Congruence and similarity: a week of ready-to-teach maths

Five lessons for Year 8 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 shape cards, a kit of eight triangles to cut out and compare, triangles to construct, a mini-check and every answer with its working.

Start here: five minutes to Monday

1. Skim the week at a glance on the next page.
2. Print the five days. Each day is two A4 sheets: a plan and a worksheet.
3. Print the two cut-out sheets, one copy per pair: the shape cards (Day 1) and the triangle kit (Days 3 and 4).
4. Open the free interactive unit on the board. Every plan says which picture to show and which buttons to press.
5. Teach straight from the plan. Timings, talk prompts, misconceptions and worked answers are all on the one page.

The idea in plain words

Two shapes are congruent when one can be moved exactly onto the other by translating, rotating or reflecting it: the same shape and the same size. They are similar when an enlargement, perhaps with those moves, maps one onto the other: the same shape, any size, with every length multiplied by one scale factor and every angle unchanged. For triangles, a few facts are enough. SSS, SAS, ASA and RHS each fix a triangle, so two triangles that share one of these sets of facts are congruent. Equal angles (AAA), sides in one ratio (SSS) or two sides in one ratio with the angle between them equal (SAS) make two triangles similar. For other shapes, sides and angles must both match.

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.

Naming the tests

S stands for a side, A for an angle, R for a right angle and H for the hypotenuse. The order matters: in SAS the angle is between the two sides. Some books use AAS for two angles and any matching side; once two angles are known the third is known too, so it comes to the same as ASA.

The week at a glance

One lesson a day for a week. Each day stands on the day before, so run them in order.

How the week builds

Day 1 sorts shapes by eye and by transformation; Days 2 and 3 find which facts fix a triangle; Day 4 does the same for similar triangles; and Day 5 uses scale factors and widens out to other shapes. It builds on transformations and angle facts from Year 7, and prepares the next unit, which proves properties of quadrilaterals with congruent triangles, and Year 9, where similar right-angled triangles lead to the sine, cosine and tangent ratios and enlargements are explored further.

Materials for the week (one trip)

• From the classroom: pencils, rulers, protractors, pairs of compasses, scissors, calculators, this pack printed.
• Cut-out sheet 1, one per pair: 12 shape cards (Day 1).
• Cut-out sheet 2, one per pair: the triangle kit, eight triangles to cut out (Days 3 and 4).
• Optional: tracing paper, to check congruence by tracing instead of cutting.

Words for the word wall

congruent, similar, corresponding, transformation, translation, rotation, reflection, enlargement, scale factor, ratio, included angle, hypotenuse, SSS, SAS, ASA, RHS, AAA, ambiguous, construct

Dear families

This week in maths, Year 8 is finding out when two shapes are exact copies (congruent) and when one is an enlargement of the other (similar), and which few measurements are enough to be sure for triangles.

Try this at home

• Look for congruent shapes: tiles, windows, the two halves of a symmetrical object.
• Compare a photo with a smaller print of it: is every length shrunk by the same factor?
• Ask: are all rectangles similar? (No: a long thin one and a square-ish one have different shapes.)
• On a sunny day, measure your shadow and a pole’s shadow, then work out the pole’s height.

My shapes this week

Fill one row a day. Name two shapes, say whether they are congruent, similar or neither, and how you know.

Printed from the free seegongsik Congruence and similarity teaching pack · seegongsik.com/au/y8/space/AC9M8SP01/pack

Congruent or similar?

Congruent shapes are exact copies; similar shapes are enlargements. Students sort twelve shape cards into congruent, similar and neither, then link each kind to the transformations that make it.

We are learning to

• tell congruent, similar and neither apart by checking both sides and angles,
• link translations, rotations and reflections to congruent images, and enlargements to similar ones,
• explain why equal sides, or equal areas, are not enough.

Success criteria

• I can say which transformations give a congruent image and which give a similar one.
• I can explain why a square and a rhombus with equal sides are not similar.

You need

Cut-out sheet 1 (the shape cards) and scissors, one set per pair; the worksheet each; tracing paper if you have it.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part A; start the second with Part B.

Watch for these ideas

• Calling a turned or flipped copy “not congruent”.
• Thinking equal sides are enough (card 7), or equal areas (card 10).
• Calling any two shapes of the same kind similar, such as any two rectangles.

