Question

state if the three side lengths can form a triangle or not. a. 10 m, 3 m, 8 m b. 1.5 ft, 4 ft, 2.5 ft c. 6 in, 1 ft, 10 in find the value of x.

Explanation:

Step1: Recall triangle - inequality theorem

The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Step2: Check part a

For side lengths \(a = 10m\), \(b = 3m\), \(c = 8m\):
\(b + c=3 + 8=11m>10m\), \(a + b = 10+3 = 13m>8m\), \(a + c=10 + 8=18m>3m\). So, they can form a triangle.

Step3: Check part b

For side lengths \(a = 1.5ft\), \(b = 4ft\), \(c = 2.5ft\):
\(a + c=1.5+2.5 = 4ft\), which is not greater than \(b = 4ft\). So, they cannot form a triangle.

Step4: Convert units for part c

Since \(1ft = 12in\), the side - lengths are \(a = 6in\), \(b = 12in\), \(c = 10in\).
\(a + c=6 + 10=16in>12in\), \(a + b=6 + 12 = 18in>10in\), \(b + c=12 + 10=22in>6in\). So, they can form a triangle.

Step5: Solve for \(x\) in the triangle

The two triangles are congruent by the Side - Side - Side (SSS) congruence criterion. Corresponding angles of congruent triangles are equal. So, \(x = 52^{\circ}\).

Answer:

a. Yes
b. No
c. Yes
x = 52°