14. Analyze the two triangles in diagram 14.

Analysis of Figure 14:

• The figure shows two triangles: triangle QRT and triangle MST.
• Triangle QRT has sides QR = 4, QT = 4, and Angle Q = 30°. Since two sides are equal, it is an isosceles triangle. The other two angles would be (180° - 30°) / 2 = 150° / 2 = 75° each.
• Triangle MST has sides MS = 8, MT = 8, and Angle M = 30°. Since two sides are equal, it is an isosceles triangle. The other two angles would be (180° - 30°) / 2 = 150° / 2 = 75° each.
• Both triangles have the same angle measures (30°, 75°, 75°) and thus are similar by the AAA similarity criterion.
• Alternatively, using the SAS similarity criterion:
• We have Angle Q = Angle M = 30°.
• The ratio of adjacent sides in triangle QRT is QR/QT = 4/4 = 1.
• The ratio of adjacent sides in triangle MST is MS/MT = 8/8 = 1.
• Since the ratios of the corresponding sides adjacent to the equal angles are equal (QR/QT = MS/MT = 1), and the included angles are equal (Angle Q = Angle M), the two triangles QRT and MST are similar by the SAS similarity criterion.