Deductive proof: a week of ready-to-teach maths
Five lessons for Year 10 Space. Print this pack and the week is prepared: each day has a one-page plan and a student worksheet with practice and a stretch section, plus twelve angle and reason cards to match, a parallelogram proof cut into twelve pieces to rebuild, a mini-check and every answer with its working.
Start here: five minutes to Monday
- Skim the week at a glance on the next page.
- Print the five days. Each day is two A4 sheets: a plan and a worksheet.
- Print the two cut-out sheets, one copy per pair. Cut the 12 angle and reason cards before Day 1 and the 12 proof pieces before Day 2, and keep each set in an envelope.
- Rulers and protractors help students see why measuring is not proving. Calculators are needed on Day 3 only.
- Open the free interactive unit on the board. Every plan says which picture to show and which buttons to press.
The idea in plain words
Measuring one diagram shows that something is true for that drawing; a proof shows it is true for every case. A deductive proof starts from what is given and moves one step at a time, and every step carries a reason: an angle fact, a congruence or similarity test, or a theorem proved before. Once a result is proved it becomes a tool for solving new problems, such as finding an angle or measuring a river without crossing it. One step without a reason and the whole proof fails, even if its conclusion is true.
One day, one lesson
The five lessons fill a week of maths, one lesson of about 60 minutes a day. Run them in order: each day stands on the one before. Every plan marks a split point for two shorter sessions, and every worksheet ends with a stretch section for students who finish early.
How we write these
A proof is set out in two columns: each claim on the left, its reason on the right. Angles are written ∠ABC or ∠B, triangles△ABC, and “is congruent to” as ≅. Diagrams are not always to scale: the reasons, not the look of a picture, carry the proof.
The week at a glance
One lesson a day for a week. Each day stands on the day before, so run them in order.
| Day | Lesson | Students learn and do | On screen |
|---|---|---|---|
| 1 | Reasons and proof | Give a reason for every angle step, prove that vertically opposite angles are equal and that a triangle’s angles add to 180°, and match six diagrams to their reasons | Measuring shows one case; proving covers all; worked proof |
| 2 | Congruent triangles | Name the congruence test or say why there is none, write a congruence proof, and rebuild a cut-up proof about a parallelogram | A proof is a chain of justified steps |
| 3 | Similar triangles | Decide whether triangles are similar and by which test, prove the midpoint theorem, and use a parallel line to find a length | Using a theorem to solve a spatial problem |
| 4 | Proving properties of quadrilaterals | Sort diagonal properties of five quadrilaterals, then prove two of them with congruent triangles | A proof is a chain of justified steps |
| 5 | Theorems at work | Chain reasons to find angles, find the gap in a flawed proof, measure a river with congruent triangles, and play the self-check quiz as a class game | One gap breaks the whole proof; self-check quiz as a class game |
How the week builds
Day 1 sets out what a reason is and what a proof looks like; Days 2 and 3 add the congruence and similarity tests as reasons; Day 4 uses them to prove properties of quadrilaterals; and Day 5 puts proved results to work. It builds on Year 9, where students used congruence and similarity and the angle facts for parallel lines, and it prepares the formal geometry and proof of senior mathematics.
Materials for the week (one trip)
- From the classroom: rulers, protractors, scissors, envelopes for the cards, this pack printed.
- Cut-out sheet 1, one per pair: 12 angle and reason cards (Day 1).
- Cut-out sheet 2, one per pair: 12 proof pieces (Day 2).
- Calculators for Day 3. No other equipment to buy.
Words for the word wall
given, prove, deduce, reason, theorem, proof, vertically opposite, alternate, co-interior, congruent, similar, SSS, SAS, AAS, RHS, bisect, perpendicular, counterexample
Dear families
This week in maths, Year 10 is learning to prove things about shapes: not just to see that something looks true, but to show it must be true every time, with a reason for every step.
Try this at home
- Ask: you measured three triangles and each one’s angles added to 180°. Have you proved it? (No: three cases are not every case.)
- Ask: why does a gate brace (a diagonal) stop a gate from sagging? (A triangle cannot change shape once its sides are fixed.)
