Triangles: Basic Concepts and Properties

Try this

Drag the ends of the two rods to where the two dashed arcs cross: they touch, and you have a triangle. Each dashed arc shows every place the end of a rod can reach.
Slowly stretch the base $a$ . The crossing point of the arcs sinks lower and lower, and at $a = b + c$ it lands on the base: the three rods lie flat along one segment. Any longer and the arcs no longer cross.
Shrink the base to $2$ , set the right rod to $5$ and the left rod to $2$ . This time the right rod is too long: $a + c < b$ , and the left rod can never reach its end, however you turn it. You need all three inequalities.
Once you have a triangle, look at the angle readings: the longest rod is opposite the largest angle.

Three segments $a$ , $b$ , $c$ make a triangle if and only if any two of them together are longer than the third. In fact you only need to check one: the two shorter ones together must be longer than the longest one.

Predict first

Two sticks have lengths $5$ and $9$ . The third stick has a whole-number length and the three must make a triangle. How many choices are there?

8

9

10

13

Choose an answer first; then you can see the result.

Why: the shortest path is straight

From “no triangle” to “triangle”

Step 1 / 4

The two rods are too short
The base is $a = 7$ and the rods are $b = 2$ and $c = 3$ . Laid flat and pointing at each other, they are as close as they can get, and there is still a gap of $2$ .

$b + c = 5 < 7 = a$
Just long enough
A little longer and it bends up
The reason

The three angles

Drag the vertices of the triangle below. Each angle changes. What about their sum?

The purple angle is the exterior angle at

C

. Which two angles is it related to?

Why the angles add up to 180°

Step 1 / 4

Three angles
We want to prove $∠ A + ∠ B + ∠ C = 18 0^{\circ}$ . Measuring shows it is true, but measuring cannot show it is true “always”.

$∠ A + ∠ B + ∠ C = ?$
Draw the line through A parallel to BC
Alternate angles are equal
Together they make a straight angle

Two direct consequences:

An exterior angle equals the sum of the two interior angles that are not next to it (the purple angle above $= ∠ A + ∠ B$ ), so it is bigger than either of them.
The average of the three angles is $6 0^{\circ}$ , so every triangle has at least one angle of at least $6 0^{\circ}$ , and at least one angle of at most $6 0^{\circ}$ . This is a pigeonhole-style idea: when the total is fixed, not everyone can be below the average.

Remove one condition

Break it

Counterexample lab: remove one condition

Conclusion: The three segments make a triangle

$a = 6$ , $b = 3.5$ , $c = 4.5$ : $3.5 + 4.5 > 6$ . Click “Close it as far as possible”.

Practice

Exercise 1Basic

An isosceles triangle has two sides of length $3$ and $7$ . Find its perimeter.

Exercise 2Basic

The three angles of a triangle are in the ratio $2 : 3 : 4$ . How many degrees is the largest angle?

Exercise 3Advanced

There are five segments of lengths $2$ , $3$ , $4$ , $5$ , $6$ . In how many ways can you choose three of them that make a triangle?

Exercise 4Challenge

In $△ A B C$ , $∠ A = 7 0^{\circ}$ , and the bisectors of $∠ B$ and $∠ C$ meet at $I$ . Find $∠ B I C$ in degrees.

Try this

Why: the shortest path is straight

From “no triangle” to “triangle”

The three angles

Why the angles add up to 180°

Remove one condition

Counterexample lab: remove one condition

Practice

Mistake Museum · Exhibits from this stop

Checking only one inequality

Splitting an isosceles triangle into cases and forgetting to check them