Congruence & Similarity
Congruent or Similar?
Draw a line to match each description to the correct term.
Classify Each Pair
Sort each pair of shapes.
Congruence Conditions
Circle the correct congruence condition.
Three sides equal (SSS) means triangles are:
Two sides and included angle equal (SAS) means:
Three angles equal (AAA) means:
Scale Factor
Find the scale factor from original to image.
Original side 4 cm, image side 12 cm. Scale factor =
Original side 10 m, image side 4 m. Scale factor =
Scale factor 2.5, original side 8 cm. Image side =
Find Missing Sides in Similar Shapes
Use the scale factor.
Similar triangles. Sides 3 cm and 9 cm correspond. Another side of small triangle = 4 cm. Matching side of large triangle =
Scale factor 1.5. Small rectangle 6 m × 4 m. Large rectangle length =
Similarity in Real Life
Show all working.
A model car is at scale 1:20. The real car is 4 m long and 1.6 m wide. What are the model's dimensions in cm?
Two similar triangles have sides 5 cm and 15 cm. Smaller triangle area = 10 cm². What is the larger triangle's area? (Area scales by scale factor squared.)
Congruence Conditions for Triangles
Draw a line to match each congruence condition to its full name.
Area Ratios in Similar Figures
If the scale factor is k, areas scale by k². Answer each question.
Scale factor = 2. Area ratio (large : small) =
Scale factor = 3. Small area = 5 cm². Large area =
Scale factor = ½. Large area = 100 m². Small area =
Scale Drawings
Show all working.
A scale drawing uses the scale 1 cm : 5 m. A room on the drawing is 4.2 cm × 3 cm. What are the real dimensions? What is the real floor area?
On a map, two towns are 8.5 cm apart. The map scale is 1 cm : 20 km. What is the actual distance between the towns?
What Does This Scale Factor Tell Us?
Sort each statement: does the scale factor describe an enlargement or a reduction?
Map Scale Problems
Use the scale 1 cm : 50 km.
Distance on map = 3.5 cm. Real distance =
Real distance = 200 km. Distance on map =
Map distance = 6 cm. Real distance =
Identifying Congruence Conditions
State which congruence condition (SSS, SAS, AAS, or RHS) applies, or write 'Not enough information'.
Triangle ABC: AB = 5 cm, BC = 7 cm, AC = 9 cm. Triangle DEF: DE = 5 cm, EF = 7 cm, DF = 9 cm. Which condition?
Two triangles share two equal angles and the side between them is equal in both. Which condition?
Two right-angled triangles both have hypotenuse 13 cm and one leg of 5 cm. Which condition?
Two triangles both have all three angles equal (60°, 70°, 50°). Are they congruent?
Scale Factor Practice — Set A
Find the scale factor and then find the missing side.
Triangle A has sides 4, 6, 8. Triangle B (similar) has corresponding sides 10, ?, ?. Find the scale factor and the missing sides.
Rectangle A is 5 cm × 3 cm. Rectangle B (similar) has length 15 cm. Find the width of B.
Two similar pentagons have one pair of corresponding sides 8 m and 20 m. What is the scale factor?
Scale Factor Mixed Practice
Find the scale factor or missing dimension.
Original side 6 cm, image side 9 cm. Scale factor =
Scale factor 0.5, original side 14 m. Image side =
Scale factor 2.5, image side 20 cm. Original side =
Similar Figures — Ratio of Areas
Area scales by the square of the scale factor: area ratio = k².
Scale factor k = 3. Original area = 8 cm². What is the image area?
Scale factor k = ½. Original area = 40 m². What is the image area?
Two similar triangles have areas 9 cm² and 36 cm². What is the scale factor?
Real-World Similarity
Show all working.
A photograph of a car is 8 cm long. The actual car is 4 m long. What is the scale factor (photo to real car)? What is the height of the car if the photo shows the car as 2 cm tall?
Two similar triangles are formed by a tree and its shadow. The tree's shadow is 10 m, and a nearby 2 m post casts a 4 m shadow. How tall is the tree?
Map Scale Problems
Show all working.
Map scale: 1 cm : 25 km. Two cities are 7.4 cm apart on the map. What is the real distance?
Real distance: 180 km. Map scale: 1 cm : 30 km. How far apart are the cities on the map?
A room is 5 m × 3 m. Draw it at a scale of 1 cm : 0.5 m. What are the scaled dimensions?
Proof: Congruence Using Conditions
Write a justification for each congruence claim.
Triangle PQR and triangle STU have PQ = ST = 8 cm, QR = TU = 10 cm, PR = SU = 12 cm. Are they congruent? State the condition.
Two isosceles right-angled triangles both have hypotenuse 10 cm. Are they congruent? State the condition.
Similarity with Triangles in Diagrams
Two triangles share a common vertex. The triangles are similar.
In triangle ABC, D is on AB and E is on AC such that DE is parallel to BC. AD = 4, DB = 6, DE = 5. Find BC.
Explain why DE parallel to BC ensures the triangles are similar.
