Answer:
1)
y = 45° (converse of alternate interior angle theorem)
z = 45° (alternate interior angle theorem)
k = 135° (exterior angle theorem)
2)
Cos θ + Sin θ = √2 Cos θ
(Cos θ + Sin θ)² = (√2 Cos θ)²
(Exponential property of equality)
Cos² θ + Sin² θ + 2 (Cos θ • Sin θ) = (√2 Cos θ)²
(Distributive property)
1 + 2(Cos θ • Sin θ) = (√2 Cos θ)²
(Definition of the pythagorean identity).
This identity is true because, if a² + b² = c². The triangle must be a right triangle, and a right triangle always has a 90° angle tangent to the triangle, and the tangent of 90° is 1, cos applied to the adjacent over hypotenuse, and sin applies to the opposite over hypotenuse, so this identity must work).
1 + 2(Cos θ • Sin θ) = 2 Cos² θ
(definition of an exponent)
1 + 2(Cos θ • Sin θ) – 1 = 2 Cos² θ – 1
(subtraction property of equality)
2(Cos θ • Sin θ) = 2 Cos² θ – 1
2(Sin θ • Cos θ) = 2 Cos² θ – 1
(Associative property of multiplication ; the input for both functions are θ so this rule works, this is an exception)
Sin (2x) = 2 Cos² θ – 1
(double angled identity of 2 sine and cos)
