1. 5 and 9
Let side a = 5. Let side b = 9. Then the triangle has
sides
a=5, b=9, and c=?. Any side of a triangle must be less
than the other two sides added together, so
a < b+c, b < a+c, and c < a+b
Substituting:
5 < 9+c, 9 < 5+c, and c < 5+9
Solving all three for c:
-4 < c, 4 < c, c < 14
The first one is of course true, so ignore it.
Put the second and third together and get
4 < c < 14
That's it. c must be more than 4 and less than 14.
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2. 7 and 14
Let side a = 7. Let side b = 14. Then the triangle
has sides
a=7, b=14, and c=?. Any side of a triangle must be less
than the other two sides added together, so
a < b+c, b < a+c, and c < a+b
Substituting:
7 < 14+c, 14 < 7+c, and c < 7+14
Solving all three for c:
-7 < c, 7 < c, c < 21
The first one is of course true, so ignore it.
Put the second and third together and get
7 < c < 21
That's it. c must be more than 7 and less than 21.
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3. 8 and 13
Let side a = 8. Let side b = 13. Then the triangle
has sides
a=8, b=13, and c=?. Any side of a triangle must be less
than the other two sides added together, so
a < b+c, b < a+c, and c < a+b
Substituting:
8 < 13+c, 13 < 8+c, and c < 8+13
Solving all three for c:
-5 < c, 5 < c, c < 21
The first one is of course true, so ignore it.
Put the second and third together and get
5 < c < 21
That's it. c must be more than 5 and less than 21.
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4. 10 and 12
Let side a = 10. Let side b = 12. Then the triangle
has sides
a=10, b=12, and c=?. Any side of a triangle must be less
than the other two sides added together, so
a < b+c, b < a+c, and c < a+b
Substituting:
10 < 12+c, 12 < 10+c, and c < 10+12
Solving all three for c:
-2 < c, 2 < c, c < 22
The first one is of course true, so ignore it.
Put the second and third together and get
2 < c < 22
That's it. c must be more than 2 and less than 22.
They are all done the same way. You can do the other half
by yourself.
Edwin