- Ask: two triangles have the same three angles. Must they be the same size? (No: one can be an enlargement of the other.)
- Ask: how could you find the width of a river without crossing it? (Ask again on Friday.)
My reasons week
Fill one row a day. Write something you know is true, and the reason that makes it true every time.
| Day | Something that is true | The reason |
|---|---|---|
| Monday | ||
| Tuesday | ||
| Wednesday | ||
| Thursday | ||
| Friday |
Reasons and proof
A proof moves from what is given to what is claimed, one step at a time, and every step needs a reason. Students give reasons for angle facts, prove that vertically opposite angles are equal and that the angles of a triangle add to 180°, and match six diagrams to their reasons.
We are learning to
- name the reason for each step: a line, a point, vertically opposite, alternate, co-interior,
- set out a proof as claims with reasons,
- explain why measuring one diagram does not prove a result.
Success criteria
- I can prove that vertically opposite angles are equal from angles on a straight line.
- I can prove that the angles of any triangle add to 180°.
You need
Protractors and rulers; the angle and reason cards (cut-out sheet 1), one set per pair; the worksheet.
Lesson flow (about 60 minutes)
| 12 min | Measure or prove? Show the board (see the box), then each student draws a triangle and measures its three angles. Ask: “Everyone’s angles came to about 180°. Have we proved it for every triangle?” |
| 28 min | Two first proofs Pairs complete Parts A and B. Ask: “What gives us the right to write this step?” |
| 20 min | Match the reasons Pairs match the twelve angle and reason cards and find each x (Part C); early finishers do Part D. |
Two shorter sessions? Stop after Part A; start the next with Part B.
Watch for these ideas
- Measuring as proof: a protractor checks one drawing.
- A reason that is the answer: “x = 70°” is not a reason; “angles on a straight line” is.
- Parallel facts without parallel lines: look for the arrows first.
Answers
- Part A: a + b = 180°: angles on a straight line add to 180°; b + c = 180°: angles on a straight line add to 180°; a = c: both equal 180° − b.
- Part B: x = b: alternate angles on parallel lines are equal; y = c: alternate angles on parallel lines are equal; x + a + y = 180°: angles on a straight line add to 180°; a + b + c = 180°: put b for x and c for y.
- Part C: the card key and each x are with the mini-check answers.
- Part D: c + d = 180° (angles on a straight line) and a + b + c = 180° (Part B), so d = 180° − c = a + b: an exterior angle equals the two opposite interior angles.
Reasons and proof
A. Prove: vertically opposite angles are equal
Write the reason for each claim. The first is done for you.
| Claim | Reason |
|---|---|
| a + b = 180° | Angles on a straight line add to 180° |
| b + c = 180° | |
| a = c |
B. Prove: the angles of a triangle add to 180°
| Claim | Reason |
|---|---|
| Draw the line through A parallel to BC | Construction |
| x = b | |
| y = c | |
| x + a + y = 180° | |
| a + b + c = 180° |
C. Match the reasons
Match each diagram card to its reason card, then find x. Write the letters and x.
| Pair 1: ____ and ____ x = ______ | Pair 2: ____ and ____ x = ______ |
| Pair 3: ____ and ____ x = ______ | Pair 4: ____ and ____ x = ______ |
| Pair 5: ____ and ____ x = ______ | Pair 6: ____ and ____ x = ______ |
There are 6 diagrams, each needing a different reason.
D. Stretch: the exterior angle
Congruent triangles
Two triangles are congruent when one fits exactly on the other. Four tests, SSS, SAS, AAS and RHS, each need only three facts, and some sets of three facts are not enough. Students name the test or say why there is none, write a congruence proof, and rebuild a cut-up proof about a parallelogram.
We are learning to
- name the congruence test a set of facts gives, or say why none does,
- match the vertices of congruent triangles in order,
- use congruent triangles to prove a new fact.
Success criteria
- I can explain why three equal angles, or two sides and an angle not between them, do not prove congruence.
- I can prove that the line from the apex of an isosceles triangle to the middle of its base is perpendicular to the base.
