Proportionality Theorem

Key Concepts

• Proportionality theorem for triangles
• Use of proportionality theorem for triangles
• Converse of Proportionality theorem for triangles
• Use of the converse of proportionality theorem of triangles

Different Postulates of Similarity of Triangles 

Angle-Angle (AA) Similarity Postulate 

If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar. 

Side Side Side (SSS) Similarity Postulate 

If the corresponding side lengths of two triangles are proportional, then the triangles are similar. 

Side Angle Side (SAS) Similarity Postulate 

If an angle of one triangle is congruent to an angle of a second triangle and the lengths of the sides, including these angles, are proportional, then the triangles are similar. 

Triangle Proportionality Theorem 

If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally. 

Proof: 

In the triangle ABC, in which DE is drawn parallel to BC. Point D lies in BA, and E is in BC. 

 Prove

 

∠BAC = ∠BDE (Corresponding angles)    

∠BCA = ∠BED (Corresponding angles) 

∠B is common 

Then by AA similarity postulate 

Triangle BAC is similar to triangle BDE. 

(Corresponding sides of similar triangles are proportional) 

(Subtracting 1 from both sides) 

Hence proved. 

• In ΔABC, D and E are points on the sides AB and AC, respectively, such that DE || BC 

        If

and AC = 15 cm. find AE. 

(Proportionality theorem for triangles) 

15 – AE = 5AE 

6AE = 15 

AE = 15/6

       = 5/2 = 2.5 

AE = 2.5 cm 

• Use the proportionality theorem of triangles to find the value of x. 

DE || BC, D is a point on side AB of triangle ABC, and E is a point on side AC of triangle ABC. 

5(9 – x) = 4x 

45 – 5x = 4x 

45 = 4x + 5x 

45 = 9x

x =

= 5 

Converse of Triangle Proportionality Theorem 

If a line divides two sides of a triangle proportionally, then it is parallel to the third side. 

Proof 

A triangle ABC and line DE intersects AB at D and AC at E such that, 

To Prove: DE is parallel to BC 

Assume that DE is not parallel to BC. 

Draw a line DE’ parallel to BC. 

Then by Basic Proportionality Theorem, DE’ || BC, 

But

Adding 1 on both sides 

But EC = E’C is possible only if E and E’ coincide. i.e., DE and DE’ are the same line. 

DE || BC, 

Hence proved. 

• If D and E are the points on the sides AB and AC of a triangle ABC, respectively, such that AD = 6 cm, BD = 9 cm, AE = 8 cm, and EC = 12 cm. Show that DE is parallel to BC. 

By the converse proportionality theorem of triangles, DE will be parallel to BC if

Therefore, DE is parallel to BC. 

Theorem 

If three parallel lines intersect two transversals, then they divide the transversals proportionally. 

Three lines k₁, k₂, k₃ such that k₁ || k₂ || k₃. Two transversals, t₁ and t_2, intersect at k₁, k₂, k₃ at points A, B, C, D, E, F. 

To prove

CONSTRUCTION: Draw an auxiliary line through the points A and D, and it intersects k₂ at M. 

Proof 

Consider the triangle ADF, then

(By proportionality theorem of triangles) ———(1)     

Consider the triangle ADC, then

(By proportionality theorem of triangles) ———(2) 

  or

(Reciprocal property of proportions) ———–(3) 

 From (1) and (3) 

Hence proved. 

Theorem 

If a ray bisects an angle of a triangle, then it divides the opposite side into segments whose lengths are proportional to the lengths of the other two sides. 

To prove 

Given ∠YXW = ∠WXZ 

CONSTRUCTION 

Draw a line AZ parallel to the bisector XW. Extend segment XY to meet AZ. 

Proof 

Given ∠YXW = ∠WXZ 

∠WXZ = ∠XZA (Alternate interior angles are equal since XW || AZ) 

∠WXZ = ∠XAZ (Corresponding angles are equal since XW || AZ) 

Therefore, base angles are equal. 

Then AX=XZ 

In triangle YAZ, XW || AZ, then by proportionality theorem of triangles, 

But AX= XZ 

Hence proved. 

• Find the value of x. 

If a ray bisects an angle of a triangle, then it divides the opposite side into segments whose lengths are proportional to the lengths of the other two sides. 

x= 10 

Exercise

• Use proportionality theorem for finding the missing length.

• Solve for x using proportionality theorem for triangles.

• Find the missing length.      

•          

• ABCD is a trapezium with AB parallel to DC, P and Q are points on non-parallel sides AD and BC, respectively, such that PQ parallel to AB Show that.

• n the given figure shows quadrilateral ABCD in which AB is parallel to DC,
  OA = 3x – 19, OB = x – 3, OC = x – 5 and OD = 3 find the value of x.
• If the bisector of an angle of a triangle bisects the opposite side, prove that the triangle is isosceles.
• In the given figure PS/SQ = PT/TR and PST = PRQ.

Prove that PQR is an isosceles triangle.

• In the given figure, XY is parallel to MN if

a.     LX = 4cm, XM = 6cm and LN = 12.5 cm. find LY

b.     LX : XM = 3:5 and LY = 3.6 cm. Find LN

Concept Map 

What we have learnt

• Proportionality Theorem for Triangles

If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally.

• Converse of Proportionality Theorem for Triangles

If a line divides two sides of a triangle proportionally, then it is parallel to the third side.

• Theorem

If three parallel lines intersect two transversals, then they divide the transversals proportionally.

• Theorem based on angle bisection

If a ray bisects an angle of a triangle, then it divides the opposite side into segments whose lengths are proportional to the lengths of the other two sides.