Class 10 Maths · Original EZT practice
Triangles (Class 10 Maths Chapter 6): similarity, criteria and the proportionality theorem
Two triangles can be different sizes and still be the same shape - that idea, similarity, is what this chapter is built on. Once you can spot it, shadow problems, map scales and prove-these-are-similar questions all turn into the same three checks. Below: the ideas in plain words, then original EZT practice with worked answers. (Note: this is original practice, not a textbook exercise key.)
10 original worked examples, 10 fresh self-check questions and 4 mixed questions, with explained answers. These are not copied textbook exercises, an official chapter index or a full answer key. No affiliation with NCERT is claimed.
Official sources for topic and edition context, checked 18 October 2026:
1. Similar figures
Two figures are similar when they have the same shape, not necessarily the same size: matching angles equal and matching sides in the same ratio. All circles are similar to one another; so are all squares and all equilateral triangles.
2. Similarity of triangles
For triangles, similarity means corresponding angles equal and corresponding sides in one common ratio. You rarely need to check all six; the AA, SSS and SAS criteria do the work.
3. Basic Proportionality Theorem (Thales)
A line drawn parallel to one side of a triangle divides the other two sides in the same ratio: in triangle ABC with DE parallel to BC, AD/DB = AE/EC. The converse also holds.
4. AA criterion
If two angles of one triangle equal two angles of another, the triangles are similar. The third angle is then forced to match because the angles of a triangle sum to 180 degrees.
5. SSS and SAS criteria
SSS: if all three side ratios match, the triangles are similar. SAS: if two side ratios match and the included angles are equal, they are similar.
6. Using similarity: shadows and scales
At the same time of day, objects and their shadows form similar triangles, so height/shadow is one fixed ratio. Perimeters of similar triangles scale in the same ratio as corresponding sides.
Choose, solve, check, then try a fresh question.
Before calculating, read the method prompt and name the condition that makes it useful. Write your own attempt. Open the steps only when you need them, and use the check to catch a wrong sign, unit or interpretation. Finish with the fresh question before reading its answer.
Decide whether two rectangles are similar
Choose the method: Check both the angle match and the side ratio; equal angles alone are not enough.
EZT practice question
Are any two rectangles always similar?
Show the worked solution
- Every rectangle has four 90-degree angles, so the angle condition is always met.
- The side ratios need not match: a 2 x 1 rectangle has ratio 2:1, while a square-cornered 4 x 4 rectangle has ratio 1:1.
Answer: No. Rectangles always match on angles, but they are similar only when their corresponding side ratios also match.
Check: A square is a rectangle, but a long thin rectangle is not similar to a square.
Watch for this: Do not stop after checking the angles; similarity needs both conditions.
Now try it yourself
Are a 3 x 6 rectangle and a 5 x 10 rectangle similar?
Check your answer
Yes.
Their side ratios are both 1:2.
Use the Basic Proportionality Theorem
Choose the method: Name the parallel line first, then match the two side segments in the same order.
EZT practice question
In triangle ABC, DE is parallel to BC. AD = 3 cm, DB = 6 cm and AE = 4 cm. Find EC.
Show the worked solution
- DE parallel to BC gives AD/DB = AE/EC.
- Substitute: 3/6 = 4/EC.
Answer: EC = 8 cm.
Check: 3/6 = 1/2 and 4/8 = 1/2, so the ratios agree.
Watch for this: Keep AD over DB and AE over EC; do not mix segments from different sides.
Now try it yourself
If AD = 5, DB = 10 and AE = 6 with DE parallel to BC, find EC.
Check your answer
12.
5/10 = 6/EC gives EC = 12.
Match corresponding sides in a parallel-line question
Choose the method: Write the ratio using one full side's two segments, then copy that order on the other side.
EZT practice question
In triangle PQR, X lies on PQ and Y lies on PR. XY is parallel to QR. PX = 2 cm, XQ = 5 cm and PY = 3 cm. Find YR.
Show the worked solution
- XY parallel to QR gives PX/XQ = PY/YR.
