Question

triangles hbn and rzk are shown. m∠bnh = 34°, m∠nhb = 85°, hn = 6.4, bh = 4.08, nb = 7.149, m∠rzk = 85°, rz = 2.04, and kz = 3.2. which additional information is needed to show △nhb ~ △kzr by angle - angle similarity?

Explanation:

Step1: Recall the Angle - Angle Similarity Criterion

For two triangles to be similar by the Angle - Angle (AA) similarity criterion, two pairs of corresponding angles must be equal.

Step2: Analyze the given angles

In \(\triangle NHB\), we know \(\angle NHB = 85^{\circ}\) and in \(\triangle KZR\), we know \(\angle RZK=85^{\circ}\). So, \(\angle NHB=\angle RZK\).
For AA similarity, we need another pair of equal angles. In \(\triangle NHB\), we can find \(\angle HBN\) using the angle - sum property of a triangle (\(\angle HBN=180^{\circ}-\angle BNH - \angle NHB=180^{\circ}-34^{\circ}-85^{\circ}=61^{\circ}\)).
We need \(\angle K R Z = 34^{\circ}\) (because \(\angle BNH = 34^{\circ}\) and for \(\triangle NHB\sim\triangle KZR\) by AA similarity, \(\angle BNH\) and \(\angle K R Z\) should be a pair of corresponding angles)

Answer:

\(m\angle KRZ = 34^{\circ}\)