Question

4 quadrilateral wxyz is a reflection of quadrilateral abcd. the lengths of the sides and the measures of the angles of quadrilateral abcd are given below. ab = 4 cm bc = 2 cm cd = 3 cm da = 2 cm m∠a = 82° m∠b = 59° m∠c = 129° m∠d = 90° predict the lengths of the sides and measures of the angles in quadrilateral wxyz. explain your reasoning. wx = ____ xy = __ yz = __ zw = __ m∠w = __ m∠x = __ m∠y = __ m∠z = ____

Explanation:

Step1: Recall Reflection Property

A reflection is a rigid transformation, so corresponding sides and angles of the original and reflected figures are congruent.

Step2: Match Corresponding Sides

• \( WX \) corresponds to \( AB \), so \( WX = AB = 4\space\text{cm} \)
• \( XY \) corresponds to \( BC \), so \( XY = BC = 2\space\text{cm} \)
• \( YZ \) corresponds to \( CD \), so \( YZ = CD = 3\space\text{cm} \)
• \( ZW \) corresponds to \( DA \), so \( ZW = DA = 2\space\text{cm} \)

Step3: Match Corresponding Angles

• \( \angle W \) corresponds to \( \angle A \), so \( m\angle W = m\angle A = 82^\circ \)
• \( \angle X \) corresponds to \( \angle B \), so \( m\angle X = m\angle B = 59^\circ \)
• \( \angle Y \) corresponds to \( \angle C \), so \( m\angle Y = m\angle C = 129^\circ \)
• \( \angle Z \) corresponds to \( \angle D \), so \( m\angle Z = m\angle D = 90^\circ \)

Answer:

\( WX = 4\space\text{cm} \), \( XY = 2\space\text{cm} \), \( YZ = 3\space\text{cm} \), \( ZW = 2\space\text{cm} \)
\( m\angle W = 82^\circ \), \( m\angle X = 59^\circ \), \( m\angle Y = 129^\circ \), \( m\angle Z = 90^\circ \)
Reasoning: Reflection is a rigid transformation, so corresponding sides and angles of \( ABCD \) and \( WXYZ \) are congruent.