Question

use rigid motions to write a proof of the converse of the isosceles triangle theorem. given: ∠j ≅ ∠l prove: jk ≅ kl reflect △jkl horizontally right across a line perpendicular to jl to form image △lkj. this shows that

Explanation:

Step1: Reflect the triangle

Reflect \(\triangle{JKL}\) horizontally right across a line perpendicular to \(\overline{JL}\) to form image \(\triangle{L'K'J'}\). A reflection is a rigid motion, so \(\triangle{JKL}\cong\triangle{L'K'J'}\). Then \(\angle{J}\cong\angle{L'}\), \(\angle{L}\cong\angle{J'}\), and \(\overline{JK}\cong\overline{L'K'}\), \(\overline{LK}\cong\overline{J'K'}\).

Step2: Use the given angle congruence

Given \(\angle{J}\cong\angle{L}\). Since \(\angle{J}\cong\angle{L'}\) (from reflection) and \(\angle{J}\cong\angle{L}\), then \(\angle{L}\cong\angle{L'}\). Also, \(\angle{L}\) and \(\angle{L'}\) are in a position that implies that the line of reflection is also a line of symmetry for the combined figure of \(\triangle{JKL}\) and \(\triangle{L'K'J'}\) with respect to the sides \(\overline{JK}\) and \(\overline{LK}\).

Step3: Conclude the side - congruence

Because of the properties of rigid motions (reflection preserves side lengths and angles) and the angle - angle - side (AAS) or angle - side - angle (ASA) congruence (if we consider the overlapping and symmetry), we can say that \(\overline{JK}\cong\overline{KL}\). In more detail, if we consider the two triangles \(\triangle{JKL}\) and its reflected image (in a way that we can pair angles and sides), since \(\angle{J}\cong\angle{L}\), and the side \(\overline{KL}\) (in the original triangle) corresponds to \(\overline{J'K'}\) (in the reflected triangle) and \(\overline{JK}\) (in the original triangle) corresponds to \(\overline{L'K'}\) (in the reflected triangle), and using the fact that the reflection maps \(\triangle{JKL}\) to a congruent triangle where the equality of angles \(\angle{J}\) and \(\angle{L}\) forces the equality of the sides opposite to these angles (by the converse of the isosceles triangle theorem which states that if two angles of a triangle are congruent, then the sides opposite those angles are congruent).

Answer:

By reflecting \(\triangle{JKL}\) across a line perpendicular to \(\overline{JL}\) (a rigid motion), using the given \(\angle{J}\cong\angle{L}\), and the properties of congruent triangles (reflection preserves angle - side relationships), we have proved that \(\overline{JK}\cong\overline{KL}\) as required by the converse of the Isosceles Triangle Theorem.