Relationship between Sides and Angles of a Triangle

How to compare the sides and angles of a triangle by using their relationship: 3 examples and their solutions.

Example 1

Example

Solution

Compare the given sides:
8 > 7 > 5.

The larger angle lies opposite the longer side.

8 is the longest side.

Then the angle opposite to the side 8, ∠A,
is the largest angle.

7 is the next longer side.

Then the angle opposite to the side 7, ∠B,
is the next larger angle.

5 is the shortest side.

Then the angle opposite to the side 5, ∠C,
is the smallest angle.

So m∠A > m∠B > m∠C.

Example 2

Example

Solution

Compare the measures of the interior angles:
66 > 62 > 52.

The longer side lies opposite the larger angle.

66º is the largest angle.

Then the side opposite to the angle 66º, AB,
is the longest side.

62º is the next larger angle.

Then the side opposite to the angle 62º, BC,
is the next longer side.

52º is the smallest angle.

Then the side opposite to the angle 52º, AC,
is the shortest side.

So AB > BC > AC.

Example 3

Example

Solution

Compare the measures of the interior angles:
67 = 67 > 46.

The longer side lies opposite the larger angle.

67º is the larger angle.

Then the side opposite to the left angle 67º, AC,
is the longer side.

The other 67º is the larger angle.

Then the side opposite to the right angle 67º, AB,
is the longer side.

46º is the smaller angle.

Then the side opposite to the angle 46º, BC,
is the shorter side.

So AC = AB > BC.