Geometry path · G09
Before this lesson: Triangle congruence, The midpoint theorem, Similar triangles
Your goal: Use the defining properties of each quadrilateral without assuming extra ones.
Start with the idea and one example. Take a break before the written task if you need it. The questions are original teaching exercises, not official past-paper questions.
Quadrilaterals and their diagonals: the key idea
A parallelogram has both pairs of opposite sides parallel. Useful converses include both pairs of opposite sides equal, one pair equal and parallel, or diagonals bisecting each other. A rectangle is a parallelogram with a right angle; a rhombus is a parallelogram with all sides equal; a square satisfies both. Equal diagonals alone do not force a rectangle unless the needed parallelogram condition is present. The midpoints of any quadrilateral form a parallelogram by applying the triangle midpoint theorem along its diagonals. Here “trapezium” means a quadrilateral with at least one pair of parallel sides; naming conventions differ internationally.
A worked example
A parallelogram has adjacent side lengths 5 and 8. Find its perimeter.
Opposite sides are equal, so the perimeter is 2(5+8)=26.
Your turn: change one thing
Why is a parallelogram with equal diagonals a rectangle?
Try this on paper before opening the explanation.
Compare your reasoning
For adjacent vertices A,B,C,D, compare triangles ABC and BAD. They share AB, have BC=AD, and AC=BD. SSS gives ∠ABC=∠BAD. These adjacent parallelogram angles sum to 180°, so both are 90°.
Pause and check
A trap to avoid: Assuming equal diagonals imply a square or mixing regional trapezium terminology.
Practise and adjust the level
Foundation checks the language; Core applies the method; Stretch asks you to choose or justify an idea. These are levels within this lesson. A session selects six of the nine questions; unused questions allow the level to change. Advanced theory still needs written practice.
Interactive practice loads here. You can also use the complete question set below.
Write a complete argument
Prove the midpoint quadrilateral is a parallelogram.
Planning hint
List the assumptions and the exact conclusion. Identify the definition or theorem in this lesson that connects them. Explain why its conditions hold before using it.
Read the full solution after your attempt
Let E,F,G,H be midpoints of AB,BC,CD,DA. In triangles ABC and ADC, EF and HG are each parallel to AC and half its length. Thus one pair of opposite sides is equal and parallel, which proves EFGH is a parallelogram. Equivalently, the other pair is parallel to BD.
My proof notebook
Write on paper, or keep a draft here. Compare your reasoning with the solution only after a real attempt. The checklist is your own review, not an automatic mark.
All nine practice questions
Prefer paper or have JavaScript switched off? The complete question set, hints and solutions are here. Interactive practice uses these same questions in an order chosen from your answers.
1. Which property defines a parallelogram?
Foundation
- All diagonals equal
- Only one right angle
- All sides different
- Both pairs of opposite sides parallel
Hint
Use the parallel-side definition.
Answer and reasoning
Both pairs of opposite sides parallel. Both opposite pairs must be parallel.
2. A square is both which two types?
Foundation
- Rectangle and rhombus
- Only a kite and triangle
- Rectangle and circle
- Rhombus and non-parallelogram
Hint
It has equal sides and right angles.
Answer and reasoning
Rectangle and rhombus. It satisfies the defining conditions of both classes.
3. Parallelogram adjacent sides 5 and 8: perimeter?
Core
- 16
- 26
- 13
- 40
Hint
Each side length occurs twice.
Answer and reasoning
26. 2·5+2·8=26.
4. Diagonals bisecting each other imply what for a quadrilateral?
Core
- It is always cyclic
- Its sides are all equal
- It is a parallelogram
- It is always a square
Hint
Use a parallelogram converse.
Answer and reasoning
It is a parallelogram. Opposite small triangles are congruent, giving opposite parallel sides.
5. Equal diagonals alone force an arbitrary quadrilateral to be a rectangle?
Stretch
- Yes
- Only if drawn upright
- Only if convex
- No
Hint
An isosceles trapezium is a counterexample.
Answer and reasoning
No. Extra conditions such as being a parallelogram are needed.
6. The midpoints of a quadrilateral form what?
Stretch
- A parallelogram
- Always a square
- Always a rectangle
- Always a rhombus
Hint
Apply the midpoint theorem across the diagonals.
Answer and reasoning
A parallelogram. Opposite midpoint segments are parallel and equal.
7. A rhombus is a parallelogram with which property?
Foundation
- All sides equal
- All angles 90° necessarily
- Unequal opposite sides
- No parallel sides
Hint
Use its side-length definition.
Answer and reasoning
All sides equal. A rhombus has four equal sides but need not have right angles.
8. A rectangle has side lengths 5 and 12. Find a diagonal.
Core
- 13
- 17
- 7
- 60
Hint
Use a right triangle across the rectangle.
Answer and reasoning
13. .
9. Why do equal adjacent angles in a parallelogram force a rectangle?
Stretch
- All parallelograms are rectangles
- Their sum is 360°
- They also sum to 180°, so each is 90°
- They must both be 60°
Hint
Use supplementary adjacent angles.
Answer and reasoning
They also sum to 180°, so each is 90°. Two equal supplementary angles are each right angles.
Choose your next step
Continue to Concurrency and collinearity. If this felt difficult, return to a prerequisite above. Every lesson stays open.
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