Triangles · Lesson 4 of 4

Chapter Summary and Practice

“Similarity, proportional sides and area relationships come together for one final geometric workout.”

Learning Objectives

• Revise the complete chapter quickly and clearly. • Recall BPT, its converse, and the three triangle similarity criteria. • Choose the correct theorem or similarity criterion in a problem. • Practise textbook-style questions with fresh values and labels. • Check your understanding before moving to the next chapter.

Chapter Summary

The Whole Chapter in One Line

Similarity is about the same shape. Parallel lines create proportional sides, and AA, SAS and SSS help us prove that two triangles have the same shape.

Similar Figures

Similar figures have the same shape but need not have the same size. Their corresponding angles are equal and their corresponding lengths are proportional.

Similarity and Congruence

Every pair of congruent figures is similar because the shape is the same. But similar figures need not be congruent because their sizes may be different. Congruence is the special case of similarity with scale factor 1.

Similar TrianglesLaTeX

△ A B C \sim △ P QR \Rightarrow ∠ A = ∠ P, ∠ B = ∠ Q, ∠ C = ∠ R

Corresponding SidesLaTeX

\frac{A B}{P Q} = \frac{B C}{QR} = \frac{C A}{R P}

Basic Proportionality Theorem

If a line is drawn parallel to one side of a triangle and intersects the other two sides, it divides those two sides in the same ratio.

BPTLaTeX

D E ∥ B C \Rightarrow \frac{A D}{D B} = \frac{A E}{E C}

The converse works in the opposite direction. If two sides of a triangle are divided in the same ratio, then the line joining those division points is parallel to the third side.

Converse of BPTLaTeX

\frac{A D}{D B} = \frac{A E}{E C} \Rightarrow D E ∥ B C

Memory Trick

BPT: parallel → proportional. Converse BPT: proportional → parallel.

Similarity Criteria

Criterion	What You Need
AA	Two pairs of corresponding angles are equal
SAS	Two pairs of corresponding sides are proportional and the included angle is equal
SSS	All three pairs of corresponding sides are proportional

AA SimilarityLaTeX

∠ A = ∠ P, ∠ B = ∠ Q \Rightarrow △ A B C \sim △ P QR

SAS SimilarityLaTeX

\frac{A B}{P Q} = \frac{A C}{P R}, ∠ A = ∠ P \Rightarrow △ A B C \sim △ P QR

SSS SimilarityLaTeX

\frac{A B}{P Q} = \frac{B C}{QR} = \frac{C A}{R P} \Rightarrow △ A B C \sim △ P QR

How to Choose the Right Idea

What the Question Gives You	Think Of
A line parallel to one side of a triangle	BPT
Equal ratios on two divided sides	Converse of BPT
Two equal corresponding angles	AA similarity
Two proportional side pairs + included equal angle	SAS similarity
Three proportional side pairs	SSS similarity

Before Writing Any Ratio

Match the vertices first. If △ABC ~ △PQR, then A ↔ P, B ↔ Q and C ↔ R. A correct theorem with the wrong correspondence can still give a wrong answer.

Guided Practice

Example 1: Use BPT to Find a Length

Problem
In △XYZ, M lies on XY and N lies on XZ such that MN ∥ YZ. If XM = 5 cm, MY = 3 cm and XN = 10 cm, find NZ.

1.Because MN ∥ YZ, use the Basic Proportionality Theorem.
2.XM/MY = XN/NZ.
3.Substitute the values: 5/3 = 10/NZ.
4.5 × NZ = 30.
5.NZ = 6 cm.

Example 2: Use the Converse of BPT

Problem
In △LMN, P lies on LM and Q lies on LN. LP = 6 cm, PM = 4 cm, LQ = 9 cm and QN = 6 cm. Is PQ parallel to MN?

1.Compare the divisions of the two sides.
2.LP/PM = 6/4 = 3/2.
3.LQ/QN = 9/6 = 3/2.
4.The ratios are equal.
5.Therefore, by the converse of BPT, PQ ∥ MN.

Example 3: Recognise AA Similarity

Problem
In △ABC and △DEF, ∠A = 48°, ∠B = 72°, ∠D = 48° and ∠E = 72°. Prove that the triangles are similar.

1.∠A = ∠D = 48°.
2.∠B = ∠E = 72°.
3.Two pairs of corresponding angles are equal.
4.Therefore, △ABC ~ △DEF by AA similarity.