You need
Rulers and protractors; the proof pieces (cut-out sheet 2), one set per pair; the worksheet.
Lesson flow (about 60 minutes)
| 12 min | Only three facts Show the board (see the box), then everyone draws a triangle with sides 5, 6 and 7 cm. Ask: “Is yours the same as your partner’s? Now draw one with angles 40°, 60° and 80°: the same again?” |
| 28 min | Name the test, then prove Pairs complete Parts A and B. Ask: “Which letters match in your congruence statement, and why does the order matter?” |
| 20 min | Rebuild the proof Pairs order the six claims and pair each with its reason (Part C); early finishers do Part D. |
Two shorter sessions? Stop after Part A; start the next with Part B.
Watch for these ideas
- Three equal angles as a test: same shape, not always the same size.
- Two sides and any angle: for SAS the angle must sit between the two sides.
- Vertices out of order: △ABC ≅ △PQR means A with P, B with Q, C with R.
Answers
- Part A: SSS; SAS; AAS; RHS; none: same shape, maybe not the same size; none: the angle is not between the two sides.
- Part B: BM = CM: M is the midpoint of BC; AM = AM: A common side; △ABM ≅ △ACM: SSS; ∠AMB = ∠AMC: Matching angles of congruent triangles; ∠AMB + ∠AMC = 180°: Angles on a straight line add to 180°; ∠AMB = 90°, so AM ⊥ BC: Two equal angles that add to 180°.
- Part C: the order and pairs are with the mini-check answers.
- Part D: with the diagonals meeting at O, △AOB ≅ △COD by AAS (alternate angles at A and C, alternate angles at B and D, and AB = CD from Part C), so AO = CO and BO = DO.
Congruent triangles
A. Which test, if any?
Each row says what is known about triangles ABC and PQR. Name the congruence test, or say why there is none. The first is done for you.
| Known | Test, or why none |
|---|---|
| AB = PQ, BC = QR, CA = RP | SSS |
| AB = PQ, ∠B = ∠Q, BC = QR | |
| ∠A = ∠P, ∠B = ∠Q, BC = QR | |
| ∠C = ∠R = 90°, AB = PQ, AC = PR | |
| ∠A = ∠P, ∠B = ∠Q, ∠C = ∠R | |
| AB = PQ, BC = QR, ∠A = ∠P |
B. Prove: AM is perpendicular to BC
In triangle ABC, AB = AC and M is the midpoint of BC. Write the reason for each claim.
| Claim | Reason |
|---|---|
| AB = AC | Given |
| BM = CM | |
| AM = AM | |
| △ABM ≅ △ACM | |
| ∠AMB = ∠AMC | |
| ∠AMB + ∠AMC = 180° | |
| ∠AMB = 90°, so AM ⊥ BC |
C. Rebuild the proof
The proof that opposite sides of a parallelogram are equal has been cut up. Put the 6 claim cards in order and pair each with its reason card.
| Step 1: claim ____ reason ____ | Step 2: claim ____ reason ____ |
| Step 3: claim ____ reason ____ | Step 4: claim ____ reason ____ |
| Step 5: claim ____ reason ____ | Step 6: claim ____ reason ____ |
D. Stretch: the diagonals
Similar triangles
Similar triangles have equal angles and their sides in one ratio. Three tests prove it: equal angles (AAA), all three sides in one ratio (SSS), and two sides in one ratio with the angle between them equal (SAS). Students test pairs of triangles, prove the midpoint theorem, and use a parallel line to find a length.
We are learning to
- decide whether two triangles are similar and name the test,
- use similar triangles to prove a result,
- find an unknown length from the scale factor.
Success criteria
- I can show why a line parallel to one side of a triangle cuts off a smaller similar triangle.
- I can prove that the segment joining two midpoints is half the third side.
You need
Calculators and rulers; the worksheet.
