The diagram shows a triangle. Given that all the angles are in degrees, find the size of the smallest angle in the
We’ve been given expressions for the sizes of the three angles of the triangle all in
terms of the variable 𝑥. In order to find the size of the smallest angle in the triangle, we first need to
find the value of 𝑥. To do so, we need to recall a key fact of our angles in triangles, which is that the
angles’ sum in any triangle is 180 degrees.
We can, therefore, form an equation by adding together the expressions for the three
angles: 𝑥 plus 35 plus four 𝑥 plus 73 plus three 𝑥 plus 16 is equal to 180. We can now simplify this equation. One 𝑥 plus four 𝑥 plus three 𝑥 is equal to eight 𝑥. And to find the sum of 35, 73, and 16, we can use a column method. In the units column, we have five plus three which is eight plus six which makes a
total of 14. In the tens column, three plus seven is 10 plus one is 11 and then plus the one we’ve
carried over gives a total of 12.
The equation simplifies to eight 𝑥 plus 124 is equal to 180. To solve this equation, we first subtract 124 from each side, giving eight 𝑥 is
equal to 56. Next, we divide by eight. And as 56 divided by eight is equal to seven, we found the value of 𝑥: 𝑥 is equal
to seven. Now that we know the value of 𝑥, we need to return to the triangle to consider which
angle is the smallest.
It’s clear which angle is the largest. It’s the angle of four 𝑥 plus 73 degrees as it’s the only obtuse angle in the
triangle. It also has the largest number of 𝑥s and the largest number added to this. So it’s clearly the biggest angle. It isn’t clear though which of the other two angles is the smaller angle. So we need to evaluate them both. For the angle of 𝑥 plus 35 degrees, this becomes seven plus 35 degrees which is 42
degrees. For the angle of three 𝑥 plus 16 degrees, this is three multiplied by seven plus 16
degrees which is 21 plus 16 degrees which is 37 degrees.
37 is smaller than 42. So the smallest angle in the triangle is 37 degrees. For a little extra confidence in our answer, we could evaluate the third angle in the
triangle even though we know it’s the largest. Four multiplied by seven plus 73 gives 101 degrees for the final angle in the
triangle. If we checked the sum of our three calculated angles, 101 plus 42 plus 37 is indeed
equal to 180. So we have the correct angle sum for the triangle. And this just gives us a little bit more confidence in our answer.