Learn theorems relating the areas of triangles and parallelograms that share the same base and lie between the same parallels.
Parallelograms on the same base and between the same parallels are equal in area. Similarly, triangles on the same base and between the same parallels are equal in area.
Two parallelograms share the same base of 10 cm and lie between the same parallel lines 6 cm apart. What can be said about their areas?
They are equal, both equal to 10×6=60 cm² (base×height)
A triangle has base 8 cm and lies between parallels 5 cm apart. If another triangle shares the same base and parallels, how do their areas compare?
They are equal in area, both = ½×8×5=20 cm²
Two parallelograms share a base of 14 cm and lie between parallels 5 cm apart. Find each area.
14×5=70 cm² each
A triangle and a parallelogram on the same base and between the same parallels have the triangle's area equal to half the parallelogram's area.
A parallelogram has area 48 cm². A triangle shares its base and lies between the same parallels. Find the triangle's area.
48÷2=24 cm²
Why is a triangle's area exactly half of a corresponding parallelogram's area on the same base and parallels?
Because the parallelogram can be split into two congruent triangles along its diagonal, each equal to the given triangle's area
A triangle has area 32 cm² on a base shared with a parallelogram between the same parallels. Find the parallelogram's area.
64 cm²
1. Two triangles share a base of 12 cm and lie between the same parallels 7 cm apart. Find each area.
2. A parallelogram has area 60 cm². Find the area of a triangle on the same base between the same parallels.
3. Two parallelograms share a base of 9 cm between parallels 4 cm apart. Find each area.
4. A triangle has area 25 cm² on a base and between parallels shared with a parallelogram. Find the parallelogram's area.
5. Why must the parallels be the 'same' for these area theorems to apply?
6. Two triangles share a base of 16 cm between parallels 9 cm apart. Find each area.
7. A parallelogram has area 84 cm². Find the area of a triangle on the same base between the same parallels.
8. Two parallelograms share a base of 11 cm between parallels 6 cm apart. Find each area.
9. A triangle has area 30 cm² sharing a base and parallels with a parallelogram. Find the parallelogram's area.
10. Why must the two figures lie between the exact same pair of parallel lines for this theorem to hold?