For Integrated Programme students: Your current school materials, teacher instructions, and assessment scope take precedence because IP topic sequence and depth vary by school. This is an Eclat IP guide, not the O-Level / SEC G3 exam-track guide.
How this chapter applies
Eclat core: congruency and similarity tests, scale drawings, constructions, bisectors, and length, area, or volume scale factors form the main route.
School-sensitive extension: proof chains that combine circle, vector, or coordinate ideas may vary by school.
2027 national comparison: K310 Topic G2 includes congruent and similar figures, enlargement, reduction, scale drawings, perpendicular and angle bisectors, triangle tests, and area or volume ratios.
Check your school: confirm accepted proof tests, construction notation, and whether scale drawings are assessed practically.
Q: What does IP EMaths Notes (Upper Sec, Year 3-4): 12) Similarity and Congruency cover? A: Use similarity ratios and congruency criteria to prove geometric relationships and scale figures accurately.
The core idea is simple: Similar shapes have matching angles and scaled sides; congruent shapes match exactly.
Use it as a working check: Write corresponding vertices in the correct order before using ratios. Similarity scales lengths by k, areas by k2
. Congruency transfers equal sides and angles directly.
Then go one layer deeper: Use the lamppost and proof examples to practise clear geometry writing: state the matching angles or sides, name the test, set up the ratio, and conclude in words.
Similarity compares shapes with proportional sides and equal angles, while congruency proves exact matches. Keep diagrams annotated with tick marks, matching angle symbols, and explicit ratio statements so the reader can follow the logic without guessing.
Keep the full topic roadmap handy via our IP Maths tuition hub so you can jump into related drills, quizzes, or diagnostics as you move through these notes.
Before writing ratios, decide what the question is asking you to prove or find:
matching diagrams
-> exact same size: prove congruency
-> same shape but scaled: prove similarity
-> asking for length or perimeter: use k
-> asking for area: use k^2
-> asking for volume: use k^3
Question target
First evidence to mark
Ratio or test to use
Common trap
Prove two triangles match exactly
Equal sides, equal angles, or a right angle plus hypotenuse and side
SSS, SAS, ASA, AAS, or RHS
Using similarity language when the answer needs exact equality
Find an unknown length in similar figures
Corresponding vertices and one known side pair
Side ratio k
Pairing a side with the wrong corresponding side
Find an area after a length scale changes
Side ratio first
Area ratio k2
Multiplying area by k instead of k2
Find a volume after a length scale changes
Side ratio first
Volume ratio k3
Squaring the scale factor for a 3D solid
Worked check: if two similar triangles have side ratio 3:5, the area ratio is 9:25, not 3:5. If two similar solids have side ratio 3:5, the volume ratio is 27:125.
Misconception check: congruent figures are always similar with scale factor 1, but similar figures are not always congruent. Only claim congruency after proving the sizes match exactly.
Similarity fundamentals
Construction and bisector checkpoint
K310 includes scale drawings, perpendicular bisectors, and angle bisectors. Keep the construction marks visible because they are evidence of the method.
Construction
Compass action
Result to state
Common trap
Perpendicular bisector of a segment
Draw equal-radius arcs from both endpoints that meet above and below the segment.
Joining the arc intersections gives a perpendicular line through the midpoint.
Changing the compass radius between endpoints.
Angle bisector
Draw one arc from the vertex, then equal-radius arcs from the two points where it cuts the arms.
Joining the vertex to the new arc intersection divides the angle equally.
Measuring by eye instead of preserving equal radii.
Scale drawing
Convert every real length using the same scale before drawing.
Convert a measured result back with the inverse scale.
Applying a length scale directly to an area or volume.
Identify corresponding parts carefully (write them in order, e.g. A↔D, B↔E).
Parallel lines create equal alternate or corresponding angles, a common trigger for similarity.
Once similarity is confirmed, state both the angle relationships and the chosen side ratio before solving for unknowns.
Congruency fundamentals
Congruency requires explicit evidence that size and shape match: show equal sides/angles or cite a standard test (e.g. SAS).
