Question 1

In a triangle $	riangle ABC$, if $\angle A = 50^\circ$, $AB = 2$ cm, and $AC = 4$ cm, and in another triangle $	riangle DEF$, $\angle D = 50^\circ$, $DE = 3$ cm, and $DF = 6$ cm, which similarity criterion can be used to prove $	riangle ABC \sim 	riangle DEF$?

Accepted Answer

Correct Option: C. Solution: The SAS (Side-Angle-Side) similarity criterion can be used because one angle of a triangle is equal to one angle of another triangle and the sides including these angles are in the same ratio.

Question 2

In triangle $	riangle PQR$, a line $ST$ is drawn parallel to $QR$ such that it divides $PQ$ and $PR$ in the same ratio. If $PS = 3 	ext{ cm}$, $SQ = 5 	ext{ cm}$, and $PT = 4 	ext{ cm}$, what is the length of $TR$?

Accepted Answer

Correct Option: A. Solution: Since $ST \parallel QR$, by the Basic Proportionality Theorem, $PS/SQ = PT/TR$. Thus, $3/5 = 4/x \Rightarrow x = 6 	ext{ cm}$.

Question 3

In two similar triangles, if the corresponding angles are equal, what can be said about their corresponding sides?

Accepted Answer

Correct Option: B. Solution: In similar triangles, corresponding sides are in the same ratio, which is a fundamental property of similar figures.

Question 4

In $	riangle ABC$, if $AD$ is a median and $	riangle ABC \sim 	riangle PQR$ with $PM$ as a median, which of the following is true?

Accepted Answer

Correct Option: A. Solution: In similar triangles, the corresponding medians are proportional to the corresponding sides, so $\frac{AB}{PQ} = \frac{AD}{PM}$.

Question 5

Which of the following is NOT a criterion for the similarity of triangles?

Accepted Answer

Correct Option: D. Solution: The criteria for similarity of triangles are AAA, SSS, and SAS. ASA is a criterion for congruence, not similarity.

Question 6

Which of the following criteria is used to determine that two triangles are similar when their corresponding angles are equal?

Accepted Answer

Correct Option: C. Solution: The AAA (Angle-Angle-Angle) criterion states that if in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio and hence the two triangles are similar.

Question 7

If in two triangles, the sides of one triangle are proportional to the sides of the other triangle, which similarity criterion does this represent?

Accepted Answer

Correct Option: B. Solution: The SSS (Side-Side-Side) similarity criterion states that if in two triangles, the sides of one triangle are proportional to the sides of the other triangle, then their corresponding angles are equal and hence the triangles are similar.

Question 8

In a triangle $	riangle ABC$, if $DE \parallel BC$ and $AD = 1.5 	ext{ cm}$, $DB = 3 	ext{ cm}$, and $DE = 1 	ext{ cm}$, find the length of $EC$.

Accepted Answer

Correct Option: A. Solution: Using the Basic Proportionality Theorem, $\frac{AD}{DB} = \frac{DE}{EC}$. Therefore, $\frac{1.5}{3} = \frac{1}{EC}$, solving gives $EC = 2.5 	ext{ cm}$.

Question 9

In $	riangle ABC$, $AD$ is a median. If $	riangle ABC \sim 	riangle PQR$ and $AD = 4 	ext{ cm}$, $PQ = 6 	ext{ cm}$, $QR = 8 	ext{ cm}$, find the length of median $PM$ in $	riangle PQR$.

Accepted Answer

Correct Option: D. Solution: Since $	riangle ABC \sim 	riangle PQR$, $\frac{AD}{PM} = \frac{AB}{PQ}$. Given $AD = 4 	ext{ cm}$, $AB = 6 	ext{ cm}$, $PQ = 6 	ext{ cm}$, solving gives $PM = 12 	ext{ cm}$.

Question 10

Consider two triangles $	riangle ABC$ and $	riangle DEF$ where $\angle A = \angle D$, $\angle B = \angle E$, and $\angle C = \angle F$. If $AB = 6 	ext{ cm}$, $BC = 9 	ext{ cm}$, $CA = 12 	ext{ cm}$, and $DE = 4 	ext{ cm}$, find the length of $EF$.

