4. in a triangle, let p be the location where two angle bisectors meet. must p be on the third angle bisector as well? explain your reasoning.

P must be on the third angle bisector as well

Step-by-step explanation:

The point P must be on the third angle bisector as well, because it is a theorem in Euclidean geometry that the three interior angle bisectors of a triangle meet in a single point.

This point is called the incenter.  In Euclid's Elements, Proposition 4 of Book IV proves that this point is also the center of the inscribed circle of the triangle.

The incenter is defined as the point where the internal angle bisectors of the triangle cross.

the height of the surface increases from the center out to the sides of the road.

step-by-step explanation:

as we move out we move away from the x-axis.

and as mentioned in the problem statement that height of a road increases when we move away from the center x.

hence proved that 3rd option

the height of the surface increases from the center out to the sides of the road.

is the correct option.