Maths
Proof organiser: angles and reasons
Learn how a proof links statements to reasons, then apply the same structure to triangle angle facts. A worked parallel-lines proof establishes that a triangle’s angles total 180°, followed by a proof of the exterior-angle result and an isosceles-triangle extension.
Use the parallel line and matching angle marks to follow the triangle proof.
Read the complete materialRead the material
Learn how a proof links statements to reasons, then apply the same structure to triangle angle facts. A worked parallel-lines proof establishes that a triangle’s angles total 180°, followed by a proof of the exterior-angle result and an isosceles-triangle extension.
Complete resource. Use the preview for diagrams and layout.
Page 1
MATHS AGES 14 TO 16 ANGLES AND PROOF Proof organiser Read the model proof, then write your own on page 2. NAME CLASS DATE LEARNING GOAL Prove a fact about angles by writing statements, each with a reason. SUCCESS CRITERIA I can say what is given. I can give a reason for every statement. I can check my proof with numbers. GIVEN DE is parallel to BC. p, q and r are the angles at A. Matching arcs mark equal angles. NO. STATEMENT REASON 1 angle DAB (p) = angle ABC alternate angles are equal (parallel lines) 2 angle EAC (r) = angle ACB alternate angles are equal (parallel lines) 3 angle DAB + angle BAC + angle EAC = 180° (p + q + r = 180°) angles on a straight line add up to 180° 4 so angle ABC + angle BAC + angle ACB = 180° substitute from 1 and 2 into 3 A check with numbers works for one triangle. The proof works for every triangle. E N T R Y 1 PROVE THAT The angles in a triangle add up to 180°. DIAGRAM p q r D E A B C CONCLUSION The angles in a triangle add up to 180°. DEGREES + DEGREES + DEGREES = DEGREES CHECK WITH NUMBERS 50 60 70 180 Proof organiser · Ages 14 to 16 Page 1 of 2
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MATHS AGES 14 TO 16 YOUR PROOF Now you prove it NAME CLASS DATE NO. STATEMENT REASON 1 angle ACB + angle ACD = ° 2 angle A + angle B + angle ACB = ° 3 so angle ACD = ° − angle ACB = angle + angle angles on a straight line add up to 180° angles in a triangle add up to 180° alternate angles are equal given subtract the same from both sides E N T R Y 2 PROVE THAT The exterior angle of a triangle equals the sum of the two opposite interior angles. GIVEN BCD is a straight line. DIAGRAM ACD A B C D A = 50°, B = 60°. Work out: ACB = ° ACD = ° CHECK WITH NUMBERS REASONS BANK NO. STATEMENT REASON 1 2 3 Use as many rows as you need. Continue on the back. A B C M Prove that the base angles of an isosceles triangle are equal. Given: AB = AC. M is the midpoint of BC. Join A to M. GO FURTHER Proof organiser · Ages 14 to 16 Page 2 of 2
A teaching sequence
Build a geometric proof by recording statements, reasons and a conclusion in a logical order.
Use the parallel line and matching angle marks to follow the triangle proof.
Read the complete materialFollow the triangle-angle proof
Identify the parallel lines and equal alternate angles in the worked diagram. Follow the statement-and-reason table to prove a triangle’s angles total 180°.
Which parallel-line fact justifies the equal alternate angles?
Distinguish a check from a proof
Use the numerical check to test one example, then explain why the proof applies to every triangle.
Why does one numerical example not establish the result for every triangle?
Prove the exterior-angle result
Use straight-line and triangle-angle facts to prove the exterior angle equals the two opposite interior angles. Complete each reason and check with A = 50° and B = 60°.
What established fact justifies each statement in your table?
Plan the isosceles extension
For the extension, use AB = AC and midpoint M on BC to plan a proof of equal base angles. Check the established fact that justifies each line before accepting the proof.
How do the equal sides and midpoint contribute to the argument?
Before you teach
Look for the thinking behind an answer, choose the support your class needs and leave room for an independent response.
What to look for
Check that each statement follows from the given information or an established angle rule, and that no conclusion is assumed before it is proved. The numerical check can test an example but does not replace a general argument.
Give a little support
Provide a labelled diagram and a short list of angle facts. Pupils can annotate equal angles before writing the proof. Ask them to complete one statement and reason at a time, and use the numerical check to catch arithmetic errors.
Take the idea further
Complete the isosceles proof by considering the two smaller triangles formed by the median from the apex. Identify which sides and angles match, then choose a congruence argument and state the conclusion.
Things to keep in mind
The organiser supports two angle results and an extension. A numerical example is not proof, and a diagram is not sufficient evidence on its own. Use the proof conventions expected by the course and distinguish a reason from a calculation.
Keep the idea. Make it yours.
Bring your topic, learning goal and class context to PLAI™, then review what you create.
Ideas to try with your class
Sample teaching perspectives on what to try, what to adapt and what to watch for.
Meet The Creators
Meet the creators whose teaching interests connect with this template. Find an approach to try with your class.
Vikram Sethi
I use diagnostic Mathematics questions to locate a specific misconception before choosing a worked example, keeping assessment precise enough to guide the next teaching step and review.
Meet the creator
Tom Ridgway
I run Cornbrook events for pupils in every year group, so participation has to be possible beyond the students who are already confident about joining in.
Meet the creator
Open Book Pause
The first page models the triangle-angle proof using parallel lines, alternate angles and a straight line; the second asks pupils to prove an exterior-angle relationship. A 50°+60°+70° check illustrates one case but the proof must cover all triangles.
Spare Pencil
For ages 14 to 16, the organiser pairs each statement with a reason and a final conclusion. It distinguishes a numerical check from deductive proof, which is essential when pupils explain why the result is general.
Pocketful of Stories
Use the completed model to identify where substitution of equal angles occurs, then ask pupils to fill the exterior-angle proof one line at a time. Matching arcs support the visual links between angles.
Margin Note
A diagram and one successful numerical example do not prove the theorem. Require a reason for every statement and check that the written argument uses only given facts and established angle rules.