Quadrilaterals · Lesson 5 of 13
Understanding Parallelograms
“Build the defining side and angle properties of parallelograms using parallel lines and congruence.”
• Define a parallelogram using both pairs of opposite sides. • Construct one from two adjacent sides and their included angle. • Find adjacent and opposite angles. • Explain why opposite sides are equal.
Imagine a rectangular frame with joints that can turn. Push one corner sideways without changing the side lengths. The corners are no longer right angles, but the two opposite side pairs can remain parallel. This suggests a larger family of shapes that includes rectangles and also includes slanted figures. To understand that family, we will begin with parallel lines and see what they force the sides and angles to do.
Keep the difference between a definition and a consequence in mind. We define the family using parallel sides. Equal opposite sides and related angles will follow from that starting point.
4.3 More Quadrilaterals with Parallel Opposite Sides
Draw two parallel lines. Draw another parallel pair across them. Their four intersections form a quadrilateral. If the two directions do not meet at right angles, the figure is not a rectangle, even though each pair of opposite sides is parallel.
A rectangle passes this parallel-side test too. Thus it belongs to the larger family; it is the special member whose corners are all right angles.
A quadrilateral in which both pairs of opposite sides are parallel.
Deduction 6 — What can we say about the angles of a parallelogram?
In the constructed figure, AB ∥ CD and AD is a transversal. Therefore the interior angles at A and D on the same side of this transversal add to 180°. With ∠A = 30°, we get ∠D = 150°.
Similarly, AD ∥ BC makes ∠A + ∠B = 180°, so ∠B = 150°. Then ∠B + ∠C = 180° gives ∠C = 30°. Opposite angles are equal, while adjacent angles are supplementary.
This works for any starting angle x. Its neighbour is 180° − x. The next angle is 180° − (180° − x) = x. The equality of opposite angles is a consequence of the two supplementary-angle relationships.
Problem
AB = 4 cm, AD = 5 cm and ∠A = 30°. Find the other angles.
Problem
Adjacent angles of a parallelogram are x and 2x. Find all four angles.
Deduction 7 — What can we say about the sides of a parallelogram?
Draw diagonal BD. In triangles ABD and CDB, ∠BAD = ∠DCB because they are opposite parallelogram angles. Also ∠ADB = ∠CBD because AD ∥ BC and BD is a transversal. BD is the common side.
The triangles are congruent by AAS, with A ↔ C, B ↔ D and D ↔ B. Therefore AB = CD and AD = CB. Writing ABD congruent to CBD would usually pair the wrong vertices: correspondence order is part of the statement.
The 4 cm and 5 cm construction therefore has CD = 4 cm and BC = 5 cm. Equal opposite sides do not mean all four sides have the same length.
Problem
In parallelogram ABCD, AB = 9 cm and BC = 6 cm. Find CD and DA.
Every rectangle is a parallelogram because its opposite sides are parallel. A parallelogram with angles 30° and 150° is a counterexample to the reverse claim: it is not a rectangle.
Quiz
A parallelogram is defined by:
If one angle of a parallelogram is 70°, an adjacent angle is:
Opposite angles of a parallelogram are:
Opposite sides of a parallelogram are:
Is every rectangle a parallelogram?
Practice Problems
- In parallelogram ABCD, ∠A = 125°. Find ∠B, ∠C and ∠D.
- AB = 9 cm and BC = 6 cm in parallelogram ABCD. Find CD and DA.
- Adjacent angles are x and 2x. Find all four angles.
- Why does every rectangle belong to the parallelogram family?
- Explain how diagonal BD helps prove opposite sides of ABCD are equal.
1. Adjacent angle B = 180° − 125° = 55°. 2. Opposite angles are equal, so C = 125° and D = 55°.
1. Opposite sides match. 2. CD = AB = 9 cm; DA = BC = 6 cm.
1. x + 2x = 180°, so x = 60°. 2. The angles alternate 60°, 120°, 60°, 120°.
1. A rectangle’s opposite sides are parallel. 2. That satisfies the definition of a parallelogram, even though the rectangle also has right angles.
1. It creates triangles ABD and CDB. Their opposite vertex angles are equal, and a second angle pair is equal by alternate angles between parallel lines. 2. BD is common, so AAS congruence applies. 3. Corresponding sides give AB = CD and AD = CB.
Key Takeaways
• Both pairs of opposite sides must be parallel in a parallelogram. • Adjacent angles total 180°; opposite angles are equal. • Opposite sides are equal, but adjacent sides need not be. • A diagonal lets congruent triangles explain the side equalities. • Rectangles and squares belong to the parallelogram family.