Accepted Answer

Correct Option: B. Solution: Since $	riangle ABC \sim 	riangle DEF$ by the AAA similarity criterion, $\frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{DF}$. Solving $\frac{6}{4} = \frac{9}{EF}$ gives $EF = 8 	ext{ cm}$.

Question 11

In two right triangles, the hypotenuse and one leg of the first triangle are $5$ cm and $3$ cm, respectively. The hypotenuse of the second triangle is $10$ cm. What should be the length of the corresponding leg of the second triangle for the triangles to be similar by the RHS criterion?

Accepted Answer

Correct Option: C. Solution: For the triangles to be similar by the RHS criterion, the sides must be proportional. The ratio of the hypotenuses is $\frac{5}{10} = \frac{1}{2}$. Therefore, the corresponding leg in the second triangle should be $3 	imes 2 = 6$ cm.

Question 12

In a triangle $	riangle ABC$, if a line $DE$ is drawn parallel to $BC$ such that it intersects $AB$ at $D$ and $AC$ at $E$, which of the following is true according to the Basic Proportionality Theorem?

Accepted Answer

Correct Option: A. Solution: According to the Basic Proportionality Theorem, if a line is drawn parallel to one side of a triangle to intersect the other two sides, the other two sides are divided in the same ratio: $AD/DB = AE/EC$.

Question 13

A vertical pole of length 6 m casts a shadow 4 m long on the ground. At the same time, a tower casts a shadow 28 m long. What is the height of the tower?

Accepted Answer

Correct Option: B. Solution: Using the concept of similar triangles, the ratio of the height of the pole to its shadow is equal to the ratio of the height of the tower to its shadow: $\frac{6}{4} = \frac{h}{28}$. Solving gives $h = 42 	ext{ m}$.

Question 14

If in two triangles $	riangle ABC$ and $	riangle DEF$, $\angle A = \angle D$, $\angle B = \angle E$, and $\angle C = \angle F$, which similarity criterion is used to conclude that $	riangle ABC \sim 	riangle DEF$?

Accepted Answer

Correct Option: B. Solution: The AAA similarity criterion is used when all corresponding angles of two triangles are equal, indicating that the triangles are similar.

Question 15

If $	riangle ABC \sim 	riangle DEF$ and $\angle A = 50^\circ$, $\angle B = 60^\circ$, what is $\angle D$?

Accepted Answer

Correct Option: A. Solution: In similar triangles, corresponding angles are equal. Therefore, $\angle D = \angle A = 50^\circ$.

Question 16

In triangle $	riangle ABC$, $AD$ is a median and $BE$ is an altitude. If $	riangle ABC \sim 	riangle DEF$ and $AD$ corresponds to $DF$, which of the following statements is true?

Accepted Answer

Correct Option: A. Solution: In similar triangles, corresponding angles are equal and corresponding sides are in the same ratio. Since $AD$ is a median and corresponds to $DF$, it is essential that $\angle A = \angle D$ and the sides $AB$ and $DE$, $BC$ and $EF$ are proportional.

Question 17

In the diagram of triangle $	riangle ABC$ and $	riangle DEF$, if $AB = 3$ cm, $BC = 6$ cm, $CA = 8$ cm, $DE = 4.5$ cm, $EF = 9$ cm, and $FD = 12$ cm, what can be concluded about the triangles?

Accepted Answer

Correct Option: A. Solution: The sides of $	riangle ABC$ and $	riangle DEF$ are in the same ratio, hence they are similar by the SSS criterion.

Question 18

Given $	riangle ABC \sim 	riangle DEF$ and $\frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{FD}$, which criterion confirms the similarity of the triangles?

Accepted Answer

Correct Option: B. Solution: The SSS (Side-Side-Side) similarity criterion states that if the sides of one triangle are proportional to the sides of another triangle, then the triangles are similar.

Question 19

In triangle $	riangle ABC$, $AD$ is an angle bisector. If $	riangle ABC \sim 	riangle DEF$ and $AD$ corresponds to $DF$, what can be said about the relationship between $\angle A$ and $\angle D$?

