Rectangular box has a horizontal base EFGH where HG = 15 cm,GF =8cm and BF=7cm. X is a point on AB such that XB=4cm. Calculate the angle...

First, the rectangular base EFGH is parallel to the
rectangular top ABCD, therefore AB=EF=GH = 15cm.

Since the
lateral sides are also rectangles, therefore BF is parallel to CG => BF = CG =
7cm.

The triangle CGE is a right angled and the angle G
measures 90 degrees.

To calculate the included angle CEG,
we'll calculate the length of the side EG, using the right angled triangle EFG, where EG
is the hypotenuse.

EG = `sqrt(15^2 +
8^2)`

EG = `sqrt(289)`

EG =
17

We'll keep only the positive result, since a length of
aide cannot be negative.

We'll return to the right angle
triangle CEG, where EG and CG are the legs of triangle. To determine the measure of the
angle CEG, we'll use tangent function:

tan CEG =
CG/EG

tan CEG = 7/17

tan CEG =
0.4117

CEG = 22.37 degrees

To
calculate the measure of GXF, we need to determine the length of FX and
GX.

The length of GX can be determine from the right angle
triangle GCX. We need first to determine CX, from the top right angle triangle
CBX.

CX = `sqrt(4^2 + 8^2)`

CX
= `sqrt(80)`

CX =
4`sqrt(5)`

Now, we'll determine XG, from the right angle
triangle GCX:

GX =
`sqrt(80+49)`

GX =
`sqrt(129)`

We'll determine the length of FX from the
triangle FBX.

FX =
`sqrt(16+49)`

FX =
`sqrt(65)`

We'll use the law of cosine to determine the
measure of GXF:

FG^2 = FX^2 + GX^2 - 2FX*GX*cos
GXF

64 = 65 + 129 -
2*65*129*cosGXF

-130 =
-16770*cosGXF

cosGXF =
130/16770

cosGXF = 0.00775

GXF
= 89.55 degrees

Therefore, the angle CEG
measures 22.37 degrees and the angle GXF measures 89.55
degrees.