Hello! In our last lesson, we focused on applying circle theorems to find unknown angles. We saw how combining different theorems and basic angle facts allows us to solve complex geometric puzzles.
Today, we're going to leverage that skill to tackle a topic you've mentioned you find challenging: geometric proofs. The learning outcome for this lesson is to construct logical arguments using circle theorems to complete 'show that' geometric proofs.
"Show that" questions are different from "find the value" questions. The answer is given to you; the examiner wants to see how you get there. This is where a clear, logical argument is essential for securing method marks, even if you make a small error elsewhere. Mastering this will transform these intimidating questions into a predictable way to score marks.
1. The Mindset for a Proof
Think of a geometric proof as building a case in a courtroom. You start with the given evidence (the information in the question) and you need to reach a specific conclusion (the "show that" statement). Every step you take must be justified with a reason—a law, in this case, a circle theorem or a basic geometric rule.
Your toolkit of "laws" is exactly what we covered in the last lesson. Here is a quick visual reminder of the main theorems.
When you write a proof, you should structure your argument clearly:
- Statement: A factual claim about an angle or line (e.g., ).
- Reason: The theorem or rule that justifies your statement (e.g., Angle in a semicircle).
This structure is non-negotiable for earning full marks.
2. Strategy: Working Forwards and Backwards
The most effective way to tackle a proof is to work from both ends and meet in the middle.
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Work Forwards from the "Given":
- Read the problem and identify all the given information (e.g., "PQ is a tangent," "ABCD is a cyclic quadrilateral," "Line AB is parallel to line DC").
- For each piece of information, write down what you can immediately deduce. For example, if "PQ is a tangent," you can deduce that it meets a radius at 90° and that the alternate segment theorem might apply.
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Work Backwards from the "To Prove":
- Look at the final statement you need to prove (e.g., "Show that triangle XYZ is isosceles").
- Ask yourself: "What is the last thing I would need to prove to show this?" To prove a triangle is isosceles, you need to show either two sides are equal or two base angles are equal. This becomes your new, intermediate goal.
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Bridge the Gap: Now, look at the deductions you made from the "Given" and the intermediate goal you identified by working backwards. Can you find a logical path that connects them?
Let's see this strategy in action.
3. Worked Example 1: Proving an Angle Relationship
This first example is a classic "prove that" question involving algebra. The key is to introduce a variable and use the theorems to build an expression that matches the target.
#36 Circle Theorems - Edexcel IGCSE Exam Questions
The following clip from a video by Mr Astbury demonstrates how to construct a proof for an angle relationship. Pay close attention to how he uses the 'alternate segment theorem' and 'opposite angles in a cyclic quadrilateral' to build his argument step-by-step.
Watch the section from 10:44 to 12:27. Notice the logical flow: Key Insight: A line is drawn to create an isosceles triangle. Step 1: The alternate segment theorem is used to label an angle as 'x'. Step 2: Properties of an isosceles triangle are used to find another angle 'x'. Step 3: 'Angles in a triangle sum to 180°' is used to find the third angle of the new triangle in terms of 'x'. Step 4: 'Opposite angles of a cyclic quadrilateral sum to 180°' is used to find the final angle, which simplifies to the required expression '2x'.
This example shows how combining multiple theorems creates a chain of logic that leads directly to the required conclusion.
4. Worked Example 2: Proving that Triangles are Similar
Another common type of "show that" question asks you to prove that two shapes are similar. For triangles, this means you must prove that all three pairs of corresponding angles are equal.
Everything you need to know about IGCSE Circle Theorems in 47 minutes
In this next clip, the Ginger Mathematician proves that two triangles formed by intersecting chords are similar. This is a very common exam-style question.
Watch the explanation from 11:34 to 14:04. As you watch, identify the three pairs of equal angles and the reason given for each pair. Note them down.
Test your understanding!
What were the three reasons used to prove the triangles were similar?
Show answer
- Vertically opposite angles are equal. (The angles at the intersection point X)
- Angles in the same segment are equal. (For the first pair of angles on the circumference)
- Angles in the same segment are equal. (For the second pair of angles on the circumference)
Since all three pairs of corresponding angles are equal (AAA), the triangles are similar.
5. Practice: Constructing a Proof from Scratch
Now it's your turn. Let's use our forwards-backwards strategy to build a proof for a typical IGCSE question.
Problem:
AB is a chord of a circle and CT, the tangent at C, is parallel to AB. Prove that CA = CB.
Your Task:
Follow the steps below to construct the argument.
Step 1: Work Forwards and Backwards
- To Prove: CA = CB.
- Work Backwards: What do you need to show to prove that CA = CB? (Hint: Think about the properties of the triangle that contains these sides).
- Given: We have two key pieces of information:
- CT is a tangent at C. What theorems does this suggest?
- CT is parallel to AB. What angle rules does this suggest?
Show hints
- To prove
CA = CB, you need to prove that triangle ABC is an isosceles triangle. To do this, you must show that the base angles are equal:∠CAB = ∠CBA. This is now your target. - "CT is a tangent" suggests using the Alternate Segment Theorem.
- "CT is parallel to AB" suggests using rules for alternate angles (Z-angles).
Step 2: Build the Bridge
Now, use the theorems you identified to link the given information to your target of proving ∠CAB = ∠CBA. Try to write down the chain of reasoning yourself before revealing the solution.
Show the full proof
Here is the complete logical argument. Notice how each statement has a reason.
- Let .
- Then .
- Reason: Alternate Segment Theorem (the angle between the tangent CT and the chord BC is equal to the angle in the alternate segment).
- Also, .
- Reason: Alternate Interior Angles (since line CT is parallel to line AB, with BC as the transversal).
- Therefore, .
- Since two angles in triangle ABC are equal, the triangle is isosceles.
- Therefore, the sides opposite these equal angles must be equal in length: CA = CB.
This completes the proof. By linking the parallel line rule and the alternate segment theorem, we were able to show the two base angles were equal.
Conclusion
You now have a structured method for tackling geometric proofs, a skill that directly addresses one of your main areas for improvement. The key is not to be intimidated, but to see it as a puzzle where you must show your work.
Key Takeaways:
- Every Statement Needs a Reason: This is how you collect method marks. Get used to writing the theorem in brackets after every geometric statement.
- Work Forwards and Backwards: Identify what you're given and what you need to prove. This helps you find the most efficient path instead of getting lost in the diagram.
- Proofs are Chains of Logic: One theorem often unlocks the door to using another. Look for these connections.
Preview of the Next Lesson:
We will now shift gears from pure geometry to Trigonometry. In the next lesson, you will apply SOHCAHTOA in right-angled triangles to find unknown sides and angles. This will bring us back to more calculation-focused problems, so have your calculator ready