Lesson flow (about 60 minutes)
| 12 min | Same shape? Show the board (see the box). Ask: “Every triangle with angles 50°, 60° and 70°: are they congruent? Similar? Why do two angles settle it?” |
| 28 min | Test and prove Pairs complete Parts A and B. Ask: “Which side matches which? How do you know before you divide?” |
| 20 min | Find the length Pairs complete Part C; early finishers do Part D. |
Two shorter sessions? Stop after Part A; start the next with Part B.
Watch for these ideas
- Pairing sides out of order: shortest with shortest, longest with longest.
- Using a part for the whole: AB is AD + DB.
- Checking two ratios of three: all three must agree for SSS.
Answers
- Part A: SSS: every side × 1.5; AAA: the third angles are 65° and 40°; SAS: both sides × 2, same angle between; not similar: 6/3 = 8/4 but 9/5 is not 2.
- Part B: ∠MAN = ∠BAC: A common angle; △AMN ∼ △ABC: SAS: two sides in one ratio, the angle between them equal; ∠AMN = ∠ABC: Matching angles of similar triangles; MN ∥ BC: Corresponding angles are equal; MN = BC ÷ 2: Matching sides are in the ratio 1 : 2.
- Part C: ∠A is common and ∠ADE = ∠ABC (corresponding angles, DE ∥ BC), so the triangles are similar (AAA) with k = 10 ÷ 4 = 2.5; BC = 5 × 2.5 = 12.5.
- Part D: ∠ADC = ∠CDB = 90° and ∠CAD = ∠BCD (each is 90° less ∠ABC), so the triangles are similar; CD ÷ 4 = 9 ÷ CD, so CD² = 36 and CD = 6.
Similar triangles
A. Similar or not?
For each pair, say whether the triangles are similar, name the test, and give the scale factor where the sides allow. The first is done for you.
| First triangle | Second triangle | Similar? Test and k |
|---|---|---|
| sides 4, 6, 8 | sides 6, 9, 12 | SSS: every side × 1.5 |
| angles 40° and 75° | angles 75° and 65° | |
| sides 5 and 7 with 50° between | sides 10 and 14 with 50° between | |
| sides 3, 4, 5 | sides 6, 8, 9 |
B. Prove the midpoint theorem
In triangle ABC, M is the midpoint of AB and N is the midpoint of AC. Sketch it, then write the reason for each claim.
| Claim | Reason |
|---|---|
| AM ÷ AB = AN ÷ AC = 1 ÷ 2 | M and N are the midpoints of AB and AC |
| ∠MAN = ∠BAC | |
| △AMN ∼ △ABC | |
| ∠AMN = ∠ABC | |
| MN ∥ BC | |
| MN = BC ÷ 2 |
C. A line parallel to one side
D. Stretch: the altitude of a right triangle
Proving properties of quadrilaterals
The special quadrilaterals share some diagonal properties and not others, and each property can be proved with congruent triangles. Students sort four diagonal properties across five shapes, then prove that a rhombus has perpendicular diagonals and that a parallelogram with equal diagonals is a rectangle.
We are learning to
- state which diagonal properties each special quadrilateral has,
- prove a property of a quadrilateral with congruent triangles,
- prove a converse: from a property back to the shape.
Success criteria
- I can prove that the diagonals of a rhombus cross at right angles.
- I can explain why a parallelogram with equal diagonals must be a rectangle.
You need
Rulers and protractors; paper strips or straws to build quadrilaterals if you have them; the worksheet.
Lesson flow (about 60 minutes)
| 12 min | What do the diagonals do? Show the board (see the box), then pairs draw a parallelogram, a rhombus and a kite with their diagonals. Ask: “Which diagonals cut each other in half? Which cross at right angles? Which are equal?” |
| 28 min | Sort, then prove Pairs complete Parts A and B. Ask: “Which two triangles will you prove congruent, and with which test?” |
| 20 min | Back the other way Pairs complete Part C; early finishers do Part D. |
Two shorter sessions? Stop after Part A; start the next with Part B.
Watch for these ideas
- Using the result in its own proof: the right angle cannot be a step on the way to it.
- Giving every parallelogram the rhombus properties: perpendicular diagonals need equal sides.
- Forgetting that a square is both a rectangle and a rhombus.
