Triangles · Lesson 2 of 4
Similarity of Triangles
“Similarity of triangles is basically geometry’s way of saying "same vibe, different size."”
• Understand clearly what it means for two triangles to be similar. • Distinguish between similar triangles and congruent triangles. • Identify corresponding vertices, angles and sides correctly. • Understand and apply the Basic Proportionality Theorem (BPT). • Understand and apply the converse of the Basic Proportionality Theorem.
Introduction
In the previous lesson, we learned that similar figures have the same shape but may have different sizes. Since a triangle is also a polygon, the same idea applies to triangles. However, triangle similarity becomes much more powerful because the angles and sides of a triangle are strongly connected to one another.
Two similar triangles are essentially the same triangle viewed at different sizes. One may be larger or smaller, but its shape has not changed.
Imagine Enlarging a Triangle
Imagine drawing a triangle on a sheet of paper and then enlarging it perfectly on a photocopier. Every side becomes longer by the same factor, but none of the angles change. The larger triangle looks exactly like the original, only bigger. These two triangles are similar.
Problem
Triangle ABC has side lengths 3 cm, 4 cm and 5 cm. Triangle PQR has corresponding side lengths 6 cm, 8 cm and 10 cm. What has happened to the triangle?
- 1.Compare the corresponding sides.
- 2.6/3 = 2.
- 3.8/4 = 2.
- 4.10/5 = 2.
- 5.Every side has been multiplied by 2.
- 6.If the corresponding angles remain equal, triangle PQR is simply an enlargement of triangle ABC.
- 7.The triangles therefore have the same shape, although PQR is twice as large in linear dimensions.
What Are Similar Triangles?
Two triangles are similar when their corresponding angles are equal and their corresponding sides are in the same ratio.
For two triangles ABC and PQR to be similar: • Their matching angles must be equal. • Their matching sides must be proportional.
Correspondence: The Most Important Part
Similarity becomes confusing mainly when students match the wrong vertices or sides. The letters in the similarity statement tell us exactly which parts correspond.
If we write triangle ABC similar to triangle PQR, the order tells us that A matches P, B matches Q and C matches R.
| Triangle ABC | Triangle PQR |
|---|---|
| A | P |
| B | Q |
| C | R |
| AB | PQ |
| BC | QR |
| CA | RP |
Never compare sides just because they look similarly placed in a drawing. First identify which vertices correspond. Then form the side ratios in exactly the same order.
Problem
If △ABC ~ △DEF, identify the side corresponding to BC and the angle corresponding to ∠A.
- 1.Write the vertices in matching order.
- 2.A ↔ D.
- 3.B ↔ E.
- 4.C ↔ F.
- 5.Therefore, side BC corresponds to EF.
- 6.Angle A corresponds to angle D.
Similar Triangles vs Congruent Triangles
Similarity and congruence are closely related, which is why they are often confused. The easiest way to separate them is to think about shape and size independently.
Two triangles are congruent when they have exactly the same shape and exactly the same size.
Two similar triangles have the same shape, but one triangle may be larger or smaller than the other.
| Property | Similar Triangles | Congruent Triangles |
|---|---|---|
| Shape | Same | Same |
| Size | May be different | Exactly same |
| Corresponding angles | Equal | Equal |
| Corresponding sides | Proportional | Equal |
| Scale factor | May be any positive value | Exactly 1 |
| Symbol | ~ | ≅ |
Congruence is really a special case of similarity. If two similar triangles have scale factor 1, their corresponding sides are equal and the triangles become congruent.
Understanding the Difference Through Scale Factor
Problem
Triangle A has sides 4 cm, 6 cm and 8 cm. Triangle B has sides 6 cm, 9 cm and 12 cm. What can we notice?
- 1.Compare corresponding sides.
- 2.6/4 = 1.5.
- 3.9/6 = 1.5.
- 4.12/8 = 1.5.
- 5.All sides have been multiplied by the same scale factor 1.5.
- 6.The triangles may therefore have the same shape while having different sizes.
- 7.They are not congruent because their corresponding side lengths are not equal.
Problem
Two triangles have corresponding sides 5 cm, 7 cm and 9 cm in both triangles. What is their scale factor?
- 1.Each corresponding side has exactly the same length.
- 2.Therefore every side ratio is 1.
- 3.The scale factor is 1.
- 4.If their corresponding parts match correctly, the triangles are congruent.
- 5.Because congruent triangles have the same shape, they are also similar.
Every pair of congruent triangles is similar. But every pair of similar triangles is not necessarily congruent.
If corresponding angles are equal but corresponding sides have different lengths, do not immediately say the triangles are different. They may be similar rather than congruent.
Why Is Similarity So Useful?