Accepted Answer

Correct Option: A. Solution: In similar triangles, corresponding angles are equal. Therefore, $\angle A$ corresponds to $\angle D$, making them equal.

Question 20

Which of the following is a condition for two triangles to be similar under the AA similarity criterion?

Accepted Answer

Correct Option: A. Solution: The AA similarity criterion states that if two angles of one triangle are equal to two angles of another triangle, then the triangles are similar.

Question 21

In a trapezium $ABCD$ where $AB \parallel DC$, points $E$ and $F$ are on non-parallel sides $AD$ and $BC$ respectively, such that $EF \parallel AB$. If $AE = 3 	ext{ cm}$, $ED = 2 	ext{ cm}$, $BF = 4 	ext{ cm}$, and $FC = x 	ext{ cm}$, find the value of $x$.

Accepted Answer

Correct Option: B. Solution: Since $EF \parallel AB \parallel DC$, by the Basic Proportionality Theorem, $$\frac{AE}{ED} = \frac{BF}{FC}$$. Substituting the given values, $$\frac{3}{2} = \frac{4}{x}$$. Solving for $x$, we get $x = 3 	ext{ cm}$.

Question 22

If $	riangle ABC$ and $	riangle DEF$ are similar triangles with $\angle A = \angle D$, $\angle B = \angle E$, and $\angle C = \angle F$, which theorem can be used to establish their similarity?

Accepted Answer

Correct Option: C. Solution: The AAA (Angle-Angle-Angle) similarity criterion states that if all corresponding angles of two triangles are equal, then the triangles are similar.

Question 23

Two triangles $	riangle ABC$ and $	riangle DEF$ are such that $\angle A = \angle D$, $\angle B = \angle E$, and $\angle C = \angle F$. If $AB = 3$ cm, $BC = 4$ cm, $CA = 5$ cm, $DE = 6$ cm, $EF = 8$ cm, and $FD = 10$ cm, are the triangles similar?

Accepted Answer

Correct Option: B. Solution: The triangles are similar by the SSS criterion since $\frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{FD} = \frac{3}{6} = \frac{4}{8} = \frac{5}{10} = \frac{1}{2}$.

Question 24

If $	riangle ABC \sim 	riangle DEF$ and $AB = 3$ cm, $BC = 4$ cm, $CA = 5$ cm, $DE = 6$ cm, what is $EF$ if $DF = 10$ cm?

Accepted Answer

Correct Option: A. Solution: Since $	riangle ABC \sim 	riangle DEF$, $\frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{DF}$. Substituting the known values, $\frac{3}{6} = \frac{4}{EF} = \frac{5}{10}$. Solving for $EF$, we get $EF = 8$ cm.

Question 25

A vertical pole of height $6 	ext{ m}$ casts a shadow of $4 	ext{ m}$ on the ground. At the same time, a tower casts a shadow of $28 	ext{ m}$. What is the height of the tower?

Accepted Answer

Correct Option: A. Solution: Using the similarity of triangles formed by the pole and its shadow, and the tower and its shadow, the height of the tower is $\frac{6}{4} 	imes 28 = 21 	ext{ m}$.

Question 26

Given $	riangle ABC \sim 	riangle DEF$, with $AB = 3 	ext{ cm}$, $BC = 4 	ext{ cm}$, $CA = 5 	ext{ cm}$, $DE = 6 	ext{ cm}$, and $EF = 8 	ext{ cm}$, find the length of $FD$.

Accepted Answer

Correct Option: B. Solution: Since $	riangle ABC \sim 	riangle DEF$, the sides are proportional: $\frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{FD}$. Solving $\frac{5}{FD} = \frac{3}{6}$ gives $FD = 10 	ext{ cm}$.

Question 27

Which of the following statements is a correct application of the converse of the Basic Proportionality Theorem?

Accepted Answer

Correct Option: B. Solution: The converse of the Basic Proportionality Theorem states that if a line divides two sides of a triangle in the same ratio, then it is parallel to the third side.

Question 28

In $	riangle ABC$, if $\angle A = \angle D$, $\angle B = \angle E$, and $\angle C = \angle F$, what can be concluded about $	riangle ABC$ and $	riangle DEF$?

