Question

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: Find side lengths of WXYZ

• Corresponding to \( AB = 4\space\text{cm} \), \( WX = AB = 4\space\text{cm} \) (since \( AB \) and \( WX \) are corresponding sides in reflection).
• Corresponding to \( BC = 2\space\text{cm} \), \( XY = BC = 2\space\text{cm} \).
• Corresponding to \( CD = 3\space\text{cm} \), \( YZ = CD = 3\space\text{cm} \).
• Corresponding to \( DA = 2\space\text{cm} \), \( ZW = DA = 2\space\text{cm} \).

Step3: Find angle measures of WXYZ

• Corresponding to \( m\angle A = 82^\circ \), \( m\angle W = m\angle A = 82^\circ \) (corresponding angles in reflection are congruent).
• Corresponding to \( m\angle B = 59^\circ \), \( m\angle X = m\angle B = 59^\circ \).
• Corresponding to \( m\angle C = 129^\circ \), \( m\angle Y = m\angle C = 129^\circ \).
• Corresponding to \( m\angle D = 90^\circ \), \( m\angle Z = m\angle D = 90^\circ \).

Answer:

\( WX = 4\space\text{cm} \), \( XY = 2\space\text{cm} \), \( YZ = 3\space\text{cm} \), \( ZW = 2\space\text{cm} \)
\( m\angle W = 82^\circ \), \( m\angle X = 59^\circ \), \( m\angle Y = 129^\circ \), \( m\angle Z = 90^\circ \)