Question

The angles of a triangle have the ratio \[3:2:1.\] what is the measure of the smallest angle?

Answer 1

Hint: First, let us assume the ratio values to with unknown integer values. Then we need to apply the triangle angle sum theorem which states that the sum of unknown values is equal to \[{180^ \circ }\] . Find \[t\] value and apply it to the conversion of ratio values. Finally, we will obtain the required smallest angle.

Complete step-by-step answer:
Given that the angles of a triangle have the ratio \[3:2:1.\]
Let assume that some integer or angle value is in \[t\] .
We include the \[t\] in ratio values.
It means that,
 \[3t,\,2t,\,t\]
First of all, we find the \[t\] value.
Now we are used to one of the mathematical theorems called the triangle angle sum theorem.
The Triangle angle sum theorem states that the sum of all the three interior angles of a triangle is always equal to \[180\] degrees.
It means \[a + b + c = {180^ \circ }\] .
We find the \[t\] value used in the triangle angle sum theorem.
And assume the \[3t,\,2t,\,t\] value \[a,\,b,\,c\] is in respectively.
 \[
  a = 3t \\
  b = 2t \\
  c = t \\
 \]
The Sum of the \[a,\,b,\,\] and \[c\] value is equal to the \[180\] degree.
 \[3t + 2t + t = {180^ \circ }\]
Add to left-hand side terms on,
 \[6t = {180^ \circ }\]
Divide to \[6\] into both sides,
 \[\dfrac{6}{6}t = \dfrac{{{{180}^ \circ }}}{6}\]
 \[t = {30^ \circ }\]
Now we get \[t\] value is \[{30^ \circ }\] .
Now find the \[a,\,b,\,c\] angles.
First find to angle \[a\] .
We apply the \[t\] value in angle \[a\] .
 \[a = 3t\]
 \[a = 3({30^ \circ })\]
 \[ = {90^ \circ }\]
So that \[a = {90^ \circ }\]
Next find to angle \[b\]
We apply the \[t\] value in angle \[b\] .
 \[b = 2t\]
 \[b = 2({30^ \circ })\]
 \[ = {60^ \circ }\]
So that \[b = {60^ \circ }\]
Next find the angle \[c\] .
We apply the \[t\] value in angle \[c\]
 \[c = t\]
 \[c = {30^ \circ }\]
So that \[c = {30^ \circ }\]
Now we find the angles of \[a,\,b,\,c\] .
The angles of a triangle \[{90^ \circ },\,{60^ \circ },\,{30^ \circ }\] are.
Now we find which one is the smallest angle. In the decreasing order to \[{30^ \circ }\] is the smallest angle.
The smallest angle of a triangle is \[{30^ \circ }\] .

Note: We include some unknown integer \[t\] ratio values. The used triangle angle sum theorem is to find the \[t\] value. Then apply the \[t\] value to ratio values to find the angles. We will get three angles and just look at the angles. And we need to pick the smallest angle among the three angles and it is the required answer.