Answers

• Part A: congruent 1, 4, 6, 9, 11; similar 2, 5, 8, 12; neither 3, 7, 10.
• Part B: 1 a rotation (a quarter turn), congruent; 2 an enlargement (scale factor 2), similar; 3 a translation, congruent; 4 a reflection, congruent.
• Part C: card 7: every side is the same length, but the square’s angles are 90° and the rhombus’s are not. Card 10: the same base and height, so the same area, but the angles differ, so the sides cannot be in one ratio.
• Part D: yes: congruent shapes are similar with scale factor 1. Similar shapes are congruent only when the scale factor is 1.

Congruent or similar?

A. Sort the cards

Sort the twelve shape cards. Check sides by counting grid squares, and angles by eye or with a corner of paper.

How did you decide for the hardest card?

B. Which move made it?

Each of shapes 1 to 4 is an image of shape P. Name the transformation and say whether the image is congruent or similar to P.

C. Two traps

Card 7 shows a square and a rhombus with all their sides equal. Card 10 shows two triangles with the same base and height. Explain why neither pair is similar.

D. Stretch: both at once?

Can two shapes be congruent and similar at the same time? Can they be similar but not congruent? Explain using the scale factor.

Tests for congruent triangles

Three facts can be enough to fix a triangle. Students construct triangles from SSS, SAS, ASA and RHS facts with a ruler, protractor and compasses, then compare with a partner: if every triangle matches, those facts fix the triangle.

We are learning to

• construct a triangle from three given facts,
• recognise the four sets of facts that fix a triangle: SSS, SAS, ASA and RHS,
• name the test that shows two triangles are congruent, matching the corners in order.

Success criteria

• I can construct a triangle with sides 4 cm, 5 cm and 6 cm using compasses.
• I can say why SAS needs the angle between the two sides.

You need

A ruler, a protractor and a pair of compasses each; the worksheet; scissors for comparing.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part A; start the second with Part B.

Watch for these ideas

• Reading an angle from the wrong scale of the protractor.
• Putting the SAS angle at the end of a side instead of between the two sides.
• Matching triangles by their letters instead of by their equal parts (Part C).

Answers

• Part A: SSS angles about 41°, 56°, 83°; SAS third side about 3.9 cm; ASA third angle 80°, other sides about 5.3 cm and 3.9 cm; RHS third side 4 cm (5² − 3² = 16). Every pair’s triangles match.
• Part B: 1. SSS; 2. SAS; 3. ASA; 4. RHS; 5. no test: the angle is not between the sides (SSA).
• Part C: no. AB = 7 cm matches QR, BC matches RP and CA matches PQ, so the statement is △ABC ≡ △QRP.
• Part D: four sides do not fix the angles: a square and a rhombus can both have four 3 cm sides.

Tests for congruent triangles

A. Construct and compare

Construct each triangle in its box. Measure the parts you were not given, then compare with a partner.

B. Name the test

For each pair, name the test that shows the triangles are congruent, or write “no test”.

C. Matching order

Kai writes △ABC ≡ △PQR for pair 1, “because both have sides 5, 6 and 7”. Is the statement right? If not, write it correctly.

D. Stretch: four sides

Is there an SSSS test for quadrilaterals? Sketch two different quadrilaterals that both have four 3 cm sides.

Facts that do not fix a triangle

Not every three facts fix a triangle. Students construct two different triangles from the same SSA facts, see that AAA fixes the shape but not the size, then cut out the triangle kit and find the congruent pairs by laying one on another.

We are learning to

• show that SSA can give two different triangles, so it is not a test,
• explain why AAA fixes the shape but not the size,
• check congruence by matching sides and angles, flipping a shape over if needed.

Success criteria

• I can construct both triangles for one set of SSA facts.
• I can find the two congruent pairs in the triangle kit and say which one I flipped.

You need

Cut-out sheet 2 (the triangle kit) and scissors, one per pair; a ruler, a protractor and compasses each.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part B; start the second with the kit.

Watch for these ideas

• Stopping at the first SSA triangle without looking for the second.
• Calling triangles with equal angles congruent.
• Rejecting a match that needs a flip: a reflection keeps a shape congruent.

Answers

• Part A: the arc of radius 4.5 cm from B cuts the base twice, so AC is about 6.9 cm or about 2.3 cm: the same three facts give two triangles.
• Part B: the two triangles have the same angles but different sides: similar, not congruent.
• Part C: A and H, and E and G; one of each pair is flipped over (E and H are printed flipped). C looks like A, but one side is longer.
• Part D: one triangle. A 7 cm arc from B is longer than AB, so it meets the base line only once on the side of A that makes a triangle.

