IM Geo.2.12 Lesson: Proofs about Quadrilaterals

Which figures (if any) are always rectangles? Which figures can be dragged to make a rectangle?

Which figures (if any) are always parallelograms? Which figures can be dragged to make a parallelogram?

• All rectangles are parallelograms.
• If a parallelogram has (at least) one right angle, then it is a rectangle.
• If a quadrilateral has 2 pairs of opposite sides that are congruent, then it is a parallelogram.
• If the diagonals of a quadrilateral both bisect each other, then the quadrilateral is a parallelogram.
• If the diagonals of a quadrilateral both bisect each other and they are perpendicular, then the quadrilateral is a rhombus.

Rewrite the conjecture to identify the given information and the statement to prove.

Write a rough draft of how you might prove your conjecture is true.

Exchange proofs with your partner. Read the rough draft of their proof. If it convinces you, write a detailed proof together following their plan. If it does not convince you, suggest changes that will make the proof convincing.

Draw 2 circles (of different sizes) that intersect in 2 places. Label the centers  and  and the points of intersection  and . Prove that segment  must be perpendicular to segment .