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Triangle is a closed polygon with three sides, three angles, and three vertices. A triangle is a closed figure formed by the intersection of three lines in a plane. The sum of all three interior angles of a triangle is always 180° and the sum of all the exterior angles is always 360°. The basics of trigonometry and Pythagoras theorem root back to the properties of triangles. Triangles can be classified on the basis of the length of each side, and internal angles. In this article, we will learn about the properties of triangles, like perimeter, area, Heron’s formula, concepts such as congruency of triangles (SAS, ASA, AAS, SSS, RHS) in detail and look at some solved examples to understand the concept.
Key Terms: Properties of triangle, heron’s formula, types of triangle, congruency, Pythagoras theorem, inequality in a triangle.
What is a triangle?
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A triangle is a polygon with three sides, three vertices, and three angles. The sum of the three interior angles of a triangle is always equal to 180 °. And the sum of all the exterior angles of a triangle is equal to 360 °. This is called the angle sum property of a triangle.
Triangle
Types of triangle
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Triangles can be classified on the basis of (i) the length of sides, and (ii) the internal angle. In total there can be 7 types of triangles. Each of them has different properties that distinguish them from each other.
Types of triangles based on length of sides
Triangles can be classified into three types on the basis of the length of sides.
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Scalene triangle
A scalene triangle is a triangle with all three sides of different lengths. No two sides of the triangle are of the same length. Hence, all three angles of the scalene triangle are different as well.
Scalene triangle
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Isosceles triangle
An isosceles triangle is a triangle where any of the two sides are equal to each other. The angles opposite to equal sides are equal as well.
Isosceles triangle
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Equilateral triangle
An equilateral triangle is a triangle with all three equal sides. All the interior angles are equal to 60°.
Equilateral triangle
Types of triangles based on internal angles
Triangles can be classified on the basis of internal angles as well. There are three types of triangles; acute triangle, obtuse triangle, and right-angled triangle.
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Acute angle triangle
In an acute angle triangle, all the interior angles are less than 90°.
Example of an acute angled triangle
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Obtuse angled triangle
In an obtuse angled triangle, any one of the three angles is greater than 90°.
Example of an obtuse angled triangle
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Right-angled triangle
A right-angled triangle is a triangle where any one of the angles of the triangle is equal to 90° (or right angle).
Example of a right-angled triangle
Properties of a triangle
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The theorems of triangles are all based on the properties of triangles. Properties of triangles are as follows:
- The sum of the three interior angles is equal to 180°.
- The sum of the three exterior angles is equal to 360°.
- The sum of two interior angles is equal to the opposite exterior angle.
- If two sides of a triangle are equal, then the angles opposite to these sides are equal as well.
- Two triangles are said to be similar if the corresponding angles of the two triangles are equal.
- If two angles of a triangle are unequal, the side opposite to the larger angle is larger too.
- In any triangle, the side opposite to the larger (greater) angle is longer.
- The area of a triangle is given by ½ x base x height.
- The sum of the length of two sides of a triangle is greater than the length of the third side of the triangle. This is called the Triangle inequality theorem.
Read More: properties of triangles.
Angles of triangle
A triangle has three internal and three external angles. There are various advanced theorems based on the angles of triangles.
The exterior angles are formed by extending one side of the triangle.
Interior and exterior angles of a triangle
- The sum of three internal angles is 180°. The sum of angles a, b, and c in the figure above is 180°.
- The sum of two interior angles is always equal to the opposite exterior angle.
Perimeter of triangle
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A triangle is made of three sides. The perimeter of a triangle is the sum of three sides of the triangle.
So, if we consider a triangle ABC, then the perimeter of this triangle would be (AB + BC + AC).
- As we have already discussed the different types of triangles, a scalene triangle has all three sides unequal, let us consider the length of the three sides are a, b, and c.
So the perimeter of a scalene triangle would be a+b+c.
- In an isosceles triangle, two of the three sides are equal. If we consider the length of the equal sides as a, and the length of the unequal side as b.
The perimeter of an isosceles triangle would be (2a + b).
- In an equilateral triangle, three sides of the triangle are of equal length. Let us consider an equilateral triangle whose each side measures a.
So, the perimeter of an equilateral triangle would be equal to 3a.
Area of triangle
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The area of a triangle is defined as the space confined within the three sides of the triangle. To calculate the area of a triangle, we need the base length and the height of the triangle.
Area of triangle
In the figure given above, b is the base of the triangle, and h is the height of the triangle.
