Three side lengths, when they fit at all, give exactly one triangle.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Three angles give many
Angles fix the shape but not the size, so 40-60-80 describes infinitely many triangles of different sizes.
Sometimes none
Sides of 2, 3 and 9 cannot close. The two short sides together are shorter than the long one.
The test
Any two sides added must be longer than the third. That is the triangle inequality.
One triangle, many, or none
Three given measurements can determine exactly one triangle, several different triangles, or no triangle at all. Which happens depends on which measurements you are given, and it is not obvious in advance.
Three sides fix the triangle — if they fit
Three side lengths give exactly one triangle, provided the two shorter ones add to more than the longest. Sides of 2, 3 and 10 make no triangle at all, because the short sides cannot reach. That condition is the triangle inequality.
Three angles fix only the shape
Angles of 60, 60 and 60 describe infinitely many triangles of every size. Angles determine shape but not size, which is exactly what similar figures are. To pin down a size you need at least one length.
Why this matters
Engineers and surveyors need to know when measurements determine a structure uniquely. A triangle fixed by three sides is rigid, which is why triangles appear in bridges and roof trusses while rectangles need diagonal bracing.
Step 2: Try It Yourself
Tap and try it out.
40° is an acute angle.
The two angles together measure 100°. Angle measures add.
Step 3: Watch an Example
One step at a time.
Watch Leo Test Three Sets
Leo checks 3-4-5, then 40-60-80 degrees, then sides 2-3-9.
- Step 1
3 and 4 add to 7, which beats 5, so 3-4-5 closes into one triangle.
Step 4: Your Turn
Practice makes it stick.
The Third Angle
Problem 1 of 2
A triangle has angles of 50° and 60°. What is the third, in degrees?
The Impossible Triangle
Problem 2 of 2
Can sides 2, 3 and 9 make a triangle? Answer 1 for yes, 0 for no.
Does It Close
1 of 4
Angles 30° and 90°. What is the third?
2 of 4
Can sides 5, 6 and 10 make a triangle? 1 for yes, 0 for no.
3 of 4
Can sides 1, 2 and 8 make a triangle? 1 for yes, 0 for no.
4 of 4
Do three given angles fix the size of a triangle? 1 for yes, 0 for no.
Step 5: Quick Check
Show what you know.
Question 1 of 1
Can sides 4, 4 and 9 make a triangle? 1 for yes, 0 for no.
What You Learned
- Three sides that satisfy the triangle inequality give exactly one triangle.
- Three angles give infinitely many, all the same shape at different sizes.
- Any two sides added must beat the third, or nothing closes.