**Proving Triangles are Similar Worksheet :**

Worksheet given in this section will be much useful for the students who would like to practice problems on proving triangles are similar.

Before look at the worksheet, if you would like to know the stuff related to proving triangles are similar,

**Problem 1 :**

In the triangles LMN and RST,

RS / LM = ST / MN = TR / NL

Prove : ΔRST ∼ ΔLMN

**Problem 2 : **

Which of the following three triangles are similar ?

**Problem 3 :**

In the diagram shown below, use the given lengths to prove that ΔRST ∼ ΔPSQ.

**Problem 4 :**

Determine whether the two triangles shown below are similar. Justify your answer.

**Problem 5 :**

As we move the tracing pin of a pantograph along a figure, the pencil attached to the far end draws an enlargement. As the pantograph expands and contracts, the three brads and the tracing pin always form the vertices of a parallelogram. The ratio of PR to PT is always equal to the ratio of PQ to PS. Also, the suction cup, the tracing pin, and the pencil remain collinear.

a. How can we prove that ΔPRQ ∼ ΔPTS ?

b. In the diagram, PR is 10 inches and RT is 10 inches. The length of the cat RQ, in the original print is 2.4 inches. Find the length TS in the enlargement.

**Problem 6 :**

To measure the width of a river, we use a surveying technique, as shown in the diagram. Use the given lengths (measured in feet) to find RQ.

**Problem 1 :**

In the triangles LMN and RST,

RS / LM = ST / MN = TR / NL

Prove : ΔRST ∼ ΔLMN

**Solution : **

Begin by graphing ΔLMN and ΔRST.

Locate P on RS so that PS = LM.

Draw PQ so that PQ ∥ RT.

Then ΔRST ∼ ΔPSQ, by the Angle Angle (AA) Similarity Postulate, and

RS / PS = ST / SQ = TR / QP

Because PS = LM, we can substitute in the given proportion and find that SQ = MN and QP = NL.

By the SSS Congruence Theorem, it follows that

ΔPSQ ≅ ΔLMN

Finally, use the definition of congruent triangles and the AA Similarity Postulate to conclude that

ΔRST ∼ ΔLMN

**Problem 2 : **

Which of the following three triangles are similar ?

**Solution : **

To decide which, if any, of the triangles are similar, we need to consider the ratios of the lengths of corresponding sides.

**Ratios of Side Lengths of ΔABC and ΔDEF : **

AB / DE = 6 / 4 = 3 / 2

CA / FD = 12 / 8 = 3 / 2

BC / EF = 9 / 6 = 3 / 2

Because all of the ratios are equal, ΔABC ∼ ΔDEF.

**Ratios of Side Lengths of ΔABC and ΔGHJ : **

AB / GH = 6 / 6 = 1

CA / JG = 12 / 14 = 6 / 7

BC / HJ = 9 / 10

Because the ratios are not equal, ΔABC and ΔGHJ are not similar.

Since ΔABC is similar to ΔDEF and ΔABC is not similar to ΔGHJ, ΔDEF is not similar to ΔGHJ.

**Problem 3 :**

In the diagram shown below, use the given lengths to prove that ΔRST ∼ ΔPSQ.

**Solution : **

Given : SP = 4, PR = 12, SQ = 5, QT = 15

Prove : ΔRST ∼ ΔPSQ

Use the SAS Similarity Theorem.

Begin by finding the ratios of the lengths of the corresponding sides.

SR / SP = (SP + PR) / SP = (4 + 12) / 4 = 16 / 4 = 4

ST / SQ = (SQ + QT) / SQ = (5 + 15) / 5 = 20 / 5 = 4

So, the lengths of sides SR and ST are proportional to the lengths of the corresponding sides of ΔPSQ.

Because ∠S is included angle in both triangles, use the SAS Similarity Theorem to conclude that

ΔRST ∼ ΔPSQ

**Problem 4 :**

Determine whether the two triangles shown below are similar. Justify your answer.

