Triangles properties and types | GMAT Geometry Tutorial

In the series on the basic building blocks of geometry, after a overview of lines, rays and segments, this time we cover the types and properties of triangles.

Definition: A triangle is a closed figure made up of three line segments.

simple triangle

A triangle consists of three line segments and three angles. In the figure above, AB, BC, CA are the three line segments and ∠A, ∠B, ∠C are the three angles.

There are three types of triangles based on sides and three based on angles.

Types of triangles based on sides

Equilateral triangle: A triangle having all the three sides of equal length is an equilateral triangle.

equilateral triangle

Since all sides are equal, all angles are equal too.

Isosceles triangle: A triangle having two sides of equal length is an Isosceles triangle.

isosceles triangle

The two angles opposite to the equal sides are equal.

Scalene triangle: A triangle having three sides of different lengths is called a scalene triangle.

scalene triangle

Types of triangles based on angles

Acute-angled triangle: A triangle whose all angles are acute is called an acute-angled triangle or Acute triangle.

acute angled triangle

Obtuse-angled triangle: A triangle whose one angle is obtuse is an obtuse-angled triangle or Obtuse triangle.

obtuse angled triangle

Right-angled triangle: A triangle whose one angle is a right-angle is a Right-angled triangle or Right triangle.

right angled triangle

In the figure above, the side opposite to the right angle, BC is called the hypotenuse.

For a Right triangle ABC,

BC2 = AB2 + AC2

This is called the Pythagorean Theorem.

In the triangle above, 52 = 42 + 32. Only a triangle that satisfies this condition is a right triangle.

Hence, the Pythagorean Theorem helps to find whether a triangle is Right-angled.

Types of triangles

types of triangles

There are different types of right triangles. As of now, our focus is only on a special pair of right triangles.

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

45-45-90 triangle:

A 45-45-90 triangle, as the name indicates, is a right triangle in which the other two angles are 45° each.

This is an isosceles right triangle.

isosceles right triangle

In ∆ DEF, DE = DF and ∠D = 90°.

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

30-60-90 triangle:

A 30-60-90 triangle, as the name indicates, is a right triangle in which the other two angles are 30° and 60°.

This is a scalene right triangle as none of the sides or angles are equal.

scalene right triangle

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

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

Basic properties of triangles

  • The sum of the angles in a triangle is 180°. This is called the angle-sum property.
  • The sum of the lengths of any two sides of a triangle is greater than the length of the third side. Similarly, the difference between the lengths of any two sides of a triangle is less than the length of the third side.
  • The side opposite to the largest angle is the longest side of the triangle and the side opposite to the smallest angle is the shortest side of the triangle.
  • triangle smallest angle

    In the figure above, ∠B is the largest angle and the side opposite to it (hypotenuse), is the largest side of the triangle.

    triangle largest angle

    In the figure above, ∠A is the largest angle and the side opposite to it, BC is the largest side of the triangle.

  • An exterior angle of a triangle is equal to the sum of its interior opposite angles. This is called the exterior angle property of a triangle.
  • triangle exterior angle

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

    According to the exterior angle property, ∠ACD = ∠CAB + ∠ABC.

Similarity and Congruency in Triangles

Figures with same size and shape are congruent figures. If two shapes are congruent, they remain congruent even if they are moved or rotated. The shapes would also remain congruent if we reflect the shapes by producing mirror images. Two geometrical shapes are congruent if they cover each other exactly.

Figures with same shape but with proportional sizes are similar figures. They remain similar even if they are moved or rotated.

Similarity of triangles

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

It is written as ∆ ABC ∼ ∆ XYZ and said as ∆ ABC ‘is similar to’ ∆ XYZ.

similar triangles

Here, ∠A = ∠X, ∠B =∠Y and ∠C = ∠Z AND

AB / XY = BC / YZ = CA / ZX

The necessary and sufficient conditions for two triangles to be similar are as follows:

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

If three sides of a triangle are proportional to the corresponding three sides of another triangle then the triangles are said to be similar.

similar triangles Side-Side-Side SSS

Here, ∆ PQR ∼ ∆ DEF as

PQ / DE = QR / EF = RP / FD

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

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

similar triangles Side-Angle-Side SAS

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 the two triangles are equal then the two triangles are similar.

Similar Triangles Angle-Angle-Angle AAA

Here ∆ TUV ∼ ∆ PQR as

∠T = ∠P, ∠U = ∠Q and ∠V = ∠R

Congruency of triangles

Two triangles are said to be congruent if all the sides of one triangle are equal to the corresponding sides of another triangle and the corresponding angles are equal.

It is written as ∆ ABC ≅ ∆ XYZ and said as ∆ ABC ‘is congruent to’ ∆ XYZ.

congruent triangles

The necessary and sufficient conditions for two triangles to be congruent are as follows:

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

If three sides of a triangle are equal to the corresponding three sides of another triangle then the triangles are said to be congruent.

congruent triangles Side-Side-Side SSS

Here, ∆ ABC ≅ ∆ XYZ as AB = XY, BC = YZ and AC = XZ.

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

If two sides and the angle included between the two sides of a triangle are equal to the corresponding two sides and the included angle of another triangle, then the triangles are congruent.

Congruent Triangles Side-Angle-Side SAS

Here, ∆ ABC ≅ ∆ XYZ as AB = XY, ∠A = ∠X and 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.

congruent triangles Angle-Side-Angle ASA

In the figure above, ∆ ABD ≅ ∆ CBD in which

∠ABD = ∠CBD, AB = CB and ∠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.

congruent triangles right-angle-hypotenuese

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

Area of a triangle:

The Area of a triangle is given by the formula

Area of a triangle = (1/2) *Base * Height

Area of triangle formula

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

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

For a right triangle, it’s easy to find the area as there is a side perpendicular to the base, so we can consider it as height.

area of right-angled triangle

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

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

Area of obtuse-angled triangle

For an obtuse triangle, we extend the base and draw a line perpendicular from the vertex to the extended base which becomes the height of the triangle.

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

Solve the following

1)

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

Triangles quiz problem 1

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° (since 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° (again, sum of all angles in a triangle)

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

 

2) Determine if the following are right triangles

Triangles quiz problem 2

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 theorem is the set of sides that makes a right triangle.

3)

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

Triangles quiz problem 3

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°


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