Answers
- Part A: bisect each other: parallelogram, rectangle, rhombus, square; equal: rectangle, square; at right angles: rhombus, square, kite; bisect angles: rhombus, square, kite.
- Part B: AB = AD: All four sides of a rhombus are equal; BO = DO: The diagonals of a parallelogram bisect each other; AO = AO: A common side; △ABO ≅ △ADO: SSS; ∠AOB = ∠AOD: Matching angles of congruent triangles; ∠AOB = ∠AOD = 90°: Two equal angles on a straight line.
- Part C: AB = DC, AD is common and BD = AC, so △ABD ≅ △DCA (SSS) and ∠BAD = ∠CDA; as co-interior angles they add to 180°, so each is 90°.
- Part D: OA = OB = OC (radii), so ∠OAC = ∠OCA = x and ∠OBC = ∠OCB = y (isosceles); 2x + 2y = 180°, so ∠ACB = x + y = 90°.
Proving properties of quadrilaterals
A. Which shapes always have it?
Write yes or no. The first row is done for you.
| Property | Parallelogram | Rectangle | Rhombus | Square | Kite |
|---|---|---|---|---|---|
| The diagonals bisect each other | yes | yes | yes | yes | no |
| The diagonals are equal | |||||
| The diagonals cross at right angles | |||||
| A diagonal bisects the angles it joins |
B. Prove: the diagonals of a rhombus are perpendicular
Sketch rhombus ABCD with its diagonals meeting at O, then write the reason for each claim.
| Claim | Reason |
|---|---|
| ABCD is a rhombus; its diagonals meet at O | Given |
| AB = AD | |
| BO = DO | |
| AO = AO | |
| △ABO ≅ △ADO | |
| ∠AOB = ∠AOD | |
| ∠AOB = ∠AOD = 90° |
C. Back the other way
D. Stretch: the angle in a semicircle
Theorems at work
A proved theorem is a tool. Students chain reasons to find angles, find the step with no valid reason in a flawed proof, and measure a river without crossing it, using congruent triangles; then the self-check quiz as a class game.
We are learning to
- chain reasons to find an unknown angle,
- find and repair a step that has no valid reason,
- use a congruence or similarity argument to solve a practical problem.
Success criteria
- I can find every angle of a parallelogram from one angle, with a reason for each.
- I can explain how two congruent triangles give the width of a river.
You need
Rulers; a long tape and four pegs to pace out the river problem outside, if you have time; the worksheet; the self-check quiz on the board.
Lesson flow (about 60 minutes)
| 12 min | Chain the reasons Show the board (see the box), then do Part A1 together. Ask: “Which fact comes first, and which one does it hand to the next step?” |
| 30 min | Gaps and rivers Pairs complete Parts A to C. Ask: “The flawed proof reached a conclusion. Why is that not enough?” |
| 18 min | Quiz Play the self-check quiz as a class game; early finishers do Part D. |
Two shorter sessions? Stop after Part B; start the next with Part C.
Watch for these ideas
- Looks-like reasons: “it looks like a right angle” proves nothing.
- Circular reasoning: using the conclusion as one of the steps.
- Mixing congruent and similar: congruence gives equal lengths, similarity lengths in a ratio.
Answers
- Part A: 1. the base angles are equal (isosceles) and all three add to 180°, so each is (180° − 40°) ÷ 2 = 70°. 2. ∠B = 115° (co-interior angles, AD parallel to BC), ∠C = 65° and ∠D = 115° (co-interior again).
- Part B: step 4: ∠ABC and ∠BAD are co-interior, so they add to 180° and are equal only when each is 90°. The conclusion is false for a parallelogram that is not a rectangle.
- Part C: ∠ABC = ∠EDC = 90° (given), BC = DC = 40 m (paced) and ∠ACB = ∠ECD (vertically opposite), so the triangles are congruent (AAS) and AB = DE = 63 m.
- Part D: the triangles are similar (AAA) with BC = 2 × CD, so AB = 2 × DE: measure DE and double it.
