Area Theorems

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When figures have their bases lying on one straight line and their opposite vertices lying on another straight line parallel to the first, they are said to be between the same parallels. Such figures

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Parallelograms on the same base and between the same parallels are equal in area. This means if two parallelograms share the same base and their opposite sides lie on parallel lines, they have equal a

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The area of a triangle is half that of a parallelogram on the same base and between the same parallels. Formula: Area of triangle = ½ × Area of parallelogram (Theorem 20)

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Yes, triangles on the same base and between the same parallels are equal in area (Theorem 21). This is because they have the same base length and the same height.

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Equal figures are figures that are equal in area. Congruent figures are always equal in area, but equal figures are not always congruent. Two shapes can have the same area but different shapes.

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Parallelograms on equal bases and between the same parallels are equal in area. This is a corollary of Theorem 19, extending the concept from same base to equal bases.

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A diagonal divides a parallelogram into two triangles of equal area. Each triangle has an area equal to half the area of the parallelogram.

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The ratio of areas equals the ratio of their bases: Area(△ABD)/Area(△ACD) = BD/CD. When triangles have the same vertex and bases on the same line, their areas are proportional to their base lengths.