Question

for each set of three measures, determine if they can be angle measures of a triangle. (a) (141^{circ}, 132^{circ}, 31^{circ}) (b) (23^{circ}, 125^{circ}, 53^{circ}) (c) (150^{circ}, 15^{circ}, 15^{circ}) (d) (20^{circ}, 50^{circ}, 90^{circ})

Explanation:

Step1: Recall the angle - sum property of a triangle

The sum of the interior angles of a triangle is \(180^{\circ}\). For each set of angles, calculate the sum.

Step2: Calculate the sum for part (a)

\(141^{\circ}+132^{\circ}+31^{\circ}=141 + 132+31=304^{\circ}\)
Since \(304^{\circ}
eq180^{\circ}\), these cannot be angle measures of a triangle.

Step3: Calculate the sum for part (b)

\(23^{\circ}+125^{\circ}+53^{\circ}=23 + 125+53=201^{\circ}\)
Since \(201^{\circ}
eq180^{\circ}\), these cannot be angle measures of a triangle.

Step4: Calculate the sum for part (c)

\(150^{\circ}+15^{\circ}+15^{\circ}=150 + 15+15=180^{\circ}\)
Since \(180^{\circ}=180^{\circ}\), these can be angle measures of a triangle.

Step5: Calculate the sum for part (d)

\(20^{\circ}+50^{\circ}+90^{\circ}=20 + 50+90=160^{\circ}\)
Since \(160^{\circ}
eq180^{\circ}\), these cannot be angle measures of a triangle.

Answer:

(a) Cannot be angle measures of a triangle
(b) Cannot be angle measures of a triangle
(c) Can be angle measures of a triangle
(d) Cannot be angle measures of a triangle