**Solution :**

By Triangle Sum Theorem, in Δ ABC,

∠A + ∠B + ∠C = 180°

21° + 105° + ∠C = 180°

126° + ∠C = 180°

Subtract 126° from both sides.

∠C = 54°

In triangles ABC and DEF, we have

∠A = ∠F = 21°

∠E = ∠C = 54°

Two angles of one triangle are congruent to two angles of another triangle.

By Angle-Angle (AA) Similarity Postulate, the triangles ABC and DEF are similar triangles.

**Problem 5 :**

As we move the tracing pin of a pantograph along a figure, the pencil attached to the far end draws an enlargement. As the pantograph expands and contracts, the three brads and the tracing pin always form the vertices of a parallelogram. The ratio of PR to PT is always equal to the ratio of PQ to PS. Also, the suction cup, the tracing pin, and the pencil remain collinear.

a. How can we prove that ΔPRQ ∼ ΔPTS ?

b. In the diagram, PR is 10 inches and RT is 10 inches. The length of the cat RQ, in the original print is 2.4 inches. Find the length TS in the enlargement.

**Solution (a) : **

We know that PR / PT = PQ / PS.

Because ∠P ≅ ∠P, we can apply the SAS Similarity Theorem to conclude that

ΔPRQ ∼ ΔPTS

**Solution (b) : **

Because the triangles are similar, we can set up a proportion to find the length of the cat in the enlarged drawing.

Write proportion :

PR / PT = RQ / TS

Substitute.

10 / 20 = 2.4 / TS

Solve for TS.

TS = 4.8

Hence, the length of the cat in the enlarged drawing is 4.8 inches.

**Problem 6 :**

To measure the width of a river, we use a surveying technique, as shown in the diagram. Use the given lengths (measured in feet) to find RQ.

**Solution : **

By the AA Similarity Postulate,

ΔPQR ∼ ΔSTR

Write proportion.

RQ / RT = PQ / ST

Substitute.

RQ / 12 = 63 / 9

Simplify.

RQ / 12 = 7

Multiply each side by 12.

12 ⋅ (RQ / 12) = 7 ⋅ 12

RQ = 84

Hence, the river is 84 feet wide.

After having gone through the stuff given above, we hope that the students would have understood how to prove that two triangles are similar.

Apart from the stuff given on this web page, if you need any other stuff in math, please use our google custom search here.

Widget is loading comments...

You can also visit our following web pages on different stuff in math.

**WORD PROBLEMS**

**Word problems on simple equations **

**Word problems on linear equations **

**Word problems on quadratic equations**

**Area and perimeter word problems**

**Word problems on direct variation and inverse variation **

**Word problems on comparing rates**

**Converting customary units word problems **

**Converting metric units word problems**

**Word problems on simple interest**

**Word problems on compound interest**

**Word problems on types of angles **

**Complementary and supplementary angles word problems**

**Trigonometry word problems**

**Markup and markdown word problems **

**Word problems on mixed fractrions**

**One step equation word problems**

**Linear inequalities word problems**

**Ratio and proportion word problems**

**Word problems on sets and venn diagrams**

**Pythagorean theorem word problems**

**Percent of a number word problems**

**Word problems on constant speed**

**Word problems on average speed **

**Word problems on sum of the angles of a triangle is 180 degree**

**OTHER TOPICS **

**Time, speed and distance shortcuts**

**Ratio and proportion shortcuts**

**Domain and range of rational functions**

**Domain and range of rational functions with holes**

**Graphing rational functions with holes**

**Converting repeating decimals in to fractions**

**Decimal representation of rational numbers**

**Finding square root using long division**

**L.C.M method to solve time and work problems**

**Translating the word problems in to algebraic expressions**

**Remainder when 2 power 256 is divided by 17**

**Remainder when 17 power 23 is divided by 16**

**Sum of all three digit numbers divisible by 6**

**Sum of all three digit numbers divisible by 7**

**Sum of all three digit numbers divisible by 8**

**Sum of all three digit numbers formed using 1, 3, 4**

**Sum of all three four digit numbers formed with non zero digits**