Question

(b) can these be the side lengths of a triangle? 30 cm, 60 cm, 40 cm yes no (c) can these be the angle measures of a triangle? 54°, 52°, 36°

Explanation:

Part (b)

Step1: Recall Triangle Inequality Theorem

The sum of any two sides of a triangle must be greater than the third side. Let the sides be \(a = 30\) cm, \(b = 40\) cm, \(c = 60\) cm.

Step2: Check \(a + b > c\)

Calculate \(30 + 40 = 70\). Since \(70>60\), this holds.

Step3: Check \(a + c > b\)

Calculate \(30 + 60 = 90\). Since \(90>40\), this holds.

Step4: Check \(b + c > a\)

Calculate \(40 + 60 = 100\). Since \(100>30\), this holds.

Since all three conditions of the triangle inequality theorem are satisfied, the given side lengths can form a triangle.

Step1: Recall Triangle Angle Sum Property

The sum of the interior angles of a triangle is \(180^\circ\). Let the angles be \(54^\circ\), \(52^\circ\), and \(36^\circ\).

Step2: Calculate the sum of the angles

Sum \(= 54^\circ + 52^\circ + 36^\circ\). First, \(54 + 52 = 106\), then \(106 + 36 = 142^\circ\).

Step3: Compare with \(180^\circ\)

Since \(142^\circ
eq180^\circ\), the given angle measures cannot be the angles of a triangle.

Answer:

Yes

Part (c)