Chapter 10.1, Problem 11E

To determine

To find the value of x and classify the triangle

Answer to Problem 11E

The value of $x={23}^{\circ }$

The triangle is a scalene triangle

Explanation of Solution

Given information:

One angle of the triangle is given as ${\left(3x+51\right)}^{\circ }$

Second angle is given as $x^{\circ }$

Third angle is given as ${37}^{\circ }$

Calculation:

The sum of all the angles of a triangle is ${180}^{\circ }$

This gives us-

  ${\left(3x+51\right)}^{\circ }+x^{\circ }+{37}^{\circ }={180}^{\circ }$

Add both numbers and both the x together will give us

  $4x^{\circ }+{88}^{\circ }={180}^{\circ }$

Subtract 88 from both the sides

  $4x^{\circ }+{88}^{\circ }-{88}^{\circ }={180}^{\circ }-{88}^{\circ }$

This gives us −

  $4x^{\circ }={92}^{\circ }$

Now, divide both sides by 4

  ${\frac{4x}{4}}^{\circ }=\frac{{92}^{\circ }}{4}$

This will give us-

  $x={23}^{\circ }$

Since, the first angle of the triangle is given as ${\left(3x+51\right)}^{\circ }$

To find the value of the angle , multiply 3 with 23 and then add 51 to it

  $3\times {23}^{\circ }+51={120}^{\circ }$

So, the first angle is ${120}^{\circ }$

Second angle of the triangle is given as x and the value of x is ${23}^{\circ }$

Conclusion- The value of $x={23}^{\circ }$ and the first angle measurement is ${120}^{\circ }$

So first angle is ${120}^{\circ }$ , Second angle is ${23}^{\circ }$ and the third angle is ${37}^{\circ }$

Since, all the three angles of the triangle are different

Hence, it is scalene triangle