Quadrilaterals
Semester 4B · Chapter 16 · Lesson 4 · MATH-4B-C16-L04
Before you begin: prerequisite check
Recall: 4A quadrilateral properties. Attempt these questions independently first.
Using the vertical reference line on the grid, draw a perpendicular through grid row 2 and another distinct parallel vertical line at least 2 grid spaces away. Label both.
Show answer and reasoning
The horizontal makes 90° with the reference and the vertical has equal direction.
Draw a line parallel to the red reference 5 grid spaces to its left, then a perpendicular at row 7.
Show answer and reasoning
Parallel means equal direction; perpendicular means a 90° intersection.
Draw a horizontal line at row 6 and explain whether it is perpendicular to the vertical red reference.
Show answer and reasoning
Vertical and horizontal grid lines form 90° angles.
Key vocabulary
- perpendicular
- meeting at a right angle
- parallel
- same direction and no intersection in a plane
- quadrilateral
- four-sided polygon
- line of symmetry
- line reflecting one matching half onto the other
Understand the concept
Quadrilaterals have four straight sides. A rectangle has four right angles, and a square is a rectangle with four equal sides. A rhombus has four equal sides but need not have right angles; a parallelogram has both pairs of opposite sides parallel. Different textbooks sometimes define trapezoid differently, so we will state that our trapezoid has exactly one pair of parallel opposite sides for these exercises. Classify shapes using properties rather than visual appearance.
Fully worked examples and immediate checks
Read each example, predict how to begin, then follow its numerical explanation. The check below it is a fresh problem.
Worked example 1
State two defining geometric properties of a parallelogram.
Use side lengths, parallelism, and right angles—not appearance.
Check 1: try it yourself
State two defining geometric properties of a square.
Show answer and reasoning
Use side lengths, parallelism, and right angles—not appearance.
Worked example 2
Must every rectangle have two pairs of parallel opposite sides?
Squares, rectangles, rhombuses, and parallelograms all have both pairs of opposite sides parallel.
Check 2: try it yourself
Must every square have two pairs of parallel opposite sides?
Show answer and reasoning
Squares, rectangles, rhombuses, and parallelograms all have both pairs of opposite sides parallel.
Worked example 3
Can a rhombus be a parallelogram? Justify.
Both opposite side pairs of the named shape are parallel.
Check 3: try it yourself
Can a rectangle be a parallelogram? Justify.
Show answer and reasoning
Both opposite side pairs of the named shape are parallel.
Guided practice
State two defining geometric properties of a rhombus.
Show answer and reasoning
Use side lengths, parallelism, and right angles—not appearance.
Describe which angle and side conditions a rhombus must satisfy.
Show answer and reasoning
State defining angle and side properties, not a sketch alone.
A rectangle has a side measuring 5 cm. State two defining angle or side properties of that type of quadrilateral.
Show answer and reasoning
The class-specific properties do not depend on the scale of a drawing.
Describe which angle and side conditions a rectangle must satisfy.
Show answer and reasoning
State defining angle and side properties, not a sketch alone.
A square has a side measuring 14 cm. State two defining angle or side properties of that type of quadrilateral.
Show answer and reasoning
The class-specific properties do not depend on the scale of a drawing.
Independent practice
State two defining geometric properties of a rectangle.
Show answer and reasoning
Use side lengths, parallelism, and right angles—not appearance.
Describe which angle and side conditions a parallelogram must satisfy.
Show answer and reasoning
State defining angle and side properties, not a sketch alone.
A square has a side measuring 11 cm. State two defining angle or side properties of that type of quadrilateral.
Show answer and reasoning
The class-specific properties do not depend on the scale of a drawing.
A rhombus has a side measuring 9 cm. State two defining angle or side properties of that type of quadrilateral.
Show answer and reasoning
The class-specific properties do not depend on the scale of a drawing.
A rectangle has a side measuring 15 cm. State two defining angle or side properties of that type of quadrilateral.
Show answer and reasoning
The class-specific properties do not depend on the scale of a drawing.
A parallelogram has a side measuring 8 cm. State two defining angle or side properties of that type of quadrilateral.
Show answer and reasoning
The class-specific properties do not depend on the scale of a drawing.
Describe which angle and side conditions a square must satisfy.
Show answer and reasoning
State defining angle and side properties, not a sketch alone.
Must every parallelogram be a rectangle? Justify.
Show answer and reasoning
A rectangle has four right angles, but a general parallelogram does not have to.
Challenge and transfer
Use a diagram or verbal justification where appropriate. The goal is transfer rather than speed.
State two defining geometric properties of a rectangle. Explain or sketch the mathematical relationship.
Show answer and reasoning
Use side lengths, parallelism, and right angles—not appearance. State which quantities or geometric properties justify the method.
A learner claims: “A square cannot also be a rectangle.” Identify what is wrong or incomplete and give a mathematically correct explanation.
Show answer and reasoning
False: a square has four right angles, so it is a special rectangle. Explain the relevant quantities or properties, not merely the final answer.
Exit check
A parallelogram has a side measuring 3 cm. State two defining angle or side properties of that type of quadrilateral.
Show answer and reasoning
The class-specific properties do not depend on the scale of a drawing.
A rectangle has a side measuring 14 cm. State two defining angle or side properties of that type of quadrilateral.
Show answer and reasoning
The class-specific properties do not depend on the scale of a drawing.
A square has a side measuring 15 cm. State two defining angle or side properties of that type of quadrilateral.
Show answer and reasoning
The class-specific properties do not depend on the scale of a drawing.
Mastery check: Solve a new instance without hints, explain why the method works, and return later to confirm retention.