Example Question - value of angle

Determining the Value of an Angle in a Quadrilateral

Given: - Quadrilateral \( ABCD \) - \( AB = BC = CD = 4 \) - \( \angle C = 32^\circ \) - \( \angle A = 115^\circ \) - \( BD \) is a diagonal To find: - \( \angle x \) in \( \triangle BCD \) Step 1: Identify that \( ABCD \) is a kite since two pairs of adjacent sides are equal. Step 2: Use the fact that in a kite, one diagonal is the perpendicular bisector of the other. Step 3: This makes \( \triangle BCD \) and \( \triangle ABD \) right triangles. Therefore, \( \angle BDC = \angle ABD = 90^\circ \). Step 4: Calculate the remaining angle in \( \triangle BCD \) (angle \( x \)) using the fact that the sum of angles in a triangle is \( 180^\circ \). \[ x + 32^\circ + 90^\circ = 180^\circ \] \[ x + 122^\circ = 180^\circ \] \[ x = 180^\circ - 122^\circ \] \[ x = 58^\circ \]