Question

1-111. examine each diagram below. which diagrams are possible? which are impossible? justify each conclusion. (hint: you can use words or mathematical thinking to justify your conclusions.)

Explanation:

Step1: Apply 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. For part a, \(5 + 14=19<20\), so it's impossible.

Step2: Check angle - sum property of intersecting lines

For part b, the vertical angles should be equal. Here \(50^{\circ}
eq48^{\circ}\), so it's impossible.

Step3: Use angle - sum property of a triangle

The sum of interior angles of a triangle is \(180^{\circ}\). For part c, \(60^{\circ}+60^{\circ}+60^{\circ}=180^{\circ}\), so it's possible.

Step4: Apply triangle - inequality theorem

For part d, \(2 + 21 = 23<25\), so it's impossible.

Step5: Use angle - sum property of a triangle

For part e, \(62^{\circ}+59^{\circ}=121^{\circ}\), and the third - angle is \(180^{\circ}-121^{\circ}=59^{\circ}\). Also, two sides are equal. So it's possible.

Answer:

Possible: c, e; Impossible: a, b, d