Isosceles Trapezoids: Identifying and defining elements

True: in every isosceles trapezoid the base angles are equal to each other.

Let's closely examine the properties of an isosceles trapezoid to determine the relationship between its opposite angles.

An isosceles trapezoid is a type of trapezoid where the non-parallel sides (legs) are of equal length. It also implies that the angles adjacent to each leg are equal, due to the symmetry of the trapezoid. However, the most crucial aspect for this problem involves understanding the sum of angles in a trapezoid and their specific relationships.

Consider an isosceles trapezoid, $ABCD$, with base $AB$ parallel to base $CD$, and legs $AD$ and $BC$ being equal.

• The key property in any trapezoid is that each pair of angles on the same side of one of the non-parallel sides ($AD$ or $BC$) are supplementary. That is, angles $\angle DAB$ and $\angle ABC$ together sum to $180^\circ$ since they are co-interior angles formed by a transversal cutting the parallel lines $AB$ and $CD$.
• Similarly, angles $\angle BCD$ and $\angle CDA$ also sum to $180^\circ$.

In addition, the total sum of the interior angles of any quadrilateral (and thus any trapezoid) is $360^\circ$. Therefore, opposite angles $\angle DAB$ and $\angle BCD$ (first set) sum to $180^\circ$, and similarly, opposite angles $\angle ABC$ and $\angle CDA$ (second set) also sum to $180^\circ$. Therefore, the sum of the opposite angles in an isosceles trapezoid is always $180^\circ$.

Thus, through application of the properties of parallel lines and angle sums, we find that the sum of opposite angles in an isosceles trapezoid is indeed always $180^\circ$.

The correct conclusion of this problem is that the statement is True.

The sum of the opposite angles in an isosceles trapezoid is always $180^\circ$.

To determine the properties of an isosceles trapezoid's diagonals, follow these steps:

• Step 1: Define an isosceles trapezoid.

An isosceles trapezoid is a quadrilateral with one pair of opposite sides that are parallel (bases), and the non-parallel sides (legs) are equal in length.

• Step 2: Analyze the properties of the diagonals in an isosceles trapezoid.

One fundamental property of isosceles trapezoids is that their diagonals are congruent. This means that the lengths of the two diagonals are equal. This property arises because the legs are equal, and consequently, the angles at the bases also exhibit special symmetry, leading to equal diagonals.

• Step 3: Determine if the diagonals intersect.

In any convex quadrilateral, which includes an isosceles trapezoid, the diagonals will intersect each other at a point inside the quadrilateral. This follows from the definition of a convex shape, where diagonals cross each other.

Hence, it follows that not only are the diagonals of an isosceles trapezoid equal in length, but they also intersect each other.

Therefore, the correct answer to the question is Yes.

The diagonals of an isosceles trapezoid are always equal to each other,

but they do not necessarily bisect each other.

(Reminder, "bisect" means that they meet exactly in the middle, meaning they are cut into two equal parts, two halves)

For example, the following trapezoid ABCD, which is isosceles, is drawn.

Using a computer program we calculate the center of the two diagonals,

And we see that the center points are not G, but the points E and F.

This means that the diagonals do not bisect.