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Geometry
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India
Class X

Proofs Using Similar Triangles

Lesson
Lesson

Recall that if two triangles are similar, then one is an enlargement or reduction of the other.

Similar triangles have two very important features:

  • All corresponding angles are equal.

  • All pairs of corresponding sides are in the same ratio.

$\frac{A}{D}$AD​$=$=$\frac{B}{E}$BE​$=$=$\frac{C}{F}$CF​

There are three main methods we can use to prove that two triangles are similar.

  • AAA (Angle, Angle, Angle) - All corresponding angles are equal.

  • SAS (Side, Angle, Side) - Two pairs of corresponding sides are in the same ratio and the corresponding angles in between are equal.

  • SSS (Side, Side, Side) - All three pairs of corresponding sides are in the same ratio.

$\frac{A}{D}$AD​$=$=$\frac{B}{E}$BE​$=$=$\frac{C}{F}$CF​

You can revise these ideas here.

Similar triangles in more general proofs

We're going to use our knowledge of similar triangles along with other geometric concepts to come up with some more general proofs.

Worked Examples

Question 1

Consider the following parallelogram.

a) Prove that $\triangle ABO\sim\triangle DOC$△ABO~△DOC.

b) Hence, deduce that the diagonals of a parallelogram bisect each other.

Solution

a) We are going to prove that $\triangle ABO$△ABO and $\triangle DOC$△DOC are similar using an AAA proof.

Firstly, we know that $\angle AOB=\angle DOC$∠AOB=∠DOC because they are vertically opposite.

Secondly, we know that $AB\parallel DC$AB∥DC, because $ABCD$ABCD is a parallelogram. Hence, $\angle OAB=\angle OCD$∠OAB=∠OCD because they are alternate angles.

Thirdly, $\angle ABO=\angle ODC$∠ABO=∠ODC for the same reason. They are alternate angles on parallel lines.

Hence, by our AAA proof, $\triangle ABO\sim\triangle DOC$△ABO~△DOC.

b) We know that pairs of corresponding sides of similar triangles are all in the same ratio.

$\frac{AB}{DC}$ABDC​$=$=$\frac{BO}{DO}$BODO​$=$=$\frac{AO}{CO}$AOCO​

But we also know that $ABCD$ABCD is a parallelogram, and parallelograms have equal opposite sides. Hence, $AB=DC$AB=DC, which means that $\frac{AB}{DC}=1$ABDC​=1. Therefore, all pairs of corresponding sides are in the same ratio $1$1, in other words, equal.

So these triangles aren't just similar. They are congruent.

So, as we can see, the diagonals split each other into two equal halves. $AO=CO$AO=CO and $DO=BO$DO=BO. Hence, the diagonals of a parallelogram bisect each other.

 

Outcomes

10.G.T.1

Definitions, examples, counterexamples of similar triangles, covering (a) If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio, (b) If a line divides two sides of a triangle in the same ratio, the line is parallel to the third side, (c) If in two triangles, the corresponding angles are equal, their corresponding sides are proportional and the triangles are similar, (d) If the corresponding sides of two triangles are proportional, their corresponding angles are equal and the two triangles are similar (e) If one angle of a triangle is equal to one angle of another triangle and the sides including these angles are proportional, the two triangles are similar.

10.G.T.2

If a perpendicular is drawn from the vertex of the right angle to the hypotenuse, the triangles on each side of the perpendicular are similar to the whole triangle and to each other.

10.G.T.3

(a) The ratio of the areas of two similar triangles is equal to the ratio of the squares on their corresponding sides, (b) b. In a right triangle, the square on the hypotenuse is equal to the sum of the squares on the other two side, (c) c. In a triangle, if the square on one side is equal to sum of the squares on the other two sides, the angles opposite to the first side is a right triangle.

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