Sum of the Length of Two Sides of a Triangle

The length of two sides of a triangle are $10 \mathrm{cm} \mathrm{and} 8 \mathrm{cm}$. Between what two measures should the length of the third side lie?

The length of two sides of a triangle are $10 \mathrm{cm} \mathrm{and} 15 \mathrm{cm}$. Between what two measures should the length of the third side lie?

Take any point $O$ in the interior of $△ABC$ as shown. State which of the following is true?

$AB+AC+BC<2(AO+BO+CO)$

Take any point $O$ in the interior of $△ABC$ as shown. State which of the following is true?

$AO+CO>AC$

Take any point $O$ in the interior of $△ABC$ as shown. State which of the following is true?

$BO+CO>BC$

Take any point $O$ in the interior of $△ABC$ as shown. State which of the following is true?

$AO+BO>AB$

We know in a triangle sum of any two sides is greater than the third side. Is the sum of any two angles of a triangle also greater than the third angle? Justify or contradict with an example.

In $△\mathrm{PQR}, \mathrm{PS}$ is the median of the triangle. Which one of the following is always true?

$PQ+QR+PR>PS$

In $△\mathrm{PQR}, \mathrm{PS}$ is the median of the triangle. Which one of the following is always true?

$PQ+QR+PR>2PS$

In $△\mathrm{PQR}, \mathrm{PS}$ is the median of the triangle. Which one of the following is always true?

$PQ+QR+PR>3PS$

Is it possible to have a triangle with the following sides?

$3 \mathrm{cm}, 4 \mathrm{cm} \mathrm{and} 5 \mathrm{cm}$

Is it possible to have a triangle with the following sides?

$1 \mathrm{cm}, 1 \mathrm{cm} \mathrm{and} 1 \mathrm{cm}$

Is it possible to have a triangle with the following sides?

$5 \mathrm{cm}, 12 \mathrm{cm} \mathrm{and} 18 \mathrm{cm}$

Is it possible to have a triangle with the following sides?

$6 \mathrm{cm}, 8 \mathrm{cm} \mathrm{and} 9 \mathrm{cm}$

The sides of a triangle have lengths (in $\mathrm{cm}$) $10,6.5$ and $a$, where $a$ is a whole number. The minimum value that $a$ can take is:

If one side of a triangle is $8 \mathrm{cm}$, then the sum of the other two sides must be greater than $8 \mathrm{cm}$. 

The difference of two sides of a triangle is less than the third side.

A triangle whose two sides measure $4 \mathrm{cm}$ and $400 \mathrm{cm}$ is not possible.

The sum of any two sides of a triangle is:

The difference between the lengths of any two sides of a triangle is smaller than the length of third side.