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An isosceles trapezium, a rectangle or square.

Q: What is a quadrilateral with two legs are congruent and the two base angles are congruent?

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An isosceles triangle has at least two equal sides and two equal angles An isosceles triangle has two or more congruent sides called legs. In an isosceles triangle with just two congruent sides, the angle formed by the legs is called the apex, and the other two angles, called base angles, are congruent. If the isosceles triangle has three congruent sides (AKA an equilateral triangle), then all three sides and angles are congruent, and there are no definitive base or vertex angles, besides...all of them. See related link below for the web address

To prove a trapezoid is isosceles, you need to show that the legs (the non-parallel sides) are congruent. This can be done by demonstrating that the base angles opposite these sides are congruent. You can use the triangle congruence postulates or the properties of parallel lines and transversals to establish the equality of these angles.

No angle is defined, only that two sides are the same length. Also, the two base angles are congruent.

Not only. They can be formed by the legs.

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If the trapezoid is an isosceles trapezoid, with congruent legs, then the base angles are congruent. Otherwise, no.

True and its base angles are congruent.

Well a triangle with two congruent sides would be called a isosceles triangle. It has a vertex, two base angles, two legs, and a base.

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A trapezoid can be proven isosceles by proving that the 2 legs are congruent (by definition), or that the 2 base angles (either upper or lower) are congruent.

An isosceles triangle has at least two equal sides and two equal angles An isosceles triangle has two or more congruent sides called legs. In an isosceles triangle with just two congruent sides, the angle formed by the legs is called the apex, and the other two angles, called base angles, are congruent. If the isosceles triangle has three congruent sides (AKA an equilateral triangle), then all three sides and angles are congruent, and there are no definitive base or vertex angles, besides...all of them. See related link below for the web address

To prove a trapezoid is isosceles, you need to show that the legs (the non-parallel sides) are congruent. This can be done by demonstrating that the base angles opposite these sides are congruent. You can use the triangle congruence postulates or the properties of parallel lines and transversals to establish the equality of these angles.

An isosceles triangle has at least two equal sides and two equal angles An isosceles triangle has two or more congruent sides called legs. In an isosceles triangle with just two congruent sides, the angle formed by the legs is called the apex, and the other two angles, called base angles, are congruent. If the isosceles triangle has three congruent sides (AKA an equilateral triangle), then all three sides and angles are congruent, and there are no definitive base or vertex angles, besides...all of them. See related link below for the web address

No angle is defined, only that two sides are the same length. Also, the two base angles are congruent.

Equal angles and equal lengths

The only real characteristic of a trapezoid is that one pair of opposite sides is parallel. For an isosceles trapezoid, in addition to one pair of opposite sides being parallel; the legs are congruent; each pair of base angles is congruent; and the diagonals are congruent.

Not only. They can be formed by the legs.