC.

2) Which of the two triangles are right-angled?

A. Both are right triangles
B. ∆ ABC is not a right triangle, ∆ DEF is a right triangle
C. ∆ ABC is a right triangle, ∆ DEF is not a right triangle
D. Both are not right triangles

Answer: B

Explanation:

The triplet that satisfies the Pythagorean condition is a right triangle.

3)If ∆ ABC = 3 (∆ DEF), which of the following is correct?

A. ∠E = ∠F = 40°, ∠D = 120° AND DE = DF = 2 and EF = 3
B. ∠E = ∠F = 40°, ∠D = 110° AND DE = DF = 2 and EF = 3
C. ∠E = ∠F = 40°, ∠D = 100° AND DE = DF = 2 and EF = 3
D. ∠E = ∠F = 40°, ∠D = 110° AND DE = DF = 3 and EF = 3

Answer: C

Explanation:

AB and AC are equal → angles opposite are equal.

Therefore ∠B = ∠C = 40° → ∠A = 100°.

∆ ABC = 3 (∆ DEF) → ∆ ABC and ∆ DEF are similar.

When two triangles are similar, their corresponding angles are equal and the corresponding sides are proportional.

→ DE = DF = 6/3 = 2 and EF = 3

→ ∠E = ∠F = 40° and ∠D = 100°

Also read:

• GMAT / GRE Mathematics Preparation Guide: Distance, Time and Speed
• GMAT / GRE Mathematics Preparation Guide: Lines & Angles
• GMAT / GRE Mathematics Preparation Guide: Permutation & Combination
• GMAT / GRE Mathematics Preparation Guide: Probability
• GMAT / GRE Mathematics Preparation Guide: Profit & Loss
• GMAT / GRE Mathematics Preparation Guide: Quadratic Equation
• GMAT / GRE Mathematics Preparation Guide: Quadrilateral
• GMAT / GRE Mathematics Preparation Guide: Ratio Analysis
• GMAT / GRE Mathematics Preparation Guide: Ratio & Proportion
• GMAT / GRE Mathematics Preparation Guide: Set Theory
• GMAT / GRE Mathematics Preparation Guide: Simple & Compound Interest