Learning Objectives
- Identify congruent and similar figures.
- Use triangle congruence criteria correctly.
- Match corresponding vertices, sides and angles.
- Use a linear scale factor to find unknown lengths in similar figures.
- Apply similarity to maps, models and indirect measurement.
Key Terms
- Congruent
- Exactly the same shape and size.
- Similar
- The same shape, with corresponding lengths in a constant ratio.
- Corresponding
- Occupying matching positions in two figures.
- Scale factor
- The multiplier from one similar figure to another.
- SSS
- Side-side-side triangle congruence.
- SAS
- Side-angle-side triangle congruence using the included angle.
- ASA/AAS
- Angle-side-angle or angle-angle-side triangle congruence.
- RHS
- Right angle-hypotenuse-side triangle congruence.
8th Edition Chapter Map
- Congruent figures and triangle tests
- Corresponding order
- Similar figures and length scale factor
- Similar triangles
- Maps, models and indirect measurement
Congruent Figures
Congruent figures can be translated, rotated or reflected and still remain congruent. All corresponding sides and angles are equal. For triangles, it is not necessary to prove every side and angle individually; use a valid congruence test.
| Criterion | Required information | Important caution |
|---|---|---|
| SSS | Three corresponding sides equal | No angle information is needed. |
| SAS | Two sides and the included angle equal | The angle must lie between the two known sides. |
| ASA/AAS | Two angles and one corresponding side equal | The third angle follows automatically. |
| RHS | Right angle, hypotenuse and one other side equal | Applies only to right-angled triangles. |
Why AAA Is Not Congruence
Equal angles guarantee the same shape but not the same size. Therefore AAA proves similarity, not congruence. Likewise, SSA is generally insufficient because two different triangles may satisfy the same data.
Writing Corresponding Order
If \triangle ABC\cong\triangle PQR, then A corresponds to P, B to Q, and C to R. This order determines which sides are equal: AB=PQ, BC=QR, and AC=PR.
Similar Figures
Similar figures have equal corresponding angles and proportional corresponding lengths. A scale factor greater than 1 enlarges, while a scale factor between 0 and 1 reduces.
Worked Example: Similar Polygons
Question: Two similar polygons have corresponding sides 6 cm and 15 cm. A side of the smaller polygon is 9 cm. Find the corresponding side of the larger.
- Scale factor from smaller to larger =15/6=2.5.
- Multiply the corresponding smaller side: 9\times2.5=22.5.
Answer: 22.5 cm.
Similar Triangles
Triangles are similar if their corresponding angles are equal. Common reasons include AAA, parallel lines producing equal alternate or corresponding angles, or sides in a common ratio. Once similarity is established, use matching sides in the same order.
Worked Example: Parallel-Line Similarity
Question: In triangle ABC, DE is parallel to BC, with D on AB and E on AC. If AD=4,AB=10 and AE=6, find AC.
- \triangle ADE\sim\triangle ABC because corresponding angles are equal.
- AD/AB=AE/AC.
- 4/10=6/AC.
- AC=15.
Answer: AC=15.
Maps And Scale Drawings
A scale of 1:50 000 means 1 unit on the map represents 50 000 of the same units in reality. Convert units after or before applying the scale, but do not mix centimetres and kilometres in one ratio.
Worked Example: Map Scale
Question: A map has scale 1:25 000. Two points are 7.2 cm apart on the map. Find the real distance in kilometres.
- Real distance =7.2\times25000=180000 cm.
- 180000 cm =1800 m =1.8 km.
Answer: 1.8 km.
Indirect Measurement
Similar triangles can determine inaccessible heights or distances using shadows, mirrors or scale diagrams. State why triangles are similar and use consistent corresponding sides.
Scope Note
This D2 chapter focuses on congruence and linear similarity. Area and volume scale factors are treated later in the 8th Edition D4 chapter devoted to areas and volumes of similar figures and solids.
Examination Guidance
- Write vertices in corresponding order before forming ratios.
- Use the included angle for SAS and the hypotenuse for RHS.
- Give a reason for triangle similarity when the question requires proof.
- Keep all lengths in the same units before applying a scale.
Common Mistakes
- Using AAA as a congruence test.
- Matching the longest side of one figure with a non-corresponding side of the other.
- Using one ratio in the smaller-to-larger direction and another in the reverse direction.
- Confusing a linear scale factor with an area scale factor.
Knowledge Check And Practice
1. State the congruence test for two right triangles with equal hypotenuse and one equal leg.
2. Similar figures have scale factor 3 from small to large. A small side is 7 cm. Find the corresponding large side.
3. A model uses scale 1:200. A length on the model is 18 cm. Find the real length in metres.
4. Why does AAA not prove congruence?
5. If \triangle ABC\sim\triangle DEF, which side corresponds to BC?
Extended Worked Practice
Finding Missing Lengths In Similar Triangles
Question: Two similar triangles have corresponding sides 6 cm and 15 cm. A second side of the smaller triangle is 8 cm. Find its corresponding side in the larger triangle.
- Linear scale factor =15/6=2.5.
- Required side =8(2.5)=20 cm.
Answer: 20 cm.
Proving Similarity By SSS
Question: Triangle A has sides 4, 6 and 8; triangle B has sides 7, 10.5 and 14. Explain why they are similar.
The ratios are 7/4=10.5/6=14/8=1.75. All three pairs of corresponding sides are proportional, so the triangles are similar by SSS.
Using Shadows
Question: A 1.6 m pole casts a 2.4 m shadow. At the same time, a tree casts a 12 m shadow. Find the tree height.
- The triangles are similar because the Sun’s rays have the same angle and both objects are vertical.
- \frac{h}{12}=\frac{1.6}{2.4}.
- h=12\times\frac{1.6}{2.4}=8.
Answer: 8 m.
