Question

5 kasey and emma are examining the triangles formed by a diagonal of a rectangle. kasey believes a diagonal will always partition a rectangle into two right triangles. her reasoning is that since every rectangle has four right angles, each triangle formed by the diagonals will have a right angle. emma believes a diagonal will always partition a rectangle into two congruent triangles. her reasoning is that since the diagonal acts as a transversal between parallel lines, the asa criterion is satisfied by the diagonal and the two angles formed by the transversal. whose reasoning is valid? a. only emmas reasoning is valid. b. only kaseys reasoning is valid. c. both kasey and emmas reasoning is valid. d. neither kasey nor emmas reasoning is valid.

Explanation:

• Kasey's reasoning: A rectangle has four right - angles. When a diagonal is drawn, each of the two triangles formed has one of the right - angles of the rectangle. So, a diagonal always partitions a rectangle into two right - triangles.
• Emma's reasoning: In a rectangle, opposite sides are parallel. Let \(ABCD\) be a rectangle with diagonal \(AC\). \(\angle BAC=\angle DCA\) (alternate interior angles as \(AB\parallel CD\) and \(AC\) is a transversal) and \(\angle BCA = \angle DAC\) (alternate interior angles as \(AD\parallel BC\) and \(AC\) is a transversal). Also, \(AC = AC\) (common side). By the ASA (Angle - Side - Angle) congruence criterion, \(\triangle ABC\cong\triangle CDA\). So, a diagonal always partitions a rectangle into two congruent triangles.

Answer:

C. Both Kasey and Emma's reasoning is valid