Accepted Answer

Correct Option: A. Solution: If all corresponding angles of two triangles are equal, the triangles are similar by the AAA similarity criterion.

Question 29

In similar triangles $	riangle ABC \sim 	riangle DEF$, which of the following is true about their corresponding sides?

Accepted Answer

Correct Option: B. Solution: In similar triangles, the corresponding sides are in the same ratio, i.e., $\frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{FD}$.

Question 30

In $	riangle PQR$, $PQ = 5$ cm, $QR = 6$ cm, and $RP = 7$ cm. In $	riangle XYZ$, $XY = 10$ cm, $YZ = 12$ cm, and $ZX = 14$ cm. Are the triangles similar?

Accepted Answer

Correct Option: B. Solution: The triangles are similar by the SSS criterion since $\frac{PQ}{XY} = \frac{QR}{YZ} = \frac{RP}{ZX} = \frac{5}{10} = \frac{6}{12} = \frac{7}{14} = \frac{1}{2}$.

Question 31

If in two triangles, the sides of one triangle are proportional to the sides of the other triangle, what can be concluded?

Accepted Answer

Correct Option: B. Solution: If the sides of one triangle are proportional to the sides of another triangle, the triangles are similar by the SSS similarity criterion.

Question 32

In a triangle $	riangle ABC$, a line $DE$ is drawn parallel to $BC$, intersecting $AB$ at $D$ and $AC$ at $E$. If $AD = 3 	ext{ cm}$, $DB = 6 	ext{ cm}$, and $AE = 4 	ext{ cm}$, find the length of $EC$.

Accepted Answer

Correct Option: A. Solution: Since $DE \parallel BC$, by the Basic Proportionality Theorem, $\frac{AD}{DB} = \frac{AE}{EC}$. Substituting the given values, $\frac{3}{6} = \frac{4}{EC}$, which gives $EC = 8 	ext{ cm}$.

Question 33

In triangle $	riangle ABC$, $D$ is a point on $BC$ such that $AD$ bisects $\angle BAC$. If $AB = 6 	ext{ cm}$, $AC = 9 	ext{ cm}$, and $BD = 4 	ext{ cm}$, find $DC$.

Accepted Answer

Correct Option: A. Solution: By the Angle Bisector Theorem, $AB/AC = BD/DC$. So, $6/9 = 4/x \Rightarrow x = 6 	ext{ cm}$.

Question 34

Two right triangles have hypotenuses of lengths $10$ cm and $15$ cm, respectively. If one leg of the first triangle is $6$ cm and the corresponding leg of the second triangle is $9$ cm, are the triangles similar according to the RHS similarity criterion?

Accepted Answer

Correct Option: A. Solution: According to the RHS similarity criterion, if the hypotenuse and one side of one right triangle are proportional to the hypotenuse and one side of another right triangle, the triangles are similar. Here, $\frac{10}{15} = \frac{6}{9} = \frac{2}{3}$, so the triangles are similar.

Question 35

In $	riangle ABC$, $\angle A = 50^\circ$, $\angle B = 60^\circ$, and $\angle C = 70^\circ$. In $	riangle DEF$, $\angle D = 50^\circ$, $\angle E = 60^\circ$, and $\angle F = 70^\circ$. Are the triangles similar?

Accepted Answer

Correct Option: A. Solution: The triangles are similar by the AAA criterion since all corresponding angles are equal: $\angle A = \angle D$, $\angle B = \angle E$, and $\angle C = \angle F$.

Question 36

Which of the following is a criterion for the similarity of two triangles?

Accepted Answer

Correct Option: D. Solution: The criteria for the similarity of two triangles include AAA, SSS, and SAS.

Question 37

According to the Basic Proportionality Theorem, if a line is drawn parallel to one side of a triangle to intersect the other two sides, what is the result?

Accepted Answer

Correct Option: A. Solution: The Basic Proportionality Theorem states that if a line is drawn parallel to one side of a triangle to intersect the other two sides, the other two sides are divided in the same ratio.

Question 38

If two triangles have sides in the ratio $\frac{3}{4}$, $\frac{6}{8}$, and $\frac{9}{12}$, what can be concluded about the triangles?