Area of triangle = ½ (base x height) = ½ b x h
| Example: Find the area of a triangle whose base is equal to 10cm and height is equal to 7cm. Solution: area of a triangle = ½ (base x height) As given in the question, Base = 10cm Height = 7cm Therefore, area = ½ (10 x 7) = ½ x 70 = 35 cm2 |
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Area of triangle using Heron’s formula
Heron’s formula, also called Hero’s formula, gives us the area of a triangle when all the sides of the triangle are of known length. Unlike other formulas, you do not need to calculate any other dimension of the triangle.
Heron’s formula for the area of triangle
| Example: Find the perimeter and the area of an isosceles triangle if two of its sides are 5 cm each and the third side is 6 cm. Solution: Perimeter of isosceles triangle = 2a + b where a is two equal sides of an isosceles triangle, and b is the unequal side. Here, a = 5cm b = 6cm Perimeter = 2x5 + 6 = 16cm Area of triangle by Heron’s formula is given by √s(s - a)(s - b)(s - c) s = 16/2 = 8cm So, area = √8(8-5)(8-5)(8-6) = √8(3)(3)(2) = √144 = 12cm2 |
Congruency of triangle
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Two triangles of the same shape and size are said to be congruent to each other. There are several criteria for congruency.
Criteria for congruency
The criteria for congruence of triangles are:
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SSS (side-side-side) congruency
If the three sides of a triangle are equal to the corresponding three sides of another triangle, the triangles are said to be congruent to each other.
Let us consider two triangles, ΔABC and ΔPQR,
If AB = PQ, BC = QR, and AC = PR, ∠ABC = ∠PQR,
Then, ΔABC ≅ ΔPQR by SSS criteria.
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SAS (side-angle-side) congruency
When the two sides and the angle between them of one triangle is equal to the corresponding angle and sides, the triangles are said to be congruent.
Let us consider two triangles, ?ABC and ?PQR,
If AB = PQ, BC = QR, and, ∠ABC = ∠PQR,
Then, ΔABC ≅ ΔPQR by SAS criteria.
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ASA (angle-side-angle) congruency
When “Two triangles are congruent if two angles and the included side of one triangle are equal to two angles and the included side of the other triangle”.
Let us consider two triangles, Δ ABC and Δ DEF
in which: ∠ B = ∠ E, BC = EF, and ∠ C = ∠ F
Then, Δ ABC ≅ Δ DEF
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AAS (angle-angle-side) congruency
“If two angles and one side of one triangle are equal to two angles and the corresponding side of the other triangle, then the two triangles are congruent”
Let us consider two triangles, Δ ABC and Δ DEF
in which: ∠ B = ∠ E, ∠ C = ∠ F and BC = EF
Then, Δ ABC ≅ Δ DEF
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RHS (right angle-hypotenuse-side) congruency
“If in two right triangles the hypotenuse and one side of one triangle are equal to the hypotenuse and one side of the other triangle, then the two triangles are congruent”.
This is similar to the SAS congruency rule, in the RHS congruency rule, the angle must be 90°.
Let us consider two right-angled triangles, ΔABC and ΔPQR,
If AB = PQ, BC = QR, and, ∠ABC = ∠PQR =90°
Then, ΔABC ≅ ΔPQR by RHS criteria.
Inequalities in triangle
- If two sides of a triangle are unequal, then the angles opposite to them are unequal as well.
- Triangle inequality theorem: The sum of lengths of two sides is always greater than the length of the third side.
Read More:
Things to remember
- Triangle is a polygon with three sides, three angles, and three vertices. It can be defined as a closed figure formed by the intersection of three lines in a plane.
- The sum of all the interior angles in a triangle is equal to 180°, and the sum of the exterior angles of a triangle is equal to 360°.
- There are three types of triangles based on the length of sides: scalene, isosceles, and equilateral.
- There are three types of triangles based on internal angles: acute triangle, obtuse triangle, and right-angled triangle.
- The sum of two interior angles is equal to the opposite exterior angle of the triangle.
- Triangle inequality theorem: The sum of any two sides of a triangle is always greater than the third side of the triangle.
- The perimeter of a triangle of sides a, b, and c is given by the formula
Perimeter of triangle = a+b+c
- The area of a triangle is the space within the sides of the triangle. It is given by the formula,
Area of triangle = ½ (base x height) = ½ b x h
- Heron’s formula or Hero’s formula: It gives us the area inside a triangle using only the length of the sides of the triangle.
