a An example of a shape with four equal sides is a square.
A parallelogram is defined as a shape with two pairs of parallel and congruent sides. As long as this holds true, the shape is a parallelogram. For a square, opposite sides are parallel.
A square fits the description of a parallelogram, which means it is a parallelogram. We have found a counterexample that disproves the conjecture.
b The tangent ratio is defined as the opposite side divided by the adjacent side for a non-right angle in a right triangle.
tan θ =Opposite/Adjacent
Let's first think about what happens when this ratio equals 1. This means the opposite and adjacent sides of the given angle are congruent, which makes it an isosceles triangle. In an isosceles right triangle, the base angles are 45^(∘).
However, if tan θ is greater than 1, then the opposite side must be greater than the adjacent side. Let's say that we add 3 cm to the vertical leg of our right triangle.
When the opposite leg of the angle is greater than the adjacent leg, the angle must be greater than 45^(∘). Therefore, the conjecture is correct.
c The name corresponding angles only refers to where two angles are located with respect to a transversal. For example, ∠ a and ∠ b below are corresponding angles.
However, corresponding angles will be congruent only if the two lines are parallel.
