Triangle : Its Properties & Its Its Types
A triangle is one of the basic shapes of geometry. It is a polygon with three sides and 3 vertices/corners. Learn about different triangles such as equilateral, isosceles, scalene triangles and their properties.
Definition: A triangle is a closed figure made up of three line segments.
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.
Since all sides are equal, all angles are equal too.
Isosceles triangle: A triangle having two sides of equal length is an 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.
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.
Obtuse-angled triangle: A triangle whose one angle is obtuse is an obtuse-angled triangle or Obtuse triangle.
Right-angled triangle: A triangle whose one angle is a right-angle is a Right-angled triangle or Right 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.
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Types of triangles
There are different types of right triangles. As of now, our focus is only on a special pair of right triangles.
- 45-45-90 triangle
- 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.
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.
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
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- 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.
In the figure above, ∠B is the largest angle and the side opposite to it (hypotenuse), is the largest side of the triangle.
In the figure above, ∠A is the largest angle and the side opposite to it, BC is the largest side of the triangle.
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- 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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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
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.
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 ?
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’).
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
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?
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°
Angle sum property
The sum of the three angles of a triangle is 180°.
e.g. If A, B and C are the angles of a triangle ABC, then ∠A + ∠B + ∠C = 180°.
Proof:
Consider a triangle ABC.
Let line XY be parallel to side BC at A.
AB is a transversal that cuts the line XY and AB, at A and B, respectively.
As the alternate interior angles are equal, ∠1 = ∠4 and ∠2 = ∠5.
∠4, ∠3 and ∠5 form linear angles, and their sum is equal to 180°.
⇒ ∠4 + ∠3 + ∠5 = 180°
⇒ ∠1 + ∠2 + ∠3 = 180°
Hence, the sum of the three angles of a triangle is 180°.
Exterior angle property
An exterior angle of a triangle is equal to the sum of its opposite interior angles.
e.g. If ∠4 is an exterior angle of ΔABC, ∠1 and ∠2 are the interior opposite angles, then ∠4 = ∠1 + ∠2.
The sum of the lengths of any two sides of a triangle is greater than the third side.
In a right-angled triangle, the side opposite the right angle is called the hypotenuse, and the other two sides are called its legs.
Pythagorean theorem
In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
If b and c are legs and a is the hypotenuse of a right angled triangle then, a2 = b2 + c2.
Converse of Pythagorean theorem
If the sum of the squares on two sides of a triangle is equal to the square of the third side, then the triangle must be a right-angled triangle.
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