Question

1. jace constructed triangle abc, where ( mangle a = 53^circ ), and ( mangle b = 112^circ ). ruthie constructed triangle rst, where ( mangle s = 112^circ ), and ( mangle t = 15^circ ). are the two triangles similar? explain.

Explanation:

Step1: Find angle C in triangle ABC

The sum of angles in a triangle is \(180^\circ\). So, \(m\angle C = 180^\circ - m\angle A - m\angle B\). Substituting the values, \(m\angle C = 180^\circ - 53^\circ - 112^\circ = 15^\circ\).

Step2: Find angle R in triangle RST

Using the angle - sum property of a triangle, \(m\angle R=180^\circ - m\angle S - m\angle T\). Substituting \(m\angle S = 112^\circ\) and \(m\angle T = 15^\circ\), we get \(m\angle R=180^\circ - 112^\circ - 15^\circ = 53^\circ\).

Step3: Compare angles of the two triangles

In triangle ABC, the angles are \(m\angle A = 53^\circ\), \(m\angle B = 112^\circ\), \(m\angle C = 15^\circ\). In triangle RST, the angles are \(m\angle R = 53^\circ\), \(m\angle S = 112^\circ\), \(m\angle T = 15^\circ\). By the AA (Angle - Angle) similarity criterion, if two angles of one triangle are congruent to two angles of another triangle, the triangles are similar. Here, \(m\angle A=m\angle R = 53^\circ\), \(m\angle B=m\angle S = 112^\circ\) and \(m\angle C=m\angle T = 15^\circ\), so the corresponding angles are equal.

Answer:

Yes, the two triangles are similar. In \(\triangle ABC\), \(m\angle C=180^{\circ}-53^{\circ}-112^{\circ} = 15^{\circ}\). In \(\triangle RST\), \(m\angle R = 180^{\circ}-112^{\circ}-15^{\circ}=53^{\circ}\). The angles of \(\triangle ABC\) are \(53^{\circ}\), \(112^{\circ}\), \(15^{\circ}\) and the angles of \(\triangle RST\) are \(53^{\circ}\), \(112^{\circ}\), \(15^{\circ}\). By the AA similarity criterion (since two pairs of corresponding angles are equal), \(\triangle ABC\sim\triangle RST\).