Triangles · Class 10 Mathematics: 171 MCQs with answers

Q1. Two figures are similar when they have

Q2. Two figures are congruent when they have

Q3. Which statement is true?

Q4. The symbol $\sim$ means

Q5. Any two circles are

Q6. Any two squares are

Q7. Any two equilateral triangles are

Q8. For two polygons to be similar, their corresponding angles must be

Q9. Which quantity is compared by a RATIO in the definition of similar polygons?

Q10. The common ratio of corresponding sides is called the

Q11. A scale factor of 1 means the figures are

Q12. Assertion (A): Every square is similar to every other square. Reason (R): Every angle of a square is a right angle.

Q13. Similarity compares two figures' shapes; congruence compares

Q14. A figure and its enlargement are

Q15. Explain the difference between similar figures and congruent figures.

Q16. State the two conditions for two polygons of the same number of sides to be similar.

Q17. The Basic Proportionality Theorem says that a line parallel to one side of a triangle

Q18. The BPT is also known as

Q19. If $DE \parallel BC$ in $\triangle ABC$, then

Q20. The part-over-whole form of the BPT is

Q21. In the BPT proof, the construction is to

Q22. When proving the BPT, the area of a triangle is written as

Q23. Triangles on the same base and between the same parallels are equal in

Q24. Which two areas are equal in the proof of the BPT?

Q25. The converse of the BPT says that if a line divides two sides in the same ratio, then it is

Q26. In the statement of the BPT, the line must cut the other two sides in

Q27. In a triangle with a line parallel to BC, $AD = 2$, $DB = 4$ and $AE = 3$. Then EC is

Q28. Assertion (A): The BPT proof does not use similar triangles. Reason (R): The similarity criteria are themselves proved using the BPT.

Q29. Which theorem number is the Basic Proportionality Theorem in the textbook?

Q30. A parallel line gives $AD = 6$, $DB = 9$ and $AE = 8$. Then EC is

Q31. The heights EN and DM exist in the proof in order to

Q32. State the Basic Proportionality Theorem.

Q33. State the converse of the Basic Proportionality Theorem.

Q34. The AA similarity criterion requires

Q35. The SSS similarity criterion requires

Q36. The SAS similarity criterion requires the equal angle to be

Q37. Two angles of one triangle equal two angles of another. The triangles are

Q38. Why does knowing two pairs of equal angles settle the third?

Q39. If $\angle A = \angle P = 50°$ and $\angle B = \angle Q = 60°$, then $\angle C$ equals

Q40. Triangles with sides $3, 4, 5$ and $6, 8, 10$ are

Q41. In $\triangle ABC \sim \triangle PQR$, the angle equal to $\angle B$ is

Q42. In $\triangle ABC \sim \triangle PQR$, the side corresponding to $CA$ is

Q43. Similar triangles have sides that are

Q44. The similarity criteria are proved from

Q45. Assertion (A): The AAA and AA criteria say effectively the same thing. Reason (R): Once two pairs of angles match, the third pair must match because the angles of a triangle sum to 180°.

Q46. Which letters name the criterion that uses one angle and two sides?

Q47. In similar triangles, corresponding angles are

Q48. Triangles with sides $9, 12, 15$ and $15, 20, 25$ are

Q49. Assertion (A): Two triangles with three pairs of equal angles must have proportional sides. Reason (R): This is exactly what the AAA similarity criterion states.

Q50. State the AA similarity criterion, and explain why two pairs of angles are enough.

Q51. State the SSS similarity criterion.

Q52. Determine whether triangles with sides 3, 4, 5 cm and 6, 8, 10 cm are similar, showing your working.

Q53. A square and a rectangle 4 by 7 are NOT similar because

Q54. A square and a rhombus of the same side length are NOT similar because

Q55. Are all rectangles similar?

Q56. In $ABCD \sim PQRS$, the side $BC$ corresponds to

Q57. In $ABCD \sim PQRS$, the angle equal to $\angle C$ is

Q58. If $\triangle ABC \sim \triangle PQR$ with $AB = 6$, $PQ = 9$ and $BC = 8$, then $QR$ is

Q59. Two similar quadrilaterals have sides $4, 6, 8, 10$ and $6, 9, 12, x$. The value of x is

Q60. Two similar polygons have a scale factor of 3. Each side of the first is

Q61. Which pair of figures is definitely similar?

