Three sides measuring 5 in., 12 in., and 14 in.
First of all, you need to know some facts about every triangle:
FIRST FACT:
If two sides of a triangle are [tex]a \ and \ b[/tex], then the third side must be less than [tex]a+b[/tex]. Why? because if its length is exactly [tex]a+b[/tex], then you can't build up a triangle, but you'll have a line segment. In other words:
[tex]a<b+c \\ \\ b<a+c \\ \\ c<a+b[/tex]
SECOND FACT:
The internal angles of any triangle add up to 180°. So if ∠A, ∠B and ∠C are the internal angles of a triangle, then:
[tex]\angle A +\angle B+ \angle C=180^{\circ}[/tex]
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So let's analyze each case:
[tex]\angle A=25^{\circ} \\ \\ \angle B=65^{\circ} \\ \\ \angle C=90^{\circ} \\ \\ \\ \angle A +\angle B+ \angle C=25^{\circ}+65^{\circ}+90^{\circ}=185^{\circ} \neq 180^{\circ}[/tex]
The sum is greater than 180 degrees, so these angles can't form a triangle
The sum is less than 180 degrees, so these angles can't form a triangle
The inequalities are true, so these sides form a triangle.
Case 4: Three sides measuring 4 ft, 8 ft, and 14 ft
[tex]a=4 \\ \\ b=8 \\ \\ c=14 \\ \\ \\ 4<8+14\rightarrow 4<22 \\ \\ 8<4+14 \rightarrow 8<18 \\ \\ 14<4+8 \rightarrow 14<12 \ NOT \ TRUE![/tex]
Since the third inequality is not true, then we can't form a triangle with these sides.
