1. **Find the size of the angle marked with a letter in each triangle.** Recall the rule: The sum of interior angles in any triangle is always $$180^\circ$$. **(a)** Triangle with angles $$58^\circ$$, $$72^\circ$$, and $$a$$: $$a = 180^\circ - (58^\circ + 72^\circ)$$ $$a = 180^\circ - 130^\circ = 50^\circ$$ **(b)** Triangle with angles $$42^\circ$$, $$b$$, and exterior angle $$128^\circ$$: Exterior angle equals sum of opposite interior angles: $$128^\circ = 42^\circ + b$$ $$b = 128^\circ - 42^\circ = 86^\circ$$ **(c)** Triangle with angles $$71^\circ$$, $$c$$, and exterior angle $$140^\circ$$: $$140^\circ = 71^\circ + c$$ $$c = 140^\circ - 71^\circ = 69^\circ$$ **(d)** Triangle with angles $$130^\circ$$, $$d$$, $$70^\circ$$, and $$e$$ on a straight line: Sum on straight line is $$180^\circ$$, so $$d + e = 180^\circ - 130^\circ = 50^\circ$$ Inside triangle sum: $$130^\circ + 70^\circ + d = 180^\circ$$ $$d = 180^\circ - 200^\circ = -20^\circ$$ (impossible, so likely $$d$$ and $$e$$ are angles on the straight line, not inside triangle). Assuming $$d$$ and $$e$$ are supplementary angles on the line: $$d + e = 50^\circ$$ (cannot solve further without more info). **(f), (g), (h)** Triangle with angles $$52^\circ$$, $$f$$, $$g$$, $$h$$ and exterior angle $$43^\circ$$: Sum interior angles $$= 180^\circ$$ Assuming $$f + g + h + 52^\circ = 180^\circ$$ and exterior angle $$43^\circ$$ is supplementary to one interior angle. Without more info, cannot solve uniquely. **(i), (j), (k)** Triangle with angles $$37^\circ$$, $$i$$, $$j$$, $$64^\circ$$, and $$k$$: Sum interior angles $$= 180^\circ$$ Assuming $$37^\circ + 64^\circ + i + j + k = 180^\circ$$ (too many angles for triangle), likely some are exterior or on different triangles. Cannot solve uniquely without more info. 2. **Work out the sizes of the missing angles in each isosceles triangle.** Recall: In isosceles triangles, two sides are equal, so the angles opposite those sides are equal. **(a), (b)** Isosceles triangle with apex angle $$40^\circ$$ and base angles $$a$$ and $$b$$: $$a = b$$ Sum of angles: $$40^\circ + a + b = 180^\circ$$ $$40^\circ + 2a = 180^\circ$$ $$2a = 140^\circ$$ $$a = b = 70^\circ$$ **(c), (d)** Isosceles triangle with apex angle $$38^\circ$$ and base angles $$c$$ and $$d$$: $$c = d$$ Sum: $$38^\circ + c + d = 180^\circ$$ $$38^\circ + 2c = 180^\circ$$ $$2c = 142^\circ$$ $$c = d = 71^\circ$$ **(e), (f)** Isosceles triangle with angle $$62^\circ$$ and equal angles $$e$$ and $$f$$: $$e = f$$ Sum: $$62^\circ + e + f = 180^\circ$$ $$62^\circ + 2e = 180^\circ$$ $$2e = 118^\circ$$ $$e = f = 59^\circ$$