Different Rules and Formulas Relating the Sides and Angles of the Triangle: Triangles are the simplest polygons that are made of three line segments. Hence, triangles have three sides and three angles. They are classified based on their sides and angles into different types.

Different rules establish a relationship between the sides and angles of a triangle, such as trigonometric relations and the Pythagoras theorem. The lengths of the sides and the measure of the angles inside a triangle are also related. These relations help us find unknown parts when a situation arises. The inequality theorem helps in assuring the calculated sides using various methods are a perfect fit for the triangle.

Properties of Triangles

• A triangle is a \(3-\) sided polygon with \(3\) internal and \(3\) external angles.
• The angle sum property of a triangle states that all the internal angles of a triangle make a sum of \(180^{\circ}\).
• The exterior angle property states that an exterior angle of a triangle is equal to the sum of its opposite interior angles.

• The angle between two sides of a triangle can be greater than \(0\) to less than \(180^{\circ}\).
• The angle between two sides in a triangle cannot be exactly \(0\) or \(180^{\circ}\) because that would become a straight line.
• Two similar triangles have the same angles but different-length sides.
• Two congruent triangles have all sides and all angles equal.

Angle-Side Relation in a Triangle        

The sides and angles in any triangle are related to themselves as given below:

• The largest side in a triangle is that which lies opposite to the largest angle.
• The smallest side in a triangle is that which lies opposite to the smallest angle.
• The mid-sized side in a triangle is that which lies opposite to the mid-sized angle.

We can also show this as

• If two sides are congruent (equal in length), the two corresponding angles are congruent (equal in measure).

Relation Between the Sides and Angles

There are several ways to calculate the sides and angles of a triangle. To determine the length or angle of a triangle, formulas, mathematical rules, or the fact that all triangle angles add up to \(180^\circ \) can be used.

Methods for determining the sides and angles of a triangle are:

• Pythagoras theorem
• Sine rule
• Cosine rule
• Angle sum property

Pythagoras Theorem

We know that the hypotenuse of a right-angled triangle is the longest side and lies opposite to the right angle.

The Pythagorean theorem employs trigonometry to determine the longest side (hypotenuse) of a right triangle. The theorem states that the square of the length of the hypotenuse equals the sum of the squares of the other two sides.

If the sides of a triangle are \(a, b\), and \(c\), and \(c\) is the hypotenuse, then Pythagoras’ theorem can be stated as

\(c^{2}=a^{2}+b^{2}\)

\(c=\sqrt{a^{2}+b^{2}}\)

To find the measure of the hypotenuse,

Step 1: Get the lengths of the two sides.

Step 2:  Square the lengths.

Step 3: Add the resultant squares.

Step 4: Take the square root of that sum to get the length of the hypotenuse.

Sine Rule

The ratio of the length of a triangle side to the sine of the opposite angle is constant for all the three sides and angles in a triangle.

\(\frac{\mathrm{a}}{\sin A}=\frac{\mathrm{b}}{\sin B}=\frac{\mathrm{c}}{\sin C}\)

If we know the length of one side and the magnitude of the opposite angle, we can apply the sine rule. If any of the remaining angles or sides are known, all of the angles and sides can be calculated.

Cosine Rule

If \(a\) and \(b\) are known and \(C\) is the included angle, the angle between the sides of a triangle, then \(C\) can be calculated using the cosine rule. The following is the formula:

\(c^{2}=a^{2}+b^{2}-2 a b \,\cos \,C\)

It can be written for other angles as

\(b^{2}=a^{2}+c^{2}-2 a c \,\cos\, B\)

\(a^{2}=b^{2}+c^{2}-2 b c \,\cos\, A\)

We can apply the cosine rule when

We know the lengths of a triangle’s two sides and the included angle. The cosine rule can then be used to calculate the length of the remaining side.

We know all of the side lengths but none of the angles.

Angle Sum Property

The angle sum property of a triangle states that the sum of the three interior angles is equal to \(180^\circ \). If we know the two angles, we can calculate the third angle. Then, using the combination of previous rules, we can find the measure of the sides.

Triangle Inequality Theorem

The rule of triangle inequality theorem states that the sum of the lengths of two sides in a triangle is greater than the length of the third side. One of the following cases holds for all triangles:

• \(a+b>c\)
• \(b+c>a\)
• \(a+c>b\)

Once, the sides of triangles are determined, we can use this theorem to verify whether our calculated lengths can be the sides of a triangle.

PRACTICE EXAM QUESTIONS AT EMBIBE

Solved Examples of Different Rules and Formulas Relating the Sides and Angles of the Triangle

We have provided solved examples on Different Rules and Formulas Relating the Sides and Angles of the Triangle below:

Q.1. The length of two sides of a triangle is \(3\) and \(4\) metres long. What is the length of its hypotenuse?
Ans: Let the sides be \(a, b\), and \(c\) and \(c\) be the hypotenuse.
Then, by Pythagoras theorem, we know that
\(c^{2}=a^{2}+b^{2}\) Or \(c=\sqrt{a^{2}+b^{2}}\)
\(\Rightarrow c=\sqrt{3^{2}+4^{2}}=\sqrt{25}=5\)
Hence, the hypotenuse of the given triangle is \(\mathbf{5}\) metres long.

