Question

1. two triangles, one with vertices a, n, f (angle at a: 29°, angle at n: 106°) and another with vertices h, s, e (angle at h: 45°, angle at s: 29°)

Explanation:

Step1: Use the triangle angle - sum theorem

The sum of the interior angles of a triangle is \(180^{\circ}\).
For \(\triangle ANF\), \(\angle A+\angle N+\angle F = 180^{\circ}\). Given \(\angle A = 29^{\circ}\), \(\angle N=106^{\circ}\), then \(\angle F=180^{\circ}-(29^{\circ} + 106^{\circ})=180^{\circ}-135^{\circ}=45^{\circ}\).
For \(\triangle HSE\), \(\angle H+\angle S+\angle E=180^{\circ}\). Given \(\angle H = 45^{\circ}\), \(\angle S = 29^{\circ}\), then \(\angle E=180^{\circ}-(45^{\circ}+29^{\circ})=106^{\circ}\).

Step2: Check for congruent angles

In \(\triangle ANF\) and \(\triangle HSE\), \(\angle A=\angle S = 29^{\circ}\), \(\angle N=\angle E = 106^{\circ}\), \(\angle F=\angle H = 45^{\circ}\).

Answer:

\(\triangle ANF\sim\triangle HSE\) (by the AA (Angle - Angle) similarity criterion, since all three pairs of corresponding angles are equal)