Facts that do not fix a triangle

A. Two triangles from SSA

Draw a long base line and mark A at its left end. Draw AB = 6 cm at 40° to the base. Set your compasses to 4.5 cm, put the point on B, and draw an arc across the base line. Join B to each place the arc meets the base.

How many triangles have AB = 6 cm, ∠A = 40° and BC = 4.5 cm? AC = ______ or ______

B. Angles only

Construct two triangles with angles 50°, 60°, 70°: one with a 4 cm side between the 50° and 60° angles, one with a 6 cm side there.

Are the two triangles congruent? Are they similar? Why?

C. The triangle kit

Cut out the eight triangles. Lay them on top of each other, flipping them over if you need to.

Congruent pairs: ______ and ______   ______ and ______

Which one looks like A but is not congruent to it? How can you tell?

D. Stretch: when SSA works

Do Part A again with BC = 7 cm, longer than AB. How many triangles now? Why?

Tests for similar triangles

Similar triangles have equal angles and every side in one ratio. Students test pairs with AAA, SSS and SAS, measure the triangle kit to find similar pairs and their scale factors, and use a scale factor to find missing sides.

We are learning to

• test two triangles for similarity with AAA, SSS or SAS,
• find a scale factor by dividing matching sides,
• use the scale factor to find a missing side, in either direction.

Success criteria

• I can show that 3, 4, 6 and 4.5, 6, 9 are similar by dividing matching sides.
• I can find a missing side of a similar triangle with the scale factor.

You need

The triangle kit from Day 3, a ruler and a calculator each; the worksheet.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part A; start the second with the kit.

Watch for these ideas

• Dividing sides that do not match: pair shortest with shortest, longest with longest.
• Adding instead of multiplying: thinking 3, 4, 5 and 5, 6, 7 are similar because each side grew by 2.
• Using SAS when the angle is not between the two sides.

Answers

• Part A: 1. SSS, every ratio 1.5; 2. AAA: the third angles are 80° and 65°, so both are 35°, 65°, 80°; 3. SAS, ratio 1.5 with the angle between; 4. not similar: 1.5, 1.5, 1.57; 5. not enough to say: the angle is not between the sides.
• Part B: A (4, 5, 6), D (6, 7.5, 9), F (2, 2.5, 3), H (4, 5, 6) are similar, and so are B (6, 8, 10), E (3, 4, 5), G (3, 4, 5) (A and H and E and G are congruent). Factors: F to A 2, F to D 3, F to H 2, E to B 2. Measured lengths depend on the print; the ratios do not.
• Part C: scale factor 6 ÷ 4 = 1.5; x = 7.5 ÷ 1.5 = 5; y = 6 × 1.5 = 9. The enlarged triangle: 15, 20, 25.
• Part D: the area is multiplied by 9: a 3, 4, 5 triangle has area 6 and a 9, 12, 15 triangle has area 54.

Kit pairs, for checking: A and D; A and F; B and E; B and G; D and F; D and H; F and H.

Tests for similar triangles

A. Which test?

Is each pair similar? Name the test, or say why not.

B. Measure the kit

Measure each kit triangle’s sides in millimetres, shortest first. Then find the similar pairs and their scale factors.

Similar pairs and scale factors:

C. Missing sides

The two triangles are similar. Find the scale factor, then x and y.

Scale factor = ______   x = ______   y = ______   A 6, 8, 10 triangle is enlarged by 2.5. New sides: ______

D. Stretch: what happens to the area?

Two triangles are similar with scale factor 3. How many times bigger is the area? Check with a 3, 4, 5 right-angled triangle.

Similar shapes in use

Scale factors measure what we cannot reach. Students find a height from shadows and the size of an enlargement, decide which shapes are always similar, and play the self-check quiz as a class game.

We are learning to

• find the similar triangles in a real situation and match their sides,
• use a scale factor to find an unknown length,
• explain the conditions for other shapes to be similar.

Success criteria

• I can find the height of a flagpole from its shadow.
• I can explain why all squares are similar, but not all rectangles.

You need

The worksheet and a calculator each; the self-check quiz at the bottom of the unit page for the game.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part B; start the second with Part C and the quiz.

Watch for these ideas

• Matching the person’s height with the pole’s shadow.
• Enlarging one side of a photo by the factor and adding the same amount to the other.
• Thinking all rectangles, or all isosceles triangles, are similar.

