Question

think about your angle bisector conjecture as you complete the puzzles in the questions below. in each puzzle, use the answer choices provided to complete the statements. you will develop an explanation for why it is possible to draw an inscribed circle in a triangle.

1. since point i is on the bisector of ∠j, it is the same distance from segment ____ as it is from segment __. this means that __ = ____.
2. since point i is on the bisector of ∠k, it is the same distance from segment ____ as it is from segment __. this means that __ = ____.
3. since point i is on all three angle bisectors, it is the same distance from all three sides of the triangle. this means that ____ = __ = __. therefore, the circle with this radius must touch the sides of the triangle at ____.

Explanation:

Step1: Analyze angle bisector property for ∠J

By the angle - bisector theorem, a point on an angle bisector is equidistant from the sides of the angle. For ∠J, the sides are \(JL\) and \(JK\). The perpendicular distances from \(I\) to \(JL\) and \(JK\) are \(IO\) and \(IM\) respectively.

Step2: Analyze angle bisector property for ∠K

For ∠K, the sides are \(KL\) and \(KJ\). The perpendicular distances from \(I\) to \(KL\) and \(KJ\) are \(IN\) and \(IM\) respectively.

Step3: Analyze the in - center property

Since \(I\) is the in - center (intersection of angle bisectors), the perpendicular distances from \(I\) to all three sides are equal. The points of tangency of the inscribed circle with the sides of the triangle are \(M\), \(N\), and \(O\) (because the radius is perpendicular to the tangent at the point of tangency).

Answer:

1. \(JL\); \(JK\); \(IO\); \(IM\)
2. \(KL\); \(KJ\); \(IN\); \(IM\)
3. \(IN\); \(IM\); \(IO\); \(M\), \(N\), and \(O\)