Theorems at work
A. Chain the reasons
B. Find the gap
A student wrote this “proof” that the diagonals of every parallelogram are equal.
| Claim | Reason |
|---|---|
| ABCD is a parallelogram | Given |
| AB = BA | A common side |
| BC = AD | Opposite sides of a parallelogram are equal |
| ∠ABC = ∠BAD | They look equal in the diagram |
| △ABC ≅ △BAD | SAS |
| AC = BD, so the diagonals are equal | Matching sides of congruent triangles |
C. Across the river
D. Stretch: a shorter walk
Diagrams and their reasons
For Day 1, one set per pair. Six cards show a diagram with an unknown angle x; six give a reason. Match each diagram to the reason that finds x.
Teacher note: the key and each x are with the mini-check answers.
Rebuild the proof
For Day 2, one set per pair. These are the claims and reasons of one proof that opposite sides of a parallelogram are equal. Put the six claims in order and give each its reason.
Teacher note: the order and pairs are with the mini-check answers.
What we know: Deductive proof
Work on your own. Give a reason for every step.
- Two angles, x and 128°, sit side by side on a straight line. Find x and give the reason. ______________________
- A line crosses two parallel lines, making a Z shape. One angle inside the Z is 74°; x is the other. Find x and give the reason. ______________________
- Triangles ABC and PQR have AB = PQ, ∠A = ∠P and AC = PR. Name the congruence test. ______________________
- Why do three equal angles not prove that two triangles are congruent? ______________________
- In triangle ABC, DE is parallel to BC with D on AB and E on AC. AD = 3, AB = 9 and DE = 4. Find BC. ______________________
- Checking a result on three carefully drawn triangles is, circle one: a proof · not a proof, only examples · a proof if the drawings are accurate
- The exterior angle at C of triangle ABC is 125° and ∠A = 50°. Find ∠B and give the reason. ______________________
- ABCD is a parallelogram with ∠A = 72°. Find ∠B and ∠C, with reasons. ______________________
How did this week feel? Circle one: I can teach it · I am getting there · I need more help
Answers and marking guide
Answers with working
- x = 180° − 128° = 52°: angles on a straight line add to 180°.
- x = 74°: alternate angles on parallel lines are equal.
- SAS: two sides and the angle between them.
- They fix the shape but not the size: an enlargement has the same angles.
- The triangles are similar (AAA) with k = 9 ÷ 3 = 3, so BC = 4 × 3 = 12.
- It is not a proof, only examples: three cases are not every case.
- ∠B = 125° − 50° = 75°: an exterior angle equals the two opposite interior angles.
- ∠B = 108° (co-interior angles add to 180°); ∠C = 72° (co-interior with ∠B, or opposite angles of a parallelogram).
A quick three-level guide
| Idea | Working towards | At standard | Beyond |
|---|---|---|---|
| Reasons | Measures, or gives the answer as its own reason | Gives a correct reason for each angle step | Chains several reasons and names each one exactly |
| Congruence | Uses AAA or two sides and any angle as a test | Names the right test and matches the vertices in order | Writes a full congruence proof and uses its result |
| Similarity | Pairs sides out of order or uses a part for the whole | Proves triangles similar and finds a length from k | Proves a theorem, such as the midpoint theorem, with similarity |
| Proof | Accepts a step because it looks right | Sets out claims and reasons from givens to conclusion | Finds the gap in a flawed proof and explains its effect |
Eight questions, four ideas. A student at standard gives a correct reason for every step and never leans on how a diagram looks.
Card keys
Angle and reason cards (Day 1): G with A, x = 70°; D with J, x = 65°; E with L, x = 120°; B with H, x = 55°; K with F, x = 105°; I with C, x = 75°. Proof pieces (Day 2), in order: U with O, then W with R, then T with M, then Q with V, then N with X, then P with S.
Weekly class record
Jot a tick as you move around the room; the mini-check fills any gaps. A tick a day is plenty.
| Name | Reasons | Congruent | Similar | Quadrilaterals | Theorems |
|---|---|---|---|---|---|
The five columns are the five days: reasons and proof, congruent triangles, similar triangles, quadrilaterals, and theorems at work.