Math Doubts

Acute Angled Triangle

acute angled triangle

Definition

A triangle which contains acute angles as interior angles is called an acute angled triangle.

A triangle can have all three angles acute angles. It means each angle is less than $90^\circ$. Hence, the triangle is known as acute angled triangle.

Properties

$\Delta UVW$ is a fundamental example triangle to study the properties of the acute angled triangle.

acute angled triangle

The sides $\overline{UV}$, $\overline{VW}$ and $\overline{WU}$ formed three interior angles at three vertices $U$, $V$ and $W$.

  1. The interior angle ($\angle UVW$) is $80^\circ$
  2. The interior angle ($\angle VWU$) is $46^\circ$
  3. The interior angle ($\angle WUV$) is $54^\circ$

The three interior angles are less than $90^\circ$. So, they are known as acute angles. Due to this property, the triangle $UVW$ is known as an acute angled triangle.

Construction

An acute angled triangle can be constructed geometrically by using geometric tools.

construction of acute angled triangle
  1. Use ruler and draw a straight line of any length horizontally. In this example, $8$ Centimeters horizontal line is drawn and the endpoints are named as $X$ and $Y$.
  2. Use protractor and coincide centre of the protractor with point $X$ and also coincide its right side base line with horizontal line. Now consider bottom scale, identity $45^\circ$ angle and then mark it. Draw a straight line from point $X$ through $45^\circ$ angle mark.
  3. Use protractor again, coincide middle point of protractor with point $Y$ and also coincide left side base line with horizontal line. Now consider top scale, identity $65^\circ$ angle and then mark it. Draw a straight line from point $Y$ through $65^\circ$ angle mark.
  4. The lines drawn from point $X$ and point $Y$ intersect at a point $Z$.

Thus, a triangle, represented as $\angle XYZ$ is constructed geometrically. The points $X$, $Y$ and $Z$ are known as vertices, and the line segments $\overline{XY}$, $\overline{YZ}$ and $\overline{ZX}$ are sides of the triangle.

Interior angles
  1. The $\angle ZXY$ is formed by the sides $\overline{XZ}$ and $\overline{XY}$. It is taken as $45^\circ$ to construct the triangle.
  2. The $\angle XYZ$ is formed by the sides $\overline{XY}$ and $\overline{YZ}$. It is taken as $65^\circ$ to construct the triangle.
  3. The $\angle YZX$ is formed by the sides $\overline{XZ}$ and $\overline{YZ}$ but the interior angle is unknown. So, measure the angle by using protractor and it is measured as $70^\circ$.

Observe each interior angle and all three angles are less than $90^\circ$. It means they are acute angles and the triangle is called as acute angled triangle by this reason.



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