Detailed examples of proving a quadrilateral is a parallelogram.

📚 Quick Study Guide

• 📏 A quadrilateral is a polygon with four sides, four angles, and four vertices.
• 🤝 A parallelogram is a quadrilateral with two pairs of parallel sides.
• ✨ Theorem 1: If both pairs of opposite sides of a quadrilateral are parallel, then the quadrilateral is a parallelogram.
• 💫 Theorem 2: If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
• 📐 Theorem 3: If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
• ✂️ Theorem 4: If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.
• ➕ Theorem 5: If one pair of opposite sides of a quadrilateral are both congruent and parallel, then the quadrilateral is a parallelogram.

Practice Quiz

1. Which of the following conditions proves that a quadrilateral is a parallelogram?

   1. Both pairs of opposite angles are supplementary.
   2. Only one pair of opposite sides are parallel.
   3. Both pairs of opposite sides are congruent.
   4. Only one diagonal bisects the other.

2. If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a:

   1. Trapezoid
   2. Rectangle
   3. Parallelogram
   4. Kite

3. In quadrilateral ABCD, AB || CD and AB ≅ CD. Which theorem proves that ABCD is a parallelogram?

   1. If both pairs of opposite sides are congruent.
   2. If both pairs of opposite angles are congruent.
   3. If one pair of opposite sides are both congruent and parallel.
   4. If the diagonals bisect each other.

4. If both pairs of opposite angles in a quadrilateral are congruent, it is necessarily a:

   1. Rectangle
   2. Square
   3. Parallelogram
   4. Trapezoid

5. The coordinates of the vertices of quadrilateral PQRS are P(1,1), Q(2,4), R(5,5), and S(4,2). Which method could be used to prove that PQRS is a parallelogram?

   1. Show that one pair of sides are both parallel and congruent.
   2. Show that the diagonals are perpendicular.
   3. Show that all sides are congruent.
   4. Show that the area is equal to the perimeter.

6. Which of the following statements is sufficient to prove that quadrilateral EFGH is a parallelogram?

   1. EF ≅ GH and FG || EH
   2. EF || GH and FG ≅ EH
   3. EF ≅ GH and FG ≅ EH
   4. EG ⊥ FH

7. Given a quadrilateral JKLM where diagonals JL and KM bisect each other at point N. What conclusion can be drawn?

   1. JKLM is a rectangle.
   2. JKLM is a square.
   3. JKLM is a parallelogram.
   4. JKLM is a trapezoid.