Accepted Answer

Correct Option: B. Solution: The triangles are similar by the SSS (Side-Side-Side) similarity criterion because their corresponding sides are in the same ratio.

Question 39

In a triangle $	riangle ABC$, if $\angle A = 60^\circ$, $\angle B = 80^\circ$, and $\angle C = 40^\circ$, which of the following triangles is similar to $	riangle ABC$ based on the AA similarity criterion?

Accepted Answer

Correct Option: A. Solution: By the AA similarity criterion, two triangles are similar if two pairs of corresponding angles are equal. Here, $	riangle PQR$ has angles equal to $	riangle ABC$.

Question 40

In triangle $	riangle ABC$, line $DE$ is drawn such that $\frac{AD}{DB} = \frac{AE}{EC}$. Which of the following is true?

Accepted Answer

Correct Option: A. Solution: By the converse of the Basic Proportionality Theorem, if $\frac{AD}{DB} = \frac{AE}{EC}$, then $DE$ is parallel to $BC$.

Question 41

Given two right triangles where the hypotenuse and one side of the first triangle are 5 cm and 3 cm respectively, and the hypotenuse and one side of the second triangle are 10 cm and 6 cm respectively. Are these triangles similar?

Accepted Answer

Correct Option: A. Solution: The sides of the triangles are in the ratio 5:10 and 3:6, which simplifies to 1:2 for both pairs of sides. Hence, the triangles are similar by the RHS Similarity Criterion.

Question 42

In a triangle $	riangle ABC$, if $DE \parallel BC$ and $D$ and $E$ are points on $AB$ and $AC$ respectively, which of the following is true?

Accepted Answer

Correct Option: A. Solution: According to the Basic Proportionality Theorem, if $DE \parallel BC$, then $\frac{AD}{DB} = \frac{AE}{EC}$.

Question 43

In two right triangles, if the hypotenuse and one side of one triangle are proportional to the hypotenuse and one side of the other triangle, what can be concluded about the triangles?

Accepted Answer

Correct Option: B. Solution: According to the RHS Similarity Criterion, if the hypotenuse and one side of one right triangle are proportional to the hypotenuse and one side of another right triangle, the triangles are similar.

Question 44

If a line divides two sides of a triangle in the same ratio, what can be concluded about the line?

Accepted Answer

Correct Option: B. Solution: According to the converse of the Basic Proportionality Theorem, if a line divides two sides of a triangle in the same ratio, then the line is parallel to the third side.

Question 45

In $	riangle ABC$, $AB = 3 	ext{ cm}$, $BC = 6 	ext{ cm}$, and $CA = 8 	ext{ cm}$. In $	riangle DEF$, $DE = 4.5 	ext{ cm}$, $EF = 9 	ext{ cm}$, and $FD = 12 	ext{ cm}$. Are these triangles similar?

Accepted Answer

Correct Option: A. Solution: The sides of $	riangle ABC$ and $	riangle DEF$ are in the same ratio: $\frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{FD} = \frac{1}{1.5}$. Therefore, the triangles are similar by the SSS similarity criterion.

Question 46

In a triangle $	riangle ABC$, a line $DE$ is drawn parallel to side $BC$ and intersects $AB$ and $AC$ at points $D$ and $E$ respectively. Which theorem justifies that $\frac{AD}{DB} = \frac{AE}{EC}$?

Accepted Answer

Correct Option: B. Solution: The Basic Proportionality Theorem states that if a line is drawn parallel to one side of a triangle to intersect the other two sides, then it divides those sides in the same ratio.

Question 47

In a triangle $	riangle ABC$, if $\angle A = 50^\circ$, $AB = 3 	ext{ cm}$, and $AC = 4 	ext{ cm}$, and in another triangle $	riangle DEF$, $\angle D = 50^\circ$, $DE = 4.5 	ext{ cm}$, and $DF = 6 	ext{ cm}$, which similarity criterion can be used to prove that $	riangle ABC \sim 	riangle DEF$?

Accepted Answer

Correct Option: C. Solution: The SAS similarity criterion is applicable here because one angle of a triangle is equal to one angle of another triangle, and the sides including these angles are proportional.

