Question

1.In triangle TRS, VZ = 6 inches. What is RZ?

Answer 1

1. C
2. B,C,E

Hope that helps.

Answer 2

Part 1)

we know that

The Centroid of a Triangle is the centre of the triangle that can be calculated as the point of intersection of all the three medians of a triangle.

The Centroid divides each median into two segments whose lengths are in the ratio $2:1$

so

$VZ=\frac{1}{3}RV$

$RV=3VZ$

we have

$VZ=6\ in$

substitute

$RV=3*6=18\ in$

Find RZ

$RZ=\frac{2}{3}RV$

$RZ=\frac{2}{3}*18=12\ in$

therefore

the answer Part 1) is the option C

$12\ in$

Part 2)

Statements

case A) ∠BEC is an exterior angle

The statement is False

Because, ∠BEC is a internal angle

case B) ∠DEC is an exterior angle.

we know that

An exterior angle is formed by one side of a triangle and the extension of another side

therefore

The statement is True

case C) ∠ABE and ∠EBC are supplementary angles.

we know that

∠ABE+∠EBC=$180\°$ -------> by supplementary angles

therefore

The statement is True

case D) ∠BCF and ∠BEC are supplementary angles  

The statement is False

Because

the only way that is true is that the triangle BEC is isosceles and that the ∠BEC is equal to the ∠BCE  

case E) ∠BEC is a remote interior angle to exterior F.∠BCF

we know that

Remote interior angles are the interior angles of a triangle that are not adjacent to a given angle. Each interior angle of a triangle has two remote exterior angles.

In this problem  ∠BEC has two remote exterior angles (∠BCF and ∠EBA)

therefore

The statement is True