Q62. If two polygons have equal corresponding angles but different side ratios, they are

Q63. Two figures with the same shape but different sizes are

Q64. Assertion (A): All rectangles are similar. Reason (R): All rectangles have four right angles.

Q65. Assertion (A): Two polygons whose corresponding sides are in the same ratio must be similar. Reason (R): Similarity of polygons requires equal angles as well as proportional sides.

Q66. In $ABCD \sim PQRS$, the side that corresponds to $DA$ is

Q67. Two similar triangles have sides in the ratio 3 : 7. The larger triangle's side matching a side of 12 is

Q68. A scale drawing of 1 : 500 means 1 cm on the drawing represents

Q69. Which is NOT required for two polygons to be similar?

Q70. Show, with an example, that equal angles alone do not make two polygons similar.

Q71. Show, with an example, that proportional sides alone do not make two polygons similar.

Q72. Are all rectangles similar? Justify your answer.

Q73. $\triangle ABC \sim \triangle PQR$ with $AB = 6$ cm, $PQ = 9$ cm and $BC = 8$ cm. Find $QR$.

Q74. Two similar quadrilaterals have sides 4, 6, 8, 10 and 6, 9, 12, x. Find the scale factor and x.

Q75. Explain why the order of the letters in $ABCD \sim PQRS$ matters.

Q76. (a) Define similar figures, and state the relationship between similarity and congruence. (b) State the two conditions for two polygons to be similar. (c) Show by a separate example that each condition on its own is insufficient.

Q77. (a) Explain what the correspondence $\triangle ABC \sim \triangle PQR$ tells you, listing the equal angles and the equal ratios. (b) Given $AB = 5$ cm, $BC = 7$ cm, $CA = 9$ cm and $PQ = 10$ cm, find $QR$ and $RP$. (c) State whether the two triangles could ever be congruent, and under what condition.

Q78. A parallel line gives $AD = 4$ cm, $DB = 6$ cm and $AE = 6$ cm. Then EC is

Q79. In $\triangle ABC$, $AD = 3$, $DB = 5$, $AE = 6$ and $EC = 10$. Then $DE$ is

Q80. In $\triangle ABC$, $AD = 2$, $DB = 3$, $AE = 4$ and $EC = 5$. Then $DE$ is

Q81. Which is a valid way to write the BPT?

Q82. To prove that a line is parallel to a side, you should use

Q83. With a parallel line, $AD = 4$, $AB = 10$ and $AE = 6$. Then AC is

Q84. Given a parallel line with $AD = 4$, $DB = 6$ and $AE = 6$, the length AC is

Q85. The ratio $\frac{ar(ADE)}{ar(BDE)}$ simplifies to $\frac{AD}{DB}$ because

Q86. A line through a vertex of a triangle

Q87. If $D$ and $E$ are the midpoints of $AB$ and $AC$, then $DE$ is parallel to BC because

Q88. The BPT holds

Q89. Assertion (A): $\frac{AD}{DB} = \frac{AE}{AC}$ follows from $DE \parallel BC$. Reason (R): The BPT may be written either as part : part or as part : whole.

Q90. Assertion (A): The converse of the BPT can be used to prove two lines parallel. Reason (R): If a line cuts two sides of a triangle in unequal proportion, it runs parallel to the remaining side.

Q91. In $\triangle ABC$, $AD = 4$, $AB = 12$, $AE = 5$ and $AC = 15$. Then DE is

Q92. If $\frac{AD}{DB} = \frac{5}{3}$ and $DE \parallel BC$, then $\frac{DB}{AB}$ is

Q93. A line cuts AB at D with $AD : DB = 3 : 7$, and cuts AC at E with $AE : EC = 3 : 7$. This line is

Q94. Assertion (A): A line through the midpoints of a triangle's two sides runs parallel to its third side. Reason (R): Both ratios are then 1 : 1, so the converse of the BPT applies.

Q95. In $\triangle ABC$, $DE \parallel BC$ with $AD = 4$ cm, $DB = 6$ cm and $AE = 6$ cm. Find EC.