Question 48

In two triangles, if the sides are in the same ratio, the triangles are always congruent.

Accepted Answer

Answer: False. Solution: The SSS similarity criterion states that if in two triangles, the sides are in the same ratio, the triangles are similar, not necessarily congruent.

Question 49

Two triangles are similar if one angle of a triangle is equal to one angle of another triangle and the sides including these angles are in the same ratio.

Accepted Answer

Answer: True. Solution: This is the SAS (Side-Angle-Side) similarity criterion, which states that if one angle of a triangle is equal to one angle of another triangle and the sides including these angles are proportional, the triangles are similar.

Question 50

If two triangles have two pairs of corresponding angles equal, the triangles are not necessarily similar.

Accepted Answer

Answer: False. Solution: This is false because if two angles of one triangle are equal to two angles of another triangle, the triangles are similar by the AA similarity criterion.

Question 51

The Converse of the Basic Proportionality Theorem states that if a line is parallel to one side of a triangle, it divides the other two sides in the same ratio.

Accepted Answer

Answer: False. Solution: This statement is false. The Converse of the Basic Proportionality Theorem actually states that if a line divides two sides of a triangle in the same ratio, then the line is parallel to the third side. The given statement describes the Basic Proportionality Theorem itself, not its converse.

Question 52

In similar triangles, the corresponding medians are always proportional to the corresponding sides.

Accepted Answer

Answer: True. Solution: In similar triangles, not only are the corresponding sides proportional, but the corresponding medians are also proportional as shown in the exercises.

Question 53

For two triangles to be similar under the SAS (Side-Angle-Side) criterion, it is sufficient for one angle of a triangle to be equal to one angle of another triangle, with the sides including these angles being proportional.

Accepted Answer

Answer: True. Solution: Theorem 6.5 states that if one angle of a triangle is equal to one angle of another triangle and the sides including these angles are proportional, then the two triangles are similar. This is the SAS similarity criterion.

Question 54

If two triangles have their corresponding angles equal, then their corresponding sides are in the same ratio, making the triangles similar.

Accepted Answer

Answer: True. Solution: According to the AAA similarity criterion, if in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio and hence the two triangles are similar.

Question 55

For triangles $	riangle ABC$ and $	riangle DEF$, if $\angle A = \angle D$, $\angle B = \angle E$, and $\angle C = \angle F$, then $	riangle ABC \sim 	riangle DEF$ by the SSS similarity criterion.

Accepted Answer

Answer: False. Solution: The AAA similarity criterion states that if the corresponding angles of two triangles are equal, then the triangles are similar. The SSS criterion involves corresponding sides being in the same ratio.

Question 56

If a line is drawn parallel to one side of a triangle to intersect the other two sides, the other two sides are divided in the same ratio.

Accepted Answer

Answer: True. Solution: This statement is true and is known as the Basic Proportionality Theorem or Thales Theorem.

Question 57

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.

Accepted Answer

Answer: True. Solution: This statement is true according to Theorem 6.1, which states that if a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally.

Question 58

For two triangles to be similar, it is necessary to check both the conditions of equal corresponding angles and proportional corresponding sides.

Accepted Answer

Answer: False. Solution: In the case of similarity of triangles, it is not necessary to check both conditions as one condition implies the other.

Question 59

The Basic Proportionality Theorem states that if a line is drawn parallel to one side of a triangle to intersect the other two sides, the other two sides are divided in the same ratio.

Accepted Answer

Answer: True. Solution: The Basic Proportionality Theorem, also known as Thales Theorem, confirms that a line parallel to one side of a triangle divides the other two sides proportionally.

Question 60

If a line divides two sides of a triangle in the same ratio, then the line is parallel to the third side.

Accepted Answer

Answer: True. Solution: This statement is the converse of the Basic Proportionality Theorem, which states that if a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.

Question 61

The AA similarity criterion states that if two angles of one triangle are equal to two angles of another triangle, then the two triangles are not similar.

Accepted Answer

Answer: False. Solution: The AA similarity criterion actually states that if two angles of one triangle are equal to two angles of another triangle, then the two triangles are similar.

