if a quadrilateral has both pair of base angles congruent---- can you prove that the shape is a trapezid
Yes, I can help you prove that a quadrilateral with both pairs of base angles congruent is a trapezoid. To do so, we can use the definition and properties of a trapezoid.
First, let's understand the definition of a trapezoid. A trapezoid is a quadrilateral with at least one pair of parallel sides. In other words, it has one pair of opposite sides that are parallel, known as the base sides.
Now, let's focus on the given information: a quadrilateral with both pairs of base angles congruent. This means that both pairs of angles formed between the base sides and the non-parallel sides are equal.
To prove that the quadrilateral is a trapezoid, we need to show that one pair of sides is parallel. We can do this by assuming the opposite and reaching a contradiction.
Assume that the quadrilateral is not a trapezoid, i.e., both pairs of sides are not parallel. This means that the base sides are not parallel, which implies that the opposite sides are not parallel either.
Since the base angles are congruent, if we draw a diagonal connecting the non-parallel sides, we get two congruent triangles. However, if the opposite sides are not parallel, these two triangles cannot be congruent by SAS (Side-Angle-Side) congruence postulate as they would have different side lengths.
Therefore, the assumption that the opposite sides are not parallel leads to a contradiction. Hence, we can conclude that a quadrilateral with both pairs of base angles congruent is indeed a trapezoid.
In summary, by assuming the opposite and reaching a contradiction, we have proved that a quadrilateral with both pairs of base angles congruent is a trapezoid.