- Congruent triangles are two triangles of the same shape and size. There are 5 congruency rules, SSS, SAS, ASA, AAS, and RHS..
Sample questions
Ques. Consider ΔABC and ΔPQR are congruent. Find the value of x and y if ∠B = ∠Q = 70°, AB = PQ, and ∠R = 45°. x = ∠C, y = ∠A. (2 marks)
Ans. As given in the question, the triangles ABC and PQR are congruent. So, ∠C = ∠R.
So, x = 45°.
We know, the sum of internal angles of a triangle is equal to 180°,
So, y = 180° - (70°+45°)
y = 180° - 115° = 65°.
Ques. In the figure given below, show that ΔPSQ = ΔPSR. (2 marks)
Ans.
In ΔPSQ and ΔPSR,
PQ = PR, (given)
PS = PS,
And, ∠PSQ = ∠PSR = 90° (given),
So, by RHS congruency rule, ΔPSQ and ΔPSR are congruent.
Ques. In triangle ABC and DEF, AB = DF and ∠A=∠ D. What will determine if the two triangles will be congruent by the SAS congruence rule? (2 marks)
Ans.
For the SAS congruency rule, two triangles are congruent if two sides and the included angle of one triangle are equal to the two sides and the included angle of the other triangle.
Given,
AB = DF
∠A=∠ D
So, for ΔABC and ΔDEF to be congruent, AC must be equal to DE.
Ques. In a triangle MNO, MP is the median. If ∠MNO = 55°, and MN = MO, find the value of ∠NMO. (2 marks)
Ans. We know that If two sides of a triangle are equal, then the angles opposite to these sides are equal as well.
As given in the question,
MN = MO, (given)
Therefore, ∠MNO = ∠NOM.
In ΔMNO, ∠NMO = 180° - (55° + 55°)
Therefore, ∠NMO = 70°.
Ques. In the given triangle PQR and ABC, PA = CR, PQ = CB, ∠QPR = ∠BCA. Check if ΔABC and ΔPQR are congruent? (3 marks)
Ans.
In ΔPQR and ΔABC,
PQ = CB (Given)
∠QPR = ∠BCA (Given)
PA = CR (Given)
AR is the common part in both the triangles, so PA + AR = CR + AR
Therefore, PR = AC
Hence, ΔPQR and ΔABC are congruent by SAS congruency rule.
Ques. In the figure given below, ΔABC and ΔPQR are congruent to each other by SAS congruency criteria. Find the value of x and y in the figure. (5 marks)
Ans.
ΔABC = ΔPQR (By SAS rule) (given)
So, AB = QR and AC = PR
⇒ 3x + 10 = 5y + 15 ……(i)
Also, ∠BAC = ∠QRP (given)
⇒ 2x + 15° = 5x – 60° ……(ii)
From eq. (ii), we have
2x + 15 = 5x – 60
⇒ 2x – 5x = -15 – 60
⇒ -3x = -7 5
⇒ x = 25
Putting the value of x in eq. (i), we have
3x + 10 = 5y + 15
⇒ 3 × 25 + 10 = 5y + 15
⇒ 75 + 10 = 5y + 15
⇒ 85 = 5y + 15
⇒ 85 – 15 = 5y
⇒ 70 = 5y
⇒ y = 14
Hence, the x=25 and y=14
Ques. A right-angled triangle of base 17cm and height 12cm. Calculate the area of the triangle. (2 marks)
Ans. area of triangle = ½ (base x height)
Here, base = 17cm and
Height = 12cm.
Therefore, area of the triangle = ½ (17 x 12)
= ½ (204) = 102cm2
Ques. In the figure given below, PR=QS and AP = BQ. Prove that ΔBQR is congruent to ΔAPS. (3 marks)
Ans.
In ΔBQR and ΔAPS,
PR = QS (given)
AP = BQ (given)
RS = RS (common)
Therefore, PR + RS = QS + RS
=> PS = QR
And ∠APS = 90° = ∠BQR (given)
So, by SAS congruency rule, ΔBQR and ΔAPS are congruent.
Ques. In the figure given below, prove that ΔADC and ΔABC are congruent. (2 marks)
Ans.
in the ΔABC and ΔADC,
∠DAC = ∠BAC (given)
∠DCA = ∠BCA (given)
Also, AC is common to both ΔABC and ΔADC.
Hence, by the ASA congruency rule, ΔABC and ΔADC are congruent to each other.
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