Question 62

If two triangles have their corresponding sides in the same ratio, then their corresponding angles are equal, making the triangles similar.

Accepted Answer

Answer: True. Solution: This is the SSS (Side-Side-Side) similarity criterion, which states that if the sides of one triangle are proportional to the sides of another, their corresponding angles are equal, and hence, the triangles are similar.

Question 63

The converse of the Basic Proportionality Theorem is not true.

Accepted Answer

Answer: False. Solution: The converse of the Basic Proportionality Theorem is indeed true. It states that if a line divides two sides of a triangle in the same ratio, then the line is parallel to the third side.

Question 64

In similar triangles, the corresponding angles are equal and the corresponding sides are in the same ratio.

Accepted Answer

Answer: True. Solution: By definition, two triangles are similar if their corresponding angles are equal and their corresponding sides are in the same ratio.

Triangles

Learning Objectives

Learning Objectives

Revision Notes & Summary

Chapter Notes

Similarity of Triangles

Basic Proportionality Theorem (Thales Theorem)

Converse of Basic Proportionality Theorem

Examples and Applications

Exercises

Summary

Exam Tips & Common Mistakes

Common Mistakes & Exam Tips

Similarity of Triangles

Basic Proportionality Theorem (Thales Theorem)

Converse of Basic Proportionality Theorem

Criteria for Similarity of Triangles

RHS Similarity Criterion

General Exam Tips

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Multiple Choice Questions

Correct Answer: C

Q2.In triangle △PQR\triangle PQR△PQR, a line STSTST is drawn parallel to QRQRQR such that it divides PQPQPQ and PRPRPR in the same ratio. If PS=3 cmPS = 3 \text{ cm}PS=3 cm, SQ=5 cmSQ = 5 \text{ cm}SQ=5 cm, and PT=4 cmPT = 4 \text{ cm}PT=4 cm, what is the length of TRTRTR?

Correct Answer: A

Q3.In two similar triangles, if the corresponding angles are equal, what can be said about their corresponding sides?

Correct Answer: B

Q4.In △ABC\triangle ABC△ABC, if ADADAD is a median and △ABC∼△PQR\triangle ABC \sim \triangle PQR△ABC∼△PQR with PMPMPM as a median, which of the following is true?

Correct Answer: A

Q5.Which of the following is NOT a criterion for the similarity of triangles?

Correct Answer: D

Q6.Which of the following criteria is used to determine that two triangles are similar when their corresponding angles are equal?

Correct Answer: C

Q7.If in two triangles, the sides of one triangle are proportional to the sides of the other triangle, which similarity criterion does this represent?

Correct Answer: B

Q8.In a triangle △ABC\triangle ABC△ABC, if DE∥BCDE \parallel BCDE∥BC and AD=1.5 cmAD = 1.5 \text{ cm}AD=1.5 cm, DB=3 cmDB = 3 \text{ cm}DB=3 cm, and DE=1 cmDE = 1 \text{ cm}DE=1 cm, find the length of ECECEC.

Correct Answer: A

Q9.In △ABC\triangle ABC△ABC, ADADAD is a median. If △ABC∼△PQR\triangle ABC \sim \triangle PQR△ABC∼△PQR and AD=4 cmAD = 4 \text{ cm}AD=4 cm, PQ=6 cmPQ = 6 \text{ cm}PQ=6 cm, QR=8 cmQR = 8 \text{ cm}QR=8 cm, find the length of median PMPMPM in △PQR\triangle PQR△PQR.

Correct Answer: D

Correct Answer: B

Q11.In two right triangles, the hypotenuse and one leg of the first triangle are 555 cm and 333 cm, respectively. The hypotenuse of the second triangle is 101010 cm. What should be the length of the corresponding leg of the second triangle for the triangles to be similar by the RHS criterion?

Correct Answer: C

Q12.In a triangle △ABC\triangle ABC△ABC, if a line DEDEDE is drawn parallel to BCBCBC such that it intersects ABABAB at DDD and ACACAC at EEE, which of the following is true according to the Basic Proportionality Theorem?

Correct Answer: A

Q13.A vertical pole of length 6 m casts a shadow 4 m long on the ground. At the same time, a tower casts a shadow 28 m long. What is the height of the tower?