Similarity may look like a simple idea about shapes, but it allows us to calculate lengths that we cannot measure directly. Once two triangles have the same shape, knowing a few lengths in one triangle can tell us corresponding lengths in the other.
| Use | How Similarity Helps |
|---|---|
| Indirect measurement | Find heights of buildings, poles or trees without measuring them directly |
| Maps and scale drawings | Represent large distances using smaller proportional drawings |
| Architecture | Create models and blueprints while preserving proportions |
| Photography and graphics | Enlarge or reduce images without changing their shape |
| Geometry proofs | Use proportional sides to calculate unknown lengths |
| Astronomy and surveying | Estimate inaccessible distances using geometric relationships |
For example, suppose a 2 m pole and a tall building cast shadows at the same time. The sunlight reaches both objects at the same angle. This can create two triangles with the same shape. If we know the pole's height and both shadow lengths, proportional sides can help us calculate the building's height without climbing it.
Similarity lets us replace a difficult large measurement with an easier small measurement and a proportion.
Equiangular Triangles
Two triangles are called equiangular when their corresponding angles are equal.
For triangles, equal corresponding angles lead to proportional corresponding sides. This special property of triangles will become the basis of the AA and AAA similarity criteria in the next lesson.
Basic Proportionality Theorem
Now consider a triangle ABC. Suppose a line DE is drawn inside the triangle so that D lies on AB, E lies on AC and DE is parallel to BC. Something very important happens: the two sides AB and AC are divided proportionally.
If a line is drawn parallel to one side of a triangle and intersects the other two sides at distinct points, then it divides those two sides in the same ratio.
The Basic Proportionality Theorem is also commonly associated with the Greek mathematician Thales and is often called Thales' theorem in this context.
What Does BPT Actually Mean?
Imagine two roads starting from point A and moving apart. A line DE cuts across them, and another parallel line BC cuts across them farther away. Because DE and BC are parallel, both roads are being cut in a consistent way. If D divides the left side in a certain proportion, E divides the right side in exactly the same proportion.
For example, if AD is half of DB, then AE will also be half of EC. If AD:DB = 3:2, then AE:EC must also equal 3:2.
Parallel line inside a triangle → proportional division of the other two sides.
Problem
In △ABC, D lies on AB and E lies on AC such that DE ∥ BC. If AD = 4 cm, DB = 6 cm and AE = 8 cm, find EC.
- 1.Since DE ∥ BC, we can use the Basic Proportionality Theorem.
- 2.AD/DB = AE/EC.
- 3.Substitute the known lengths.
- 4.4/6 = 8/EC.
- 5.Simplify 4/6 to 2/3.
- 6.2/3 = 8/EC.
- 7.2 × EC = 24.
- 8.EC = 12 cm.
- 9.Therefore, EC = 12 cm.
Another Useful Form of BPT
Sometimes a problem gives the complete sides AB and AC instead of the lower portions DB and EC. Because AB = AD + DB and AC = AE + EC, BPT can also lead to a proportion involving the whole sides.
Problem
In △ABC, DE ∥ BC. If AD = 6 cm, AB = 15 cm and AC = 20 cm, find AE.
- 1.Since DE ∥ BC, proportionality applies.
- 2.Use AD/AB = AE/AC.
- 3.6/15 = AE/20.
- 4.2/5 = AE/20.
- 5.5AE = 40.
- 6.AE = 8 cm.
- 7.Therefore, AE = 8 cm.
Why Is BPT True?
The formal proof uses areas of triangles. The main idea is easier than it first appears. Triangles that share the same height have areas proportional to their bases.
Because the two area ratios have equal denominators in area value, the corresponding side ratios must be equal.
For understanding, remember the geometric idea first: parallel lines create proportional divisions. The area argument is the formal proof of why that relationship must always hold.
Converse of the Basic Proportionality Theorem
BPT starts with a parallel line and concludes that two sides are divided proportionally. The converse simply reverses this logic.
If a line divides any two sides of a triangle in the same ratio, then that line is parallel to the third side.
BPT and Its Converse: Do Not Mix Them Up
| Theorem | What You Know | What You Conclude |
|---|---|---|
| BPT | DE ∥ BC | AD/DB = AE/EC |
| Converse of BPT | AD/DB = AE/EC | DE ∥ BC |
BPT: parallel ⇒ proportional. Converse BPT: proportional ⇒ parallel.
Problem
In △ABC, D lies on AB and E lies on AC. AD = 6 cm, DB = 9 cm, AE = 8 cm and EC = 12 cm. Determine whether DE is parallel to BC.
- 1.Calculate the ratio on side AB.
- 2.AD/DB = 6/9 = 2/3.
- 3.Calculate the ratio on side AC.
- 4.AE/EC = 8/12 = 2/3.
- 5.The two ratios are equal.
- 6.Therefore, by the converse of BPT, DE ∥ BC.
