WILL GIVE BRAINLIEST! (See attachment)A. If AB is 20 and BC is 21, based only on the appearance of the figure, what is a good estimate for the length of AC? How do you know?B. If the actual length of AC is exactly what you computed in part A above, what is the measure of angle ABC? How do you know?C. Given the information from A and B above, if AE is 10 and A.F is half of AC, what is the special name for segment EF as it relates to triangle ABC?D. Given the result from C above, if the measure of angle EFB is 43.6°, what is the measure of angle FBC? How do you know?E. If Triangle DCB is congruent to triangle ABC, what kind of shape is ABCD? How do you know?

Part A:

From the appearance of the figure above, triangle A B C forms a right angle at B with A B and B C being the legs and A C being the hypothenuse.

Given that A B is 20 and B C is 21, by the pythagoras theorem,

[tex]A B^2+B C^2=A C^2 \\ \\ \Rightarrow20^2+21^2=A C^2 \\ \\ \Rightarrow400+441=A C^2 \\ \\ \Rightarrow841=A C^2 \\ \\ \Rightarrow A C=\sqrt{841}=29[/tex]



Part B:

From the appearance of the figure above, triangle A B C forms a right angle at B with A B and B C as the legs and A C as the hypothenuse.

If this is true, then the measure of angle A B C is 90 degrees.



Part C:

If A E is 10 and A F is half of A C, the special name for segment E F as it relates to triangle A B C is the midsegment.

The midsegment of a triangle is a line segment which joins the midpoints of two sides of the triangle.



Part D:

Given from part C above that line segment EF is a midsegment of triangle ABC, from the triangle midsegment theorem, line segment EF is parallel to side BC.

Thus, line BF is a transverse of parallel lines EF and BC which makes angle EFB alternate to angle FBC.

Since alternate angles are equal, given that angle EFB is 43.6°, then the measure of angle FBC  is also 43.6° because they are alternate angles.



Part E:

If Triangle DCB is congruent to triangle ABC, then angle B is congruent to angle C.
Given from part B that angle B = 90 degrees, then angle C = 90 degrees. Thus, the two adjacent angles of the quadrilateral ABCD = 90 degrees. Whih shows that the shape ABCD is a rectangle.