Q.2. What is the measure of the sides of the right triangular plot whose hypotenuse is \(10 \text {m}\) and has a base angle of \(30^\circ \)?
Ans: We are asked to find the sides of a right triangular plot which has a hypotenuse of \(10 \mathrm{~m}\) and a base angle of \(30^{\circ}\)
In a right-angled triangle,
\(\sin \theta=\frac{\text { Length of the opposite side }}{\text { Length of the hypotenuse side }}\)
Let the opposite side be \(x\).
Then, we have
\(\sin 30^{\circ}=\frac{x}{10}\)
\(\Rightarrow \frac{1}{2}=\frac{x}{10}\)
\(\Rightarrow x=5 \mathrm{~m}\)
Also, \(\cos \theta=\frac{\text { Length of adjacent side }}{\text { Length of Hypotenuse side }}\)
Let the adjacent side be \(y\).
\(\Rightarrow \cos 30^{\circ}=\frac{y}{10}\)
\(\Rightarrow \frac{\sqrt{3}}{2}=\frac{y}{10}\)
\(\Rightarrow y=5 \sqrt{3} \mathrm{~m}\)
Hence, the other two sides of the triangular park are \(5\) in and \(5 \sqrt{3}\) metres

Q.3. A triangle has the measure of the angles in the ratio \(5: 7: 8\). Find the angles.
Ans: Given ratio of sides is \(5: 7: 8\)
Let the angles be then, \(5 x, 7 x\) and \(8 x\).
By the angle-sum property of a triangle, the sum of the three interior angle is equal to \(180^{\circ}\).
\(\Rightarrow 5 x+7 x+8 x=180^{\circ}\)
\(\Rightarrow 20 x=180^{\circ}\)
\(\Rightarrow x=\frac{180^{\circ}}{20}=9^{\circ}\)
Hence, the three angles are \(5 \times 9^{\circ}=45^{\circ}, 7 \times 9^{\circ}=63^{\circ}\) and \(8 \times 9^{\circ}=72^{\circ}\)

Q.4. In triangle \(A B C, B=21^{\circ}, C=46^{\circ}\) and \(A B=9 \mathrm{~cm}\). Solve this triangle to get all sides.
Ans: We are provided with two angles and one side. So, we can use the sine rule to find the other sides.
By the angle-sum property of a triangle, the sum of the three interior angle is equal to \(180^\circ \).
\(\Rightarrow A=180^{\circ}-\left(21^{\circ}+46^{\circ}\right)=113^{\circ}\)
Now, \(c=A B=9\)
Using the sine rule
\(\frac{a}{\sin 113^{\circ}}=\frac{b}{\sin 21^{\circ}}=\frac{9}{\sin 46^{\circ}}\)
from which \(b=\sin 21^{\circ} \times \frac{9}{\sin 46^{\circ}}=4.484 \mathrm{~cm}\).
Similarly, \(a=\sin 113^{\circ} \times \frac{9}{\sin 46^{\circ}}=11.517 \mathrm{~cm}\)

Q.5. In triangle \(ABC, AB=42 \mathrm{~cm}, B C=37 \mathrm{~cm}\) and \(A C=26 \mathrm{~cm}\). Find the measure of angle \(A\).
Ans: We are given three sides of the triangle and asked to find the emasure of \(\angle A\). So, we can use the cosine rule.
\(a^{2}=b^{2}+c^{2}-2 b c \,\cos\, A\)
We have \(a=37, b=26\) and \(c=42\)
\(\Rightarrow 37^{2}=26^{2}+42^{2}-2(26)(42) \cos A\)
\(\Rightarrow \cos A=26^{2}+42^{2}-\frac{37^{2}}{(2)(26)(42)}=\frac{1071}{2184}=0.4904\)
\(\Rightarrow A=\cos ^{-1} 0.4904=60.63^{\circ}\)
Hence, the measure of angle \(A\) is \(60.63^{\circ}\)

Summary

Triangles are the simplest polygon with three sides, three angles and three vertices. The sides and angles of a triangle are related to itself in many ways. Trigonometric relations can be used to find the sides or angles in a right-angled triangle. We can use different rules to connect the sides and angles of other triangles and find the unknowns when a situation comes.

For any triangle, the sum of interior angles and exterior angles is equal to \(180^\circ \). All unknown angles can be found using these properties. We can make use of the Pythagoras theorem, sine rule, cosine rule and angle sum property to find the missing angles or sides. The triangle inequality theorem can be utilized to verify whether the calculated lengths can be the sides of the triangle.

FAQs on Different Rules and Formulas Relating the Sides and Angles of the Triangle

A few frequently asked questions regarding different rules and formulas relating to the sides and angles of the triangle are given below:

Q.1. How many sides are there in a triangle?
Ans: A triangle has three sides. It is the smallest polygon.   

Q.2. Which triangle has two sides?
Ans: No triangle has two sides. A triangle is the simplest polygon and cannot be constructed with only two sides.

Q.3. What do the \(3\) angles of a triangle add up to?
Ans: The sum of the three interior angles in a triangle is \(180^\circ \).

Q.4. How to find the angle of a triangle given \(3\) sides?
Ans: When three sides are given, we can use the cosine rule to find the measure of the angles. If \(a\) and \(b\) are known, and \(C\) is the included angle, the angle between the sides of a triangle, then \(C\) can be calculated using the cosine rule as \(c^{2}=a^{2}+b^{2}-2 a b \,\cos\, C\). It can be written for other angles as well as \(b^{2}=a^{2}+c^{2}-2 a c \,\cos\, B, a^{2}=b^{2}+c^{2}-2 b c \,\cos\, A\)

Q.5. What is the sine rule in a triangle?
Ans: The ratio of the length of a triangle side to the sine of the opposite angle is constant for all three sides and angles.
\(\frac{\mathrm{a}}{\sin A}=\frac{\mathrm{b}}{\sin B}=\frac{\mathrm{c}}{\sin C}\)
If we know the length of one side and the magnitude of the opposite angle, we can apply the sine rule. All angles and sides can be calculated if any of the remaining angles or sides are known.

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