Question

fill in the missing angles and then state if the triangles in each pair are similar. if so, state how you know they are similar and complete the similarity statement.
3.
4.
5.

Explanation:

3.

Step1: Find \(\angle E\)

In \(\triangle EGN\), using the angle - sum property of a triangle (\(\angle E+\angle G+\angle ENG = 180^{\circ}\)). Since \(\angle G = 53^{\circ}\) and \(\angle ENG=46^{\circ}\), then \(\angle E=180-(53 + 46)=81^{\circ}\). \(\angle ANB=\angle ENG = 46^{\circ}\) (vertically - opposite angles). In \(\triangle ABN\), \(\angle A+\angle B+\angle ANB = 180^{\circ}\). Let \(\angle B = 53^{\circ}\) (alternate - interior angles if lines \(EG\) and \(AB\) are parallel). Then \(\angle A=180-(53 + 46)=81^{\circ}\)

Step2: Check similarity

Since \(\angle E=\angle A = 81^{\circ}\), \(\angle G=\angle B = 53^{\circ}\), and \(\angle ENG=\angle ANB = 46^{\circ}\), by the AA (Angle - Angle) similarity criterion, \(\triangle EGN\sim\triangle ABN\)

4.

Step1: Find \(\angle P\)

In \(\triangle PTS\), \(\angle P+\angle T+\angle PST=180^{\circ}\). Given \(\angle T = 32^{\circ}\) and \(\angle PST = 101^{\circ}\), then \(\angle P=180-(32 + 101)=47^{\circ}\). Since \(PT\parallel SV\) and \(PS\parallel UV\) (by the arrow - head markings), \(\angle U=\angle P = 47^{\circ}\) (corresponding angles) and \(\angle V=\angle T = 32^{\circ}\) (corresponding angles)

Step2: Check similarity

Since \(\angle P=\angle U\), \(\angle T=\angle V\), by the AA (Angle - Angle) similarity criterion, \(\triangle PTS\sim\triangle VUS\)

5.

Step1: Find \(\angle H\) and \(\angle M\)

In \(\triangle JHG\), since \(JH = JG\) (marked equal), \(\angle H=\angle G\). Using the angle - sum property \(\angle H+\angle J+\angle G = 180^{\circ}\). \(\angle H=\angle G=\frac{180 - 96}{2}=42^{\circ}\). In \(\triangle AMP\), since \(AM = AP\) (marked equal), \(\angle M=\angle P\). \(\angle M=\angle P = 42^{\circ}\) (because \(\angle A=180-(2\times42)=96^{\circ}\))

Step2: Check similarity

Since \(JH = JG\), \(AM = AP\), \(\angle G=\angle P = 42^{\circ}\), \(\angle J=\angle A = 96^{\circ}\), by the SAS (Side - Angle - Side) similarity criterion (the sides are in proportion as \(JH/AM=JG/AP\) and the included angles \(\angle J=\angle A\)), \(\triangle JHG\sim\triangle AMP\)

Answer:

1. \(\angle E = 81^{\circ}\), \(\angle A = 81^{\circ}\), \(\triangle EGN\sim\triangle ABN\) (by AA similarity)
2. \(\angle P = 47^{\circ}\), \(\angle U = 47^{\circ}\), \(\angle V = 32^{\circ}\), \(\triangle PTS\sim\triangle VUS\) (by AA similarity)
3. \(\angle H = 42^{\circ}\), \(\angle M = 42^{\circ}\), \(\triangle JHG\sim\triangle AMP\) (by SAS similarity)