Q96. In $\triangle ABC$, $D$ is on $AB$ and $E$ on $AC$ with $AD = 3$ cm, $DB = 5$ cm, $AE = 6$ cm and $EC = 10$ cm. Is $DE \parallel BC$?

Q97. In $\triangle ABC$, $AD = 2$ cm, $DB = 3$ cm, $AE = 4$ cm and $EC = 5$ cm. Determine whether $DE \parallel BC$.

Q98. In $\triangle ABC$, $DE \parallel BC$ with $AD = 4$ cm, $AB = 10$ cm and $AE = 6$ cm. Find AC.

Q99. Explain why $\frac{AD}{DB} = \frac{AE}{AC}$ is an incorrect statement of the BPT.

Q100. In $\triangle ABC$, D and E are the midpoints of AB and AC. Show that $DE \parallel BC$.

Q101. Triangles with sides $3, 4, 5$ and $6, 8, 11$ are

Q102. In $\triangle ABC$ and $\triangle PQR$, $\angle A = \angle P$, $AB = 4$, $AC = 6$, $PQ = 6$ and $PR = 9$. The triangles are

Q103. In two triangles, $\angle B = \angle Q$ and $\frac{AB}{PQ} = \frac{AC}{PR}$. Does SAS apply?

Q104. $\triangle ABC \sim \triangle PQR$ with $AB = 6$, $BC = 9$ and $PQ = 4$. Then QR is

Q105. If $\angle A = \angle Q$, $\angle B = \angle R$ and $\angle C = \angle P$, the correct statement is

Q106. Which criterion applies when three pairs of sides are proportional?

Q107. Which criterion applies when two pairs of angles are equal?

Q108. A tree and its shadow, and a stick and its shadow, form similar triangles by

Q109. Two isosceles triangles with equal vertical angles are

Q110. The similarity criteria differ from the congruence criteria in that

Q111. Two triangles have $\frac{AB}{PQ} = \frac{BC}{QR} = \frac{2}{3}$ but nothing is known about CA and RP. The triangles are

Q112. Assertion (A): Two triangles with one equal angle and two proportional sides must be similar. Reason (R): The SAS criterion requires the equal angle to be included between the proportional sides.

Q113. Assertion (A): All equilateral triangles are similar to one another. Reason (R): An equilateral triangle has three equal sides.

Q114. $\triangle ABC \sim \triangle XYZ$ with $AB = 10$, $XY = 4$ and $CA = 15$. Then ZX is

Q115. Triangles with sides $6, 8, 10$ and $9, 12, 14$ are

Q116. Two triangles are similar with a scale factor of 3. The ratio of their perimeters is

Q117. State the SAS similarity criterion, and explain the importance of the word 'including'.

Q118. Determine whether triangles with sides 3, 4, 5 cm and 6, 8, 11 cm are similar.

Q119. In $\triangle ABC$ and $\triangle PQR$, $\angle A = \angle P$, $AB = 4$ cm, $AC = 6$ cm, $PQ = 6$ cm and $PR = 9$ cm. Prove that the triangles are similar.

Q120. $\triangle ABC \sim \triangle PQR$ with $AB = 6$ cm, $BC = 9$ cm and $PQ = 4$ cm. Find QR.

Q121. In $\triangle ABC$, $DE \parallel BC$ with D on AB and E on AC. Prove that $\triangle ADE \sim \triangle ABC$.

Q122. $\triangle ABC \sim \triangle DEF$ with $AB = 4$, $DE = 6$ and the perimeter of $\triangle ABC$ equal to 18. The perimeter of $\triangle DEF$ is

Q123. Two similar triangles have sides in the ratio 2 : 5. The ratio of their perimeters is

Q124. A photograph 6 cm by 8 cm is enlarged so that its shorter side becomes 15 cm. Its longer side becomes

Q125. $ABCD \sim PQRS$ with $AB = 8$, $PQ = 12$ and $RS = 9$. Then $CD$ is

Q126. Which condition makes two polygons similar all on its own?