Answers

• Part A: scale factor 9 ÷ 2.4 = 3.75; height 1.6 × 3.75 = 6 m.
• Part B: factor 25 ÷ 10 = 2.5; long side 15 × 2.5 = 37.5 cm. The 25 by 30 print is not similar: 30 ÷ 25 = 1.2 but 15 ÷ 10 = 1.5.
• Part C: always for squares, circles, equilateral triangles and regular hexagons, one shape at any size; sometimes for rectangles, rhombuses, isosceles triangles and right-angled triangles, whose angles or side ratios can differ. For example, two rectangles are similar only when length ÷ width is the same.
• Part D: 297 ÷ 210 ≈ 1.414 and 210 ÷ 148.5 ≈ 1.414: the same ratio, so A5 is similar to A4, with scale factor about 1.414.

Similar shapes in use

A. Shadows

A person 1.6 m tall casts a shadow 2.4 m long. At the same moment a flagpole casts a shadow 9 m long. Why are the two triangles similar? How tall is the flagpole?

B. A photo enlargement

A photo 10 cm by 15 cm is enlarged so that its short side is 25 cm. How long is the long side? Is a print 25 cm by 30 cm similar to the photo?

C. Always similar, or only sometimes?

D. Stretch: paper sizes

A4 paper is 210 mm by 297 mm. Folding it in half across the long side gives A5. Is A5 similar to A4? Show the ratios.

Twelve shape cards

For Day 1, one set per pair

Cut out the twelve cards. Sort them into congruent, similar but not congruent, and neither. Count grid spaces to check the sides.

Teacher note: congruent 1, 4, 6, 9, 11; similar 2, 5, 8, 12; neither 3, 7, 10. No row or column of cards gives a pile away.

The triangle kit

For Days 3 and 4, one per pair

Cut out the eight triangles. On Day 3, find the congruent pairs by laying one on another (flip them over if you need to). On Day 4, measure the sides to find the similar ones.

What we know: Congruence and similarity

Work on your own. Show your working, and name the test you use.

1. Which transformations always give an image congruent to the shape? ______________________
2. Are a 4 cm by 6 cm rectangle and a 6 cm by 9 cm rectangle similar? Show how you know. ______________________
3. In △ABC, AB = 8 cm, BC = 6 cm and CA = 9 cm. In △DEF, DE = 9 cm, EF = 8 cm and FD = 6 cm. Name the test and write the congruence statement. ______________________
4. In △PQR, PQ = 7 cm, ∠Q = 65° and QR = 4 cm. In △XYZ, XY = 7 cm, ∠Y = 65° and YZ = 4 cm. Are they congruent? Which test? ______________________
5. Why is SSA (two sides and an angle that is not between them) not a test for congruence? ______________________
6. Two triangles are similar. One has sides 5 cm, 7 cm, 8 cm; the matching sides of the smaller one are 4 cm, x cm and 6.4 cm. Find the scale factor and x. ______________________
7. One triangle has angles of 40° and 75°; another has angles of 65° and 40°. Are they similar? Explain. ______________________
8. Mia says, “All isosceles triangles are similar.” Is that right? Explain with an example. ______________________

Answers and marking guide

Answers with working

1. Translations, rotations and reflections (and any mix of them).
2. Yes: 6 ÷ 4 = 9 ÷ 6 = 1.5, and all the angles are 90°.
3. SSS: AB = EF, BC = FD and CA = DE, so △ABC ≡ △EFD.
4. Yes, by SAS: the 65° angle is between the two equal pairs of sides, so △PQR ≡ △XYZ.
5. The third side can swing to two places, so the same facts can make two different triangles.
6. Scale factor 4 ÷ 5 = 0.8, a reduction; x = 7 × 0.8 = 5.6 (check: 8 × 0.8 = 6.4).
7. Yes, by AAA: the third angles are 65° and 75°, so both have 40°, 75° and 65°.
8. No: for example angles 40°, 70°, 70° and 100°, 40°, 40° are both isosceles, but their angles differ.

A quick three-level guide

Eight questions, four ideas. A student at standard names a test for each triangle question and checks that sides match in order.

Weekly class record

Jot a tick as you move around the room; the mini-check fills any gaps. A tick a day is plenty.

The five columns are the five days: congruent or similar, congruence tests, SSA and AAA, similarity tests, and similar shapes in use.