Use congruency to transfer results: if △ABC≅△DEF, then AB=DE, ∠C=∠F, etc.
Congruent triangles often lead to symmetry arguments or allow you to justify perpendicular and midpoint statements in geometry proofs.
Congruency proof-writing checkpoint
A clean congruency proof should collect three matching facts before it names the test. Write the facts in the same vertex order as the triangle statement.
Evidence pattern
What to write before the test
Test you may use
Common trap
Three matching sides
AB=DE, BC=EF, CA=FD
SSS
Listing side lengths without showing which sides correspond.
Two sides and the included angle
AB=DE, ∠ABC=∠DEF, BC=EF
Two angles and one matching side
Two equal angles plus one corresponding side
ASA or AAS
Forgetting that angle order fixes the matching vertices.
Right angle, hypotenuse, and one side
Both triangles are right-angled, hypotenuse pair equal, one side pair equal
RHS
Calling it RHS when the equal side is actually another hypotenuse.
Worked check: if ∠ABD=∠ECD, ∠ADB=∠EDC, and BD=CD, then the matching order is △ABD≅△ECD by AAS. After that, you may transfer matching parts, such as AB=EC.
Misconception check: a proof is not complete just because the diagram looks symmetric. Name the equal parts, name the test, then state the conclusion that follows.
Worked example - Height via similar triangles
A lamppost casts a shadow 6.5m long when a 1.8m tall student stands nearby and casts a shadow 2.0m long. Assuming the lamppost, student, and shadow tips form similar triangles, find the lamppost height.
State similarity: the lamppost triangle and student triangle share the angle of elevation and have right angles at the bases, so △Lamp∼△Student by AA.
Let h denote the lamppost height and write the side ratio:
6.5mh=2.0m1.8m.
Solve for h:
h=6.5m×2.0m1.8m=6.5m×0.9=5.85m.
Conclude with a sentence: the lamppost is 5.85m tall (to 3 s.f.).
Worked example - Congruency proof in an isosceles triangle
Given △ABC with AB=AC. Point D lies on BC such that AD⊥BC. Prove △ABD≅△ACD, then state one angle conclusion.
Mark the equal sides: AB=AC because the triangle is isosceles.
Mark the shared side: AD=AD.
Mark the included right angles: ∠ADB=∠ADC=90∘ because AD⊥BC.
Therefore △ABD≅△ACD by RHS.
Corresponding parts of congruent triangles are equal, so ∠ABC=∠BCA. The altitude AD also bisects the base, so BD=DC.
Trap check: do not claim congruency before matching the correct corresponding parts. Similarity proves same shape; congruency needs enough evidence for same shape and same size.
Ratio consequences in practice
Perimeter scaling: Perimeter1=k×Perimeter2.
Area scaling: if k=47, then Area1=k2×Area2=1649×Area2.
When a diagram includes a scale drawing, multiply lengths by the scale factor and areas by its square to convert between model and real sizes.
Worked example - Parallel slice in a triangle
In triangle ABC, point D lies on AB with AD=6cm and AB=9cm. A line through D parallel to BC meets AC at E. If AC=12cm, find AE and the ratio of the areas of triangles ADE and ABC.
Since DE∥BC, triangles ADE and ABC are similar by AA.
Corresponding side ratio: Result:ABAD=96=32=ACAE
Solve for AE: AE=32×12=8cm
Area ratio equals the square of the side ratio: Result:[△ABC][△ADE]=(32)2=94
So AE=8cm and the area ratio is 4:9.
Practice Quiz
Check your command of similarity ratios, area/volume scaling, and congruency tests with fast prompts.
Try this
Two similar triangles have side ratio 5:8. If the smaller triangle has area 45cm2, find the area of the larger triangle.
In △PQR, PS is drawn parallel to QR meeting PR at S. Given PS=4.2cm, QR=9.0cm, and PR=7.0cm, find RS and the ratio of the areas of △PSR to △PQR.
.
.
Perimeter ratio equals k.
Area ratio equals k2.
For similar solids (cones, pyramids) the volume ratio equals k3.
SAS
Using an angle that is not between the two named sides.
