Learn with Amar
Teaching video
A teaching recording for this chapter has not been published yet. Start with the worked example below and explore the related animations where available.
Grade 10 chapters and video availability01 · Read and understand
What you will learn
- Distinguish a conditional, its converse, and its contrapositive.
- Justify the method and check its domain, units, or logical conditions.
Before you start
Algebraic equations, ratios, angle and area facts, and basic coordinate geometry.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
Analyze the statement 'If a quadrilateral is a square, then it is a rectangle.'
Why this math matters
Read theorem hypotheses carefully before applying a familiar conclusion backward. The example connects a stated condition to a conclusion and then checks whether the result satisfies that condition.
Set up the model
A useful answer starts with clear assumptions:
- Use Euclidean geometry and the angle, parallelism, similarity, or congruence conditions stated; a sketch alone does not establish them.
- Treat provided measurements as exact classroom-model values unless an approximation or uncertainty is stated.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
A true statement need not have a true converse
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Analyze the statement 'If a quadrilateral is a square, then it is a rectangle.'
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Represent the conditions
Conditional: square implies rectangle
A square satisfies the four-right-angle definition of rectangle.
Develop the calculation
Converse: rectangle implies square
A 2-by-5 rectangle disproves the reversed implication.
Check and interpret
Contrapositive: not a rectangle implies not a square; this is true
The contrapositive is logically equivalent to the original conditional, unlike its converse.
The result
Contrapositive: not a rectangle implies not a square; this is true
The contrapositive is logically equivalent to the original conditional, unlike its converse.
Common mistakes to catch
- Reversing an implication requires a separate justification.
- State inclusive shape definitions to avoid classification ambiguity.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
Form the converse of 'If a triangle is equilateral, then it is isosceles,' using at least two equal sides for isosceles.
Show a hint
Swap hypothesis and conclusion.
Reveal answer and explanation
If a triangle is isosceles, then it is equilateral
That converse is false because exactly two equal sides need not make all three equal.
Practice 2
Is 'If not square, then not rectangle' the contrapositive of the main statement?
Show a hint
A contrapositive also reverses the order.
Reveal answer and explanation
No; it is the inverse and is false
A nonsquare rectangle gives a counterexample to that inverse.
Take the idea with you
Read theorem hypotheses carefully before applying a familiar conclusion backward.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Prove the triangle angle sum with a parallel line
Completion is your own study record, not a test score. It stays in this browser, does not sync to another device, and can be removed by clearing browser data.