Answer

Verified

422.4k+ views

Hint: The given problem is related to the angle sum property of a triangle. The sum of the three interior angles of a triangle is equal to ${{180}^{o}}$ .

Complete step-by-step answer:

We will consider a triangle $\Delta XYZ$. The angles formed by any two sides of the triangle and lying on the inside of the triangle are called internal angles of a triangle. In $\Delta XYZ$, the angles $\measuredangle XYZ,\measuredangle YXZ$ and $\measuredangle XZY$ are the internal angles.

The angle sum property of a triangle states that the sum of internal angles of a triangle will always be equal to ${{180}^{o}}$. This property is valid for all the triangles. It is a necessary (but not sufficient) condition for a polygon to be a triangle. If the sum of internal angles of a polygon is not equal to ${{180}^{o}}$, then that polygon can never be a triangle. But the inverse is not necessarily true, i.e. if the sum of internal angles of a polygon is equal to ${{180}^{o}}$ , then it is not necessary that the polygon is a triangle. According to the angle sum property of a triangle, $\measuredangle XYZ+\measuredangle YXZ+\measuredangle XZY={{180}^{o}}$ .

Now, the triangle given in the question is $\Delta ABC$ . It is also given that the measures of angles $\measuredangle A$ and $\measuredangle B$ are ${{75}^{o}}$and ${{65}^{o}}$ respectively. According to the angle sum property of a triangle, the sum of internal angles of $\Delta ABC$ is equal to ${{180}^{o}}$, i.e. $\measuredangle A+\measuredangle B+\measuredangle C={{180}^{o}}$ . But we are given that $\measuredangle A={{75}^{o}}$ and $\measuredangle B={{65}^{o}}$. So, ${{75}^{o}}+{{65}^{o}}+\measuredangle C={{180}^{o}}$ .

$\Rightarrow {{140}^{o}}+\measuredangle C={{180}^{o}}$

$\Rightarrow \measuredangle C={{180}^{o}}-{{140}^{o}}={{40}^{o}}$

So, the measure of $\measuredangle C$ is equal to ${{40}^{o}}$.

Note: Always make sure that calculation mistakes are not present while solving the questions. Generally, students make calculation mistakes while solving a problem in a hurry. So, try to avoid calculation mistakes.

Complete step-by-step answer:

We will consider a triangle $\Delta XYZ$. The angles formed by any two sides of the triangle and lying on the inside of the triangle are called internal angles of a triangle. In $\Delta XYZ$, the angles $\measuredangle XYZ,\measuredangle YXZ$ and $\measuredangle XZY$ are the internal angles.

The angle sum property of a triangle states that the sum of internal angles of a triangle will always be equal to ${{180}^{o}}$. This property is valid for all the triangles. It is a necessary (but not sufficient) condition for a polygon to be a triangle. If the sum of internal angles of a polygon is not equal to ${{180}^{o}}$, then that polygon can never be a triangle. But the inverse is not necessarily true, i.e. if the sum of internal angles of a polygon is equal to ${{180}^{o}}$ , then it is not necessary that the polygon is a triangle. According to the angle sum property of a triangle, $\measuredangle XYZ+\measuredangle YXZ+\measuredangle XZY={{180}^{o}}$ .

Now, the triangle given in the question is $\Delta ABC$ . It is also given that the measures of angles $\measuredangle A$ and $\measuredangle B$ are ${{75}^{o}}$and ${{65}^{o}}$ respectively. According to the angle sum property of a triangle, the sum of internal angles of $\Delta ABC$ is equal to ${{180}^{o}}$, i.e. $\measuredangle A+\measuredangle B+\measuredangle C={{180}^{o}}$ . But we are given that $\measuredangle A={{75}^{o}}$ and $\measuredangle B={{65}^{o}}$. So, ${{75}^{o}}+{{65}^{o}}+\measuredangle C={{180}^{o}}$ .

$\Rightarrow {{140}^{o}}+\measuredangle C={{180}^{o}}$

$\Rightarrow \measuredangle C={{180}^{o}}-{{140}^{o}}={{40}^{o}}$

So, the measure of $\measuredangle C$ is equal to ${{40}^{o}}$.

Note: Always make sure that calculation mistakes are not present while solving the questions. Generally, students make calculation mistakes while solving a problem in a hurry. So, try to avoid calculation mistakes.

Recently Updated Pages

The branch of science which deals with nature and natural class 10 physics CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Define absolute refractive index of a medium

Find out what do the algal bloom and redtides sign class 10 biology CBSE

Prove that the function fleft x right xn is continuous class 12 maths CBSE

Find the values of other five trigonometric functions class 10 maths CBSE

Trending doubts

List out three methods of soil conservation

Difference Between Plant Cell and Animal Cell

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Which are the different sources of History other than class 8 social science CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Aeroplanes fly in which of the following layers of class 7 social science CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Mention the importance of the Himalayas class 10 social science CBSE

Quite a few species of plants and animals are endangered class 9 social science CBSE