Correct Answer: B

Correct Answer: B

Q15.If △ABC∼△DEF\triangle ABC \sim \triangle DEF△ABC∼△DEF and ∠A=50∘\angle A = 50^\circ∠A=50∘, ∠B=60∘\angle B = 60^\circ∠B=60∘, what is ∠D\angle D∠D?

Correct Answer: A

Q16.In triangle △ABC\triangle ABC△ABC, ADADAD is a median and BEBEBE is an altitude. If △ABC∼△DEF\triangle ABC \sim \triangle DEF△ABC∼△DEF and ADADAD corresponds to DFDFDF, which of the following statements is true?

Correct Answer: A

Q17.In the diagram of triangle △ABC\triangle ABC△ABC and △DEF\triangle DEF△DEF, if AB=3AB = 3AB=3 cm, BC=6BC = 6BC=6 cm, CA=8CA = 8CA=8 cm, DE=4.5DE = 4.5DE=4.5 cm, EF=9EF = 9EF=9 cm, and FD=12FD = 12FD=12 cm, what can be concluded about the triangles?

Correct Answer: A

Q18.Given △ABC∼△DEF\triangle ABC \sim \triangle DEF△ABC∼△DEF and ABDE=BCEF=CAFD\frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{FD}DEAB​=EFBC​=FDCA​, which criterion confirms the similarity of the triangles?

Correct Answer: B

Q19.In triangle △ABC\triangle ABC△ABC, ADADAD is an angle bisector. If △ABC∼△DEF\triangle ABC \sim \triangle DEF△ABC∼△DEF and ADADAD corresponds to DFDFDF, what can be said about the relationship between ∠A\angle A∠A and ∠D\angle D∠D?

Correct Answer: A

Q20.Which of the following is a condition for two triangles to be similar under the AA similarity criterion?

Correct Answer: A

Correct Answer: B

Q22.If △ABC\triangle ABC△ABC and △DEF\triangle DEF△DEF are similar triangles with ∠A=∠D\angle A = \angle D∠A=∠D, ∠B=∠E\angle B = \angle E∠B=∠E, and ∠C=∠F\angle C = \angle F∠C=∠F, which theorem can be used to establish their similarity?

Correct Answer: C

Correct Answer: B

Q24.If △ABC∼△DEF\triangle ABC \sim \triangle DEF△ABC∼△DEF and AB=3AB = 3AB=3 cm, BC=4BC = 4BC=4 cm, CA=5CA = 5CA=5 cm, DE=6DE = 6DE=6 cm, what is EFEFEF if DF=10DF = 10DF=10 cm?

Correct Answer: A

Q25.A vertical pole of height 6 m6 \text{ m}6 m casts a shadow of 4 m4 \text{ m}4 m on the ground. At the same time, a tower casts a shadow of 28 m28 \text{ m}28 m. What is the height of the tower?

Correct Answer: A

Q26.Given △ABC∼△DEF\triangle ABC \sim \triangle DEF△ABC∼△DEF, with AB=3 cmAB = 3 \text{ cm}AB=3 cm, BC=4 cmBC = 4 \text{ cm}BC=4 cm, CA=5 cmCA = 5 \text{ cm}CA=5 cm, DE=6 cmDE = 6 \text{ cm}DE=6 cm, and EF=8 cmEF = 8 \text{ cm}EF=8 cm, find the length of FDFDFD.

Correct Answer: B

Q27.Which of the following statements is a correct application of the converse of the Basic Proportionality Theorem?

Correct Answer: B

Q28.In △ABC\triangle ABC△ABC, if ∠A=∠D\angle A = \angle D∠A=∠D, ∠B=∠E\angle B = \angle E∠B=∠E, and ∠C=∠F\angle C = \angle F∠C=∠F, what can be concluded about △ABC\triangle ABC△ABC and △DEF\triangle DEF△DEF?

Correct Answer: A

Q29.In similar triangles △ABC∼△DEF\triangle ABC \sim \triangle DEF△ABC∼△DEF, which of the following is true about their corresponding sides?

Correct Answer: B