Q127. A model of a tower is built to a scale of 1 : 200. If the model is 8 cm tall, the tower is

Q128. If $\triangle ABC \sim \triangle PQR$ and $\triangle PQR \sim \triangle XYZ$, then $\triangle ABC$ and $\triangle XYZ$ are

Q129. Two similar figures have a scale factor of $\frac{3}{4}$. A side of 20 in the first corresponds to

Q130. Assertion (A): Congruent figures and similar figures are two entirely separate ideas with nothing in common. Reason (R): Congruence is similarity in which the scale factor equals 1.

Q131. The order of the letters in $\triangle ABC \sim \triangle PQR$ matters because

Q132. Assertion (A): Two figures that are similar to the same third figure are similar to each other. Reason (R): Equal angles and constant side ratios carry through from one similarity to the next.

Q133. A photograph measuring 6 cm by 8 cm is enlarged so that its shorter side becomes 15 cm. Find its new longer side.

Q134. $\triangle ABC \sim \triangle DEF$ with $AB = 4$ cm and $DE = 6$ cm. If the perimeter of $\triangle ABC$ is 18 cm, find the perimeter of $\triangle DEF$.

Q135. A model of a building is made to a scale of 1 : 200. The model is 8 cm tall. Find the height of the building in metres.

Q136. $ABCD \sim PQRS$. The sides of $ABCD$ are $AB = 8$ cm, $BC = 12$ cm, $CD = 10$ cm and $DA = 6$ cm, and $PQ = 12$ cm. (a) Find the scale factor, stating which figure you are taking first. (b) Find $QR$, $RS$ and $SP$. (c) Find the perimeters of both quadrilaterals and the ratio between them, and comment on what you notice.

Q137. A surveyor uses a scale drawing of a rectangular field. On the drawing the field measures 7 cm by 12 cm, and the drawing is made to a scale of 1 : 2500. (a) Explain why the drawing and the field are similar figures. (b) Find the actual dimensions of the field in metres. (c) The surveyor also draws a square plot of side 4 cm on the same drawing. Explain whether the square plot on the drawing is similar to the rectangular field on the drawing, justifying your answer.

Q138. In $\triangle ABC$, $DE \parallel BC$ with $AD = x$, $DB = x - 2$, $AE = x + 2$ and $EC = x - 1$. The value of x is

Q139. In $\triangle ABC$, $DE \parallel BC$ and $\frac{AD}{DB} = \frac{3}{4}$. Then $\frac{AD}{AB}$ is

Q140. If $DE \parallel BC$ and $\frac{AD}{AB} = \frac25$, then $\frac{AE}{EC}$ is

Q141. The perpendiculars DM and EN drawn in the proof

Q142. A line cuts $AB$ at $D$ and $AC$ at $E$ with $AD = 6$, $AB = 15$, $AE = 8$ and $AC = 20$. Then $DE$

Q143. In $\triangle ABC$, $DE \parallel BC$ with $AD = 2.4$ cm, $AE = 3.2$ cm and $EC = 4.8$ cm. Then DB is

Q144. Why can the BPT not be proved using similar triangles?

Q145. Adding 1 to both sides of $\frac{AD}{DB} = \frac{AE}{EC}$ gives

Q146. Assertion (A): In the BPT proof, the step $ar(BDE) = ar(DEC)$ needs a stated reason. Reason (R): Triangles on the same base and between the same parallels are equal in area.

Q147. If $D$ and $E$ divide $AB$ and $AC$ in the ratio $k : 1$ for the same k, then DE is

Q148. In $\triangle ABC$, $DE \parallel BC$ and $\frac{AD}{DB} = \frac34$. Find $\frac{AD}{AB}$.

Q149. In $\triangle ABC$, $DE \parallel BC$ with $AD = x$, $DB = x - 2$, $AE = x + 2$ and $EC = x - 1$. Find x.

Q150. Why does the proof of the BPT not use similar triangles?

Q151. Prove the Basic Proportionality Theorem: if a line runs parallel to one side of a triangle and meets the other two sides at two separate points, the other two sides are divided in the same ratio. Set out your answer under the headings Given, To prove, Construction and Proof.

Q152. (a) State the converse of the Basic Proportionality Theorem. (b) In $\triangle ABC$, $D$ lies on $AB$ and $E$ on $AC$ with $AD = 6$ cm, $AB = 15$ cm, $AE = 8$ cm and $AC = 20$ cm. Determine whether $DE \parallel BC$. (c) In the same triangle, a point $F$ on $AC$ has $AF = 9$ cm. Determine whether $DF \parallel BC$, and explain the difference.