Example 4: Recognise SAS Similarity

Problem
In △ABC, AB = 8 cm, AC = 12 cm and ∠A = 55°. In △PQR, PQ = 10 cm, PR = 15 cm and ∠P = 55°. Prove that the triangles are similar.

1.AB/PQ = 8/10 = 4/5.
2.AC/PR = 12/15 = 4/5.
3.So the two pairs of sides around the given angles are proportional.
4.Also, ∠A = ∠P = 55°.
5.Therefore, △ABC ~ △PQR by SAS similarity.

Example 5: Recognise SSS Similarity

Problem
The sides of △ABC are 5 cm, 7 cm and 9 cm. The corresponding sides of △PQR are 10 cm, 14 cm and 18 cm. Prove that the triangles are similar.

1.Compare all three corresponding sides.
2.5/10 = 1/2.
3.7/14 = 1/2.
4.9/18 = 1/2.
5.All three ratios are equal.
6.Therefore, △ABC ~ △PQR by SSS similarity.

Quick check

A line inside a triangle is parallel to one side. Which theorem should come to mind first?

Quick check

If AD/DB = AE/EC, what can the converse of BPT help you prove?

Quick check

Two pairs of corresponding angles are equal. Which similarity criterion is enough?

Quick check

For SAS similarity, which angle must be equal?

Quick check

Which condition is sufficient for SSS similarity?

Practice Questions

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.
In △PQR, S lies on PQ and T lies on PR. PS = 4.8 cm, SQ = 3.2 cm, PT = 7.2 cm and TR = 4.8 cm. Determine whether ST ∥ QR. Give a reason.
In △XYZ, M lies on XY and N lies on XZ. XM = 3.5 cm, MY = 5.5 cm, XN = 7 cm and NZ = 11 cm. Determine 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 △LMN, P and Q are the midpoints of LM and LN respectively. Using the converse of BPT, prove that PQ ∥ MN.
ABCD is a trapezium with AB ∥ CD. Its diagonals AC and BD meet at O. Prove that OA/OC = OB/OD.
The diagonals of quadrilateral PQRS meet at O. If PO/OR = QO/OS, prove that PQ ∥ RS.
In △ABC and △PQR, ∠A = ∠P = 62° and ∠B = ∠Q = 71°. Prove that the triangles are similar and state the criterion used.
In △ABC, AB = 9 cm, AC = 12 cm and ∠A = 45°. In △DEF, DE = 12 cm, DF = 16 cm and ∠D = 45°. Prove that △ABC ~ △DEF.
The sides of △ABC are 6 cm, 8 cm and 10 cm. The sides of △DEF are 9 cm, 12 cm and 15 cm. Prove that the triangles are similar and state the criterion used.
In △PQR, S lies on PR and T lies on QR. If ∠P = ∠RTS and ∠R is common to the two triangles, prove that △RPQ ~ △RTS.
Two right triangles △ABC and △AMP are right-angled at B and M respectively. If ∠A is common to both triangles, prove that △ABC ~ △AMP.
A vertical pole 5 m high casts a shadow 3 m long. At the same time, a tower casts a shadow 21 m long. Assuming the sun rays make the same angle with the ground, find the height of the tower.
In △ABC, D lies on BC such that ∠ADC = ∠BAC. If the required angle relations establish similarity, show that AC² = BC × CD.
In two similar triangles △ABC and △PQR, AD and PM are corresponding medians. Show that AB/PQ = AD/PM.

Before You Finish the Chapter

Ask yourself: • Can I recognise when BPT applies? • Can I use the converse to prove two lines are parallel? • Can I tell AA, SAS and SSS apart immediately? • Can I match corresponding vertices before writing ratios? • Can I explain each step instead of only writing the final equation?

Key Takeaways

• Similar figures have the same shape: their corresponding angles are equal and their corresponding sides are proportional. • Congruent figures are always similar, but similar figures are congruent only when their scale factor is 1. • The Basic Proportionality Theorem uses a parallel line to prove proportional division of two sides, while its converse uses equal ratios to prove that a line is parallel. • Use AA when two pairs of corresponding angles are equal, SAS when two side pairs are proportional with an equal included angle, and SSS when all three side pairs are proportional. • Always match the corresponding vertices before writing angle equalities, side ratios or a similarity statement. • After proving two triangles similar, use the proportionality of corresponding sides to calculate unknown lengths.

Previous · Lesson 3

Criteria for Similarity of Triangles

End of chapter

Chapter Summary

Basic Proportionality Theorem

Similarity Criteria

How to Choose the Right Idea

Guided Practice

Quiz

Practice Problems

Key Takeaways