Properties of triangles and quadrilaterals
Derive and illustrate properties of triangles, quadrilaterals, and circles using appropriate language, including interior angles, diagonals, symmetry, and relationships between side lengths
What a learner can do afterwards
- List and verify properties of a parallelogram (opposite sides parallel/equal, opposite angles equal, diagonals bisect each other)
- Explain why a square is a special case of both a rectangle and a rhombus
- Derive the relationship between the radius, diameter, and circumference of a circle
The lesson
A parallelogram is a four-sided shape where both pairs of opposite sides are parallel. In a parallelogram, opposite sides are always equal in length, and opposite angles are always equal to each other. The two diagonals also do something special: they bisect each other, meaning each diagonal cuts the other exactly in half at the point where they cross.
Picture a parallelogram ABCD, where AB is opposite CD, and BC is opposite AD. If AB measures 6 cm, then CD also measures 6 cm, since opposite sides are equal. If angle A measures 70 degrees, then angle C, its opposite angle, also measures 70 degrees. Now draw both diagonals, AC and BD: they cross at one point, and that point splits each diagonal into two equal halves.
A rectangle is a parallelogram with four right angles. A rhombus is a parallelogram with four equal sides. A square fits both descriptions at once: it has four right angles, like a rectangle, and four equal sides, like a rhombus. That is why a square counts as a special case of both a rectangle and a rhombus, rather than being a separate shape on its own.
For any circle, the diameter is always twice the radius: d = 2r. The circumference, the distance around the circle, is always about 3.14 times the diameter, written as C = pi times d. That number, pi, is the same for every circle, no matter how big or small.
Parallelograms have equal opposite sides and angles with diagonals that bisect each other, a square is a rectangle and a rhombus at once, and a circle's circumference is always pi times its diameter.
Watch it
Where it sits
This opens up
Nothing builds on it yet.
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.