Q153. (a) Write the Basic Proportionality Theorem in both its part : part and part : whole forms. (b) Show how the second follows from the first. (c) In $\triangle ABC$ with $DE \parallel BC$, $\frac{AD}{DB} = \frac{2}{5}$. Find $\frac{AD}{AB}$ and $\frac{EC}{AC}$.

Q154. A surveyor needs the width of a river. She marks a point $A$ on one bank, and points $B$ and $C$ on the far bank so that $BC$ is the width to be found. She then marks $D$ on $AB$ and $E$ on $AC$ with $DE \parallel BC$, and measures $AD = 12$ m, $DB = 18$ m and $DE = 20$ m. (a) Explain which theorem lets her use these measurements. (b) Find $\frac{AD}{AB}$. (c) Given that $\frac{DE}{BC} = \frac{AD}{AB}$ for parallel lines, find the width $BC$.

Q155. $\triangle ABC \sim \triangle DEF$ with $AB = 5$, $BC = 8$, $CA = 10$ and $DE = 7.5$. Then EF is

Q156. In $\triangle ABC$, $DE \parallel BC$. Then $\triangle ADE$ and $\triangle ABC$ are similar by

Q157. A vertical stick 1 m long casts a shadow 0.8 m long while a tower casts a shadow 24 m long. The tower's height is

Q158. Two triangles have sides $2, 3, 4$ and $4, 6, 8$. They are similar with scale factor

Q159. If $\triangle ABC \sim \triangle PQR$ and their perimeters are 24 and 36, the ratio of $AB$ to $PQ$ is

Q160. In a right triangle, if the hypotenuse and one side are proportional to those of another right triangle, the triangles are similar by

Q161. Which is NOT sufficient to prove two triangles similar?

Q162. $\triangle ABC \sim \triangle PQR$ with a scale factor of 1. The triangles are

Q163. Assertion (A): The similarity criteria cannot be proved using similar triangles. Reason (R): They are themselves proved from the Basic Proportionality Theorem.

Q164. Two triangles have $\angle A = \angle D = 90°$ and $\angle B = \angle E$. They are similar by

Q165. A vertical stick 1 m long casts a shadow 0.8 m long. At the same time a tower casts a shadow 24 m long. Find the height of the tower.

Q166. If $\angle A = \angle Q$, $\angle B = \angle R$ and $\angle C = \angle P$, write the correct similarity statement and explain why the order matters.

Q167. Explain why triangles need only one of the two similarity conditions, while polygons in general need both.

Q168. (a) State the AA, SSS and SAS similarity criteria for triangles. (b) For each of the following, say which criterion applies, or that none does: (i) $\angle A = \angle P$, $\angle B = \angle Q$; (ii) sides 4, 6, 8 and 6, 9, 12; (iii) $\angle B = \angle Q$ with $\frac{AB}{PQ} = \frac{AC}{PR}$. (c) Explain your answer to (iii) in particular.

Q169. In $\triangle ABC$, a line $DE$ is drawn with D on AB and E on AC such that $DE \parallel BC$. (a) Prove that $\triangle ADE \sim \triangle ABC$, naming the criterion used. (b) Given $AD = 3$ cm, $DB = 5$ cm and $BC = 12$ cm, find DE. (c) State the relationship between $\frac{DE}{BC}$ and $\frac{AD}{AB}$, and explain why it holds.

Q170. (a) Explain why the AAA criterion and the AA criterion amount to the same thing. (b) Two isosceles triangles have equal vertical angles. Prove that they are similar. (c) Two isosceles triangles have equal base angles. Are they necessarily similar? Justify.

Q171. Ravi wants to find the height of a lamp post. He stands a metre rule vertically in the ground and measures its shadow as 0.4 m. At the same moment the lamp post's shadow is 3.2 m long. (a) Explain why the two triangles formed are similar, naming the criterion. (b) Find the height of the lamp post. (c) Ravi's friend suggests measuring the shadows at different times of day. Explain why that would make the method fail.