High school geometry test questions: proving quadrilaterals are parallelograms.

📚 Quick Study Guide

• 📏 Definition: A parallelogram is a quadrilateral with both pairs of opposite sides parallel.
• ✨ Opposite Sides Theorem: If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
• 📐 Opposite Angles Theorem: If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
• ↔️ Consecutive Angles Theorem: If consecutive angles of a quadrilateral are supplementary, then the quadrilateral is a parallelogram. (Supplementary means they add up to $180^\circ$)
• ✂️ Diagonals Theorem: If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.
• ↩️ One Pair Parallel and Congruent: If one pair of opposite sides of a quadrilateral are both parallel and congruent, then the quadrilateral is a parallelogram.

Practice Quiz

1. Which of the following conditions is sufficient to prove that a quadrilateral is a parallelogram?
   1. Only one pair of opposite sides is parallel.
   2. Only one pair of opposite angles is congruent.
   3. Both pairs of opposite sides are congruent.
   4. Only one pair of opposite sides is congruent and parallel.
2. In quadrilateral ABCD, diagonals AC and BD bisect each other at point E. What is sufficient to conclude about ABCD?
   1. ABCD is a rectangle.
   2. ABCD is a square.
   3. ABCD is a parallelogram.
   4. ABCD is a trapezoid.
3. In quadrilateral PQRS, $\angle P = 100^\circ$ and $\angle Q = 80^\circ$. What additional information is needed to prove that PQRS is a parallelogram?
   1. $\angle R = 100^\circ$
   2. $\angle R = 80^\circ$
   3. $\angle S = 90^\circ$
   4. PQ || RS
4. Given quadrilateral WXYZ with WX = YZ and XY = WZ, what can be concluded about WXYZ?
   1. WXYZ is a rectangle.
   2. WXYZ is a square.
   3. WXYZ is a parallelogram.
   4. WXYZ is a trapezoid.
5. In quadrilateral EFGH, EF || GH and EF = GH. What can be concluded about EFGH?
   1. EFGH is a rectangle.
   2. EFGH is a square.
   3. EFGH is a parallelogram.
   4. EFGH is a trapezoid.
6. The coordinates of the vertices of quadrilateral ABCD are A(1, 2), B(5, 2), C(5, 5), and D(1, 5). What is the most precise classification of ABCD?
   1. Parallelogram
   2. Rectangle
   3. Square
   4. Trapezoid
7. Which statement is NOT sufficient to prove that a quadrilateral is a parallelogram?
   1. Both pairs of opposite sides are parallel.
   2. Both pairs of opposite angles are congruent.
   3. The diagonals bisect each other.
   4. Only one pair of opposite sides is congruent.