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Answer: 0.5 < x < 16.5.
Step-by-step explanation:
Given: Two sides of triangle: 8.0 units and 8.5 units
Measure of third side = x
According to the triangle's inequality,
- Sum of any two sides > third side. (i)
- Difference between the sides < third side. (ii)
If x is the third side, then
x < 8+8.5 [Using (i)]
i.e. x< 16.5
Also, x > 8.5-8 [Using (ii)]
i.e. x> 0.5
Hence, Range of possible sizes for side x would be 0.5 < x < 16.5.
Answer for part A is 
which in English means "angle A is congruent to angle D"
The reason why we know that angle A and angle D are congruent is because they are both right angles. By the right angle congruence theorem (the reason just below the blank for part A), we can say that angle A = angle D. They are both 90 degree angles.
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Answer for the blank in part B is "Given" without quotation marks of course. This is the first "given" while the second given is coming up in part C
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For part C, the other given is
so that's what will go in the blank. It may seem silly, but all we're doing is repeating verbatim what is given to us. With any proof, you always start with what you are given and somehow connect it with what you're trying to prove
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Finally, the answer for the blank in part D is "AAS Congruence Theorem" (without quotes)
The AAS congruence theorem allows us to prove two triangles congruent if we know if two pairs of angles are congruent plus if we know two sides are congruent as well.
The angle portions are covered by part A and part C. The side portions are part B's territory. Combining A, B, & C allows us to use the AAS theorem
Note: It is NOT the ASA theorem because the two angles do not sandwich the sides. In other words, the side AB is not between the two angles A and C. Similarly, DE is not between the two angles D and F. The order is very important as ASA is slightly different than AAS.
The reason why we know that angle A and angle D are congruent is because they are both right angles. By the right angle congruence theorem (the reason just below the blank for part A), we can say that angle A = angle D. They are both 90 degree angles.
------------------------------------------------
Answer for the blank in part B is "Given" without quotation marks of course. This is the first "given" while the second given is coming up in part C
------------------------------------------------
For part C, the other given is
------------------------------------------------
Finally, the answer for the blank in part D is "AAS Congruence Theorem" (without quotes)
The AAS congruence theorem allows us to prove two triangles congruent if we know if two pairs of angles are congruent plus if we know two sides are congruent as well.
The angle portions are covered by part A and part C. The side portions are part B's territory. Combining A, B, & C allows us to use the AAS theorem
Note: It is NOT the ASA theorem because the two angles do not sandwich the sides. In other words, the side AB is not between the two angles A and C. Similarly, DE is not between the two angles D and F. The order is very important as ASA is slightly different than AAS.