Scale Factor in 3D
Apply scaling to 3D shapes.
A model building has scale factor 1:200. The real building is 60 m tall. How tall is the model in cm?
A model ship is built at scale 1:50. The model is 40 cm long. What is the length of the real ship in metres?
Two similar boxes have side lengths 3 cm and 9 cm. What is the volume ratio?
Which Congruence Condition?
Sort each triangle pair by the correct congruence condition (or 'not congruent').
Photography and Similar Triangles
A photographer stands at a point and photographs a tree.
The photographer is 50 m from a tree. She holds a 30 cm ruler at arm's length (60 cm from her eye). The ruler appears to be the same height as the tree. How tall is the tree?
Floor Plan Design
Design a floor plan for a room using a scale of 1 cm : 0.5 m.
Draw a rectangular room 6 m × 4 m at scale. Mark the position of a 1 m wide door and two 1.5 m wide windows.
Calculate the real area of the room. Calculate the scaled area. What is the ratio of real area to scaled area?
Similar Figures and Trigonometry Preview
In similar triangles, matching angle ratios are always equal.
Two similar right-angled triangles have angles 30°, 60°, 90°. The smaller has hypotenuse 2 and short leg 1. The larger has hypotenuse 10. What is its short leg? What fraction of the hypotenuse is the short leg?
Proof Activity: Proving Triangles Congruent
Write a formal geometric proof.
Triangle ABC has AB = AC (isosceles). M is the midpoint of BC. Prove triangles ABM and ACM are congruent. State the condition.
Error Analysis
Find and fix the errors.
Student: 'Triangles with sides 3, 4, 5 and 6, 8, 10 are congruent because all sides are proportional.' What is wrong? What is the correct term?
Student: 'Scale factor is 2, so the area doubles.' What is wrong? What is the correct statement?
Investigation: Similar Polygons
Investigate properties of similar polygons beyond triangles.
Are all regular hexagons similar? Justify using the definition of similarity (same angles, proportional sides).
Are all right-angled triangles similar? Explain why or why not with an example.
Cross-Strand: Scale Factor and Pythagoras
Combine scale factor and Pythagoras in one problem.
Two similar right-angled triangles: the smaller has legs 6 cm and 8 cm. The larger has one leg of 15 cm corresponding to the 6 cm leg. Find the scale factor. Find the other leg of the larger triangle using Pythagoras or the scale factor.
Finding Unknown Sides in Similar Triangles
Use the scale factor to find all missing sides. Show working.
Triangles are similar. Sides of small triangle: 5 cm, 7 cm, 9 cm. One side of large triangle: 14 cm (corresponding to 7 cm). Find the other two sides.
Similar rectangles: small has sides 4 m and 6 m. Large has one side of 18 m. Find the other side.
Similarity — Which Pair Is Similar?
Determine which pair of triangles is similar.
Triangle 1: angles 40°, 60°, 80°. Triangle 2: angles 40°, 60°, 80°. Similar?
Triangle 1: sides 3, 4, 5. Triangle 2: sides 6, 9, 12. Similar?
Triangle 1: sides 2, 4, 6. Triangle 2: sides 3, 6, 9. Similar?
Similarity in Architecture
Answer the questions about a scaled architectural drawing.
A house plan uses scale 1 cm : 2 m. The plan shows a wall 7.5 cm long. What is the real length of the wall?
On the same plan, the living room is 4 cm × 3 cm. What is the real floor area?
The real backyard is 14 m × 9 m. How large should it appear on the plan?
Indirect Measurement Using Shadows
A vertical stick casts a shadow at the same time as a nearby building.
A 1.8 m stick casts a 2.4 m shadow. A building casts a 40 m shadow at the same time. How tall is the building? Show the similarity ratio.
Reflection and Summary
Answer each question to summarise your learning.
What are the four congruence conditions for triangles? Give one example for each.
Explain the difference between congruent and similar in your own words. Give a real-world example of each.
If two similar shapes have a scale factor of 5, what is the ratio of their areas? Their volumes?
Scale Drawing Investigation
Use a ruler to draw each shape at the given scale.
Draw a rectangle representing a tennis court (23.8 m × 10.97 m) at a scale of 1 cm : 2 m. Label all dimensions.
What is the area of your scaled drawing? What is the real area?
Applying Similarity: Estimating Heights
Use similar triangles to estimate the height of a tall object.
On a sunny day, a 1.5 m person casts a 3 m shadow. A tree casts a 20 m shadow at the same time. Estimate the height of the tree using similar triangles.
What assumption did you make about the angle of the sun? Is this assumption valid?
Congruence and Similarity at Home
Explore these concepts using objects and maps.
- 1Find a map of Australia or your state. Measure the distance between two cities on the map. Use the scale bar to calculate the real distance. Compare to an online source.
- 2Find two similar photographs of the same object at different sizes (e.g. print a photo in two sizes). Measure corresponding lengths and calculate the scale factor.
- 3Draw a floor plan of your bedroom at a scale of 1 cm : 50 cm. Label all dimensions on the drawing and on the real room.