Question

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quadrilateral ( hgfj ) is a rhombus. ( gf = 16 mathrm{~cm} ) and ( m angle ghk = 42 ^ { circ } ).
what is ( jf ) and ( m angle kgh )?
enter your answers in the boxes.
( jf = square mathrm { cm } )
( m angle kgh = square ^ { circ } )

Explanation:

Step1: Use the property of a rhombus (all sides are equal)

Since \(HGFJ\) is a rhombus, \(GF = JF\). Given \(GF = 16\) cm, so \(JF=16\) cm.

Step2: Use the property of diagonals of a rhombus (diagonals are perpendicular - \(m\angle GKH = 90^{\circ}\)) and triangle angle - sum property (\(m\angle GKH+m\angle KGH + m\angle GHK=180^{\circ}\))

We know \(m\angle GHK = 42^{\circ}\) and \(m\angle GKH = 90^{\circ}\). Substitute into the formula \(m\angle KGH=180^{\circ}-(m\angle GKH + m\angle GHK)\).

$$m\angle KGH=180^{\circ}-(90^{\circ}+ 42^{\circ})=48^{\circ}$$

Answer:

\(JF = 16\) cm, \(m\angle KGH = 48^{\circ}\)