Quadrilaterals · Lesson 1 of 13
Quadrilaterals and Rectangles
“Recognise quadrilaterals and build the rectangle definition from right angles and side relationships.”
• Recognise a closed four-sided polygon. • Name vertices, sides, opposite sides and angles. • Check the conditions for a rectangle. • Explain why turning a shape does not change its type.
Look at the edge of a notebook, a window frame and a slightly slanted cardboard sign. Each outline may have four straight sides, yet they need not have the same shape. Counting the sides gives us a starting point. To say more, we need to look at how the sides meet, which sides face one another and whether the corners are right angles. These small checks let you name a figure from its properties rather than from a quick impression.
You will begin with quadrilaterals and the familiar rectangle. Keep a ruler and a piece of paper nearby: drawing a figure, labelling it and explaining one feature is often more useful than trying to remember a whole list at once.
Trace a boundary made of four straight segments. If it closes without crossing itself, it is a quadrilateral. Its corners are vertices. An open four-segment chain is not a polygon, and a boundary with a curved edge is not made entirely of line segments.
A simple closed polygon with four sides, four vertices and four interior angles.
Name consecutive vertices A, B, C and D as you walk around the boundary. The sides are AB, BC, CD and DA. AB and CD are opposite sides because they share no endpoint. AB and BC are adjacent sides because they meet at B.
The angle at B is formed by BA and BC and can be written ∠ABC. The middle letter names its vertex. A diagonal, such as AC, joins two non-adjacent vertices; it is not a boundary side.
4.1 Rectangles and Squares
Place the corner of a sheet against each corner of a drawn rectangle. Each fits a right angle. Now compare opposite sides: the top matches the bottom, and the left matches the right. These are the two conditions we begin with.
A long, narrow rectangle and a wide rectangle belong to the same family. A rectangle turned on its corner still has the same side lengths and right angles. Position on the page is not part of the definition.
A quadrilateral whose four angles are right angles (90°) and whose opposite sides are equal in length. Later, we will prove that the right-angle condition alone forces the opposite sides to be equal.
Problem
A closed outline has four straight sides and no crossing. Is it a quadrilateral?
Problem
ABCD has angles of 90° at every vertex, AB = CD = 8 cm and BC = DA = 5 cm. Name the figure.
Problem
A slanted frame has opposite sides of 8 cm and 5 cm, but one angle is 60°. Is it a rectangle?
Quiz
Which condition directly defines a rectangle?
How many sides does every quadrilateral have?
A rectangle is turned so that a corner points upward. What happens to its type?
Opposite sides equal by itself guarantees a rectangle.
Which feature would stop a shape from being a quadrilateral?
Practice Problems
- List the sides and vertices of quadrilateral ABCD.
- Which sides are opposite AB and BC in rectangle ABCD?
- A rectangle has AB = 7 cm and BC = 3 cm. Find CD and DA.
- A rectangle is rotated so that one corner points upward. Does it remain a rectangle? Explain.
- Draw two closed four-sided figures with equal opposite sides, only one of which is a rectangle. Explain the difference.
1. Follow the boundary A → B → C → D → A. 2. The sides are AB, BC, CD and DA; the vertices are A, B, C and D.
1. Opposite sides do not meet at a vertex. 2. CD is opposite AB; DA is opposite BC.
1. Opposite sides have equal lengths. 2. CD = AB = 7 cm; DA = BC = 3 cm.
1. Rotation changes the direction in which the figure faces. 2. Its lengths and angles do not change, so it still has four right angles and equal opposite sides.
1. Draw an ordinary rectangle, then a slanted figure with two pairs of parallel sides. 2. Both have equal opposite sides, but the slanted one can have angles of 60° and 120°. 3. Only the first has four right angles; side equalities alone do not decide the name.
Key Takeaways
• A quadrilateral is a closed, non-crossing polygon with four straight sides. • Adjacent sides share a vertex; opposite sides do not. • A rectangle has four right angles and equal opposite sides. • A diagonal joins opposite vertices. • Turning a figure does not change its geometric properties.
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Next · Lesson 2
The Diagonals of a Rectangle