Problem
In △PQR, S lies on PQ and T lies on PR. PS = 4 cm, SQ = 5 cm, PT = 6 cm and TR = 9 cm. Can ST be parallel to QR?
- 1.Compare the divisions of the two sides.
- 2.PS/SQ = 4/5.
- 3.PT/TR = 6/9 = 2/3.
- 4.The ratios are not equal.
- 5.Therefore, the converse of BPT cannot establish ST ∥ QR.
- 6.In this configuration, the required proportional division condition is not satisfied.
An Important Midpoint Connection
BPT also explains a result you may remember from earlier classes. If a line passes through the midpoint of one side of a triangle and is parallel to another side, it must bisect the third side.
Similarly, if D and E are midpoints of two sides, then AD/DB = AE/EC = 1. By the converse of BPT, DE must be parallel to the third side.
How to Approach BPT Problems
| Step | What to Do |
|---|---|
| 1 | Locate the large triangle. |
| 2 | Check whether a line inside it is given parallel to one side. |
| 3 | Identify the two sides that have been divided. |
| 4 | Write matching ratios in the same order. |
| 5 | Substitute the lengths and solve. |
| 6 | If ratios are given and you need to prove parallelism, use the converse of BPT. |
Keep the direction of every ratio consistent. If you write AD/DB on one side, write AE/EC on the other. Do not suddenly reverse only one ratio and write EC/AE.
• Thinking similar triangles must have equal side lengths. • Confusing similarity with congruence. • Matching sides by appearance instead of corresponding vertices. • Writing the similarity statement in the wrong vertex order. • Comparing non-corresponding sides. • Using BPT when no parallel line is given or established. • Using the converse of BPT without first proving equal side ratios. • Reversing one ratio but not the other. • Forgetting that congruent triangles are also similar with scale factor 1.
Quiz
Which statement best describes similar triangles?
Corresponding sides of similar triangles are:
Which statement is always true?
In △ABC, if D lies on AB, E lies on AC and DE ∥ BC, which relation follows directly from BPT?
If a line divides two sides of a triangle in the same ratio, what does the converse of BPT tell us?
Which phrase best summarizes the Basic Proportionality Theorem?
What is the scale factor between two congruent triangles?
Practice Problems
- In △ABC, D lies on AB and E lies on AC such that DE ∥ BC. If AD = 2 cm, DB = 4 cm and AE = 3 cm, find EC.
- In △PQR, S lies on PQ and T lies on PR such that ST ∥ QR. If PS = 3.6 cm, SQ = 5.4 cm and PT = 4.8 cm, find TR.
- In △ABC, D and E lie on AB and AC respectively. AD = 4.5 cm, DB = 3 cm, AE = 6 cm and EC = 4 cm. Determine whether DE ∥ BC and justify your answer.
- In △XYZ, M and N lie on XY and XZ respectively. XM = 5 cm, MY = 7 cm, XN = 10 cm and NZ = 14 cm. State whether MN ∥ YZ.
- In △ABC, D is the midpoint of AB. Through D, a line parallel to BC meets AC at E. Prove that E is the midpoint of AC.
- In △PQR, S is the midpoint of PQ and T is the midpoint of PR. Using the converse of BPT, prove that ST ∥ QR.
- In a trapezium ABCD with AB ∥ DC, points E and F lie on AD and BC respectively and EF ∥ AB. Using proportionality, establish a relation between AE/ED and BF/FC.
- In △ABC, D lies on AB and E lies on AC such that DE ∥ BC. If AD = 6 cm, AB = 15 cm and AC = 25 cm, find AE.
- In △LMN, P lies on LM and Q lies on LN. If LP/PM = LQ/QN and ∠LPQ = 72°, prove that PQ ∥ MN and identify an angle equal to ∠LPQ.
- A line through points D and E on sides AB and AC of △ABC divides the sides so that AD:DB = 5:3 and AE:EC = 10:6. What can you conclude about DE and BC? State the theorem used.
Key Takeaways
• Similar triangles have the same shape but may have different sizes. • Corresponding angles of similar triangles are equal and corresponding sides are proportional. • Congruent triangles have the same shape and the same size, so their corresponding sides are equal. • Congruence is a special case of similarity with scale factor 1. • Correct correspondence of vertices is essential when writing side ratios. • Similarity is useful for indirect measurement, scale drawings, maps, models and finding inaccessible lengths. • The Basic Proportionality Theorem says that a line parallel to one side of a triangle divides the other two sides proportionally. • If DE ∥ BC in △ABC, then AD/DB = AE/EC. • The converse of BPT says that proportional division of two sides implies that the joining line is parallel to the third side. • A useful memory rule is: BPT means parallel ⇒ proportional, while converse BPT means proportional ⇒ parallel.