- Substitute: 2/5 = 3/YR.
Answer: YR = 7.5 cm.
Check: 2/5 = 0.4 and 3/7.5 = 0.4.
Watch for this: Do not pair PX with PY on one side unless the theorem's order is preserved.
Now try it yourself
If PX = 4, XQ = 6 and PY = 8, find YR.
Check your answer
12.
4/6 = 8/YR, so YR = 12.
Prove similarity with AA
Choose the method: Match two angle pairs; the third pair follows automatically.
EZT practice question
Triangle ABC has angles 50 degrees and 70 degrees. Triangle DEF has angles 50 degrees and 70 degrees. Are they similar, and what is the third angle?
Show the worked solution
- Two angle pairs match, so AA similarity applies.
- The third angle is 180 - 50 - 70 = 60 degrees in each triangle.
Answer: Yes, by AA. The third angle is 60 degrees in both triangles.
Check: 50 + 70 + 60 = 180 in each triangle.
Watch for this: You do not need all three angle pairs to use AA; two matching pairs are enough.
Now try it yourself
Two triangles have angles 35 degrees and 85 degrees. What is the third angle, and are they similar?
Check your answer
60 degrees; yes, by AA.
The third angle is forced to be 60 degrees in both.
Reject a false AA match
Choose the method: Compare the actual angle values, not just the fact that each triangle has three angles.
EZT practice question
Two triangles have angles 40 degrees and 65 degrees, and 40 degrees and 75 degrees. Are they similar?
Show the worked solution
- One angle matches: 40 degrees.
- The second listed angles differ: 65 degrees is not 75 degrees.
Answer: No. AA fails, and the triangles are not similar on the given information.
Check: Their third angles are 75 degrees and 65 degrees, so the angle sets are different.
Watch for this: Do not assume two triangles are similar because one angle matches.
Now try it yourself
Triangle angles are 30 and 70 degrees, and 30 and 80 degrees. Are the triangles similar?
Check your answer
No.
Only one angle pair matches, so AA does not apply.
Use SSS similarity
Choose the method: Put corresponding sides in the same order, then compare all three ratios.
EZT practice question
Are triangles with sides 6, 8, 10 and 9, 12, 15 similar?
Show the worked solution
- Compare corresponding sides: 6/9, 8/12 and 10/15.
- Each ratio simplifies to 2/3.
Answer: Yes. All three side ratios are 2/3, so the triangles are similar by SSS.
Check: The second triangle is 1.5 times the first in every side.
Watch for this: Match shortest with shortest and longest with longest before calculating ratios.
Now try it yourself
Are sides 4, 5, 6 and 8, 10, 12 similar?
Check your answer
Yes.
Every side in the second triangle is twice the corresponding side in the first.
Use SAS similarity
Choose the method: Check two side ratios and make sure the equal angle is between those sides.
EZT practice question
In triangles ABC and DEF, AB = 4, AC = 6, DE = 6, DF = 9, and angle A = angle D = 40 degrees. Are the triangles similar?
Show the worked solution
- AB/DE = 4/6 = 2/3 and AC/DF = 6/9 = 2/3.
- The equal 40-degree angles are the included angles between those side pairs.
Answer: Yes. Two side ratios match and the included angles are equal, so SAS similarity holds.
Check: The same 2/3 ratio appears on both sides around the equal angle.
Watch for this: SAS does not apply if the equal angle is not between the compared sides.
Now try it yourself
AB = 5, AC = 8, DE = 10, DF = 16 and angle A = angle D. Are the triangles similar?
Check your answer
Yes, by SAS.
5/10 = 8/16 = 1/2, and the equal angle is included.
Read a scale factor from matching sides
Choose the method: Reduce each corresponding-side ratio and check that it is the same every time.
EZT practice question
Are triangles with sides 5, 12, 13 and 2.5, 6, 6.5 similar?
Show the worked solution
- Compare 5 with 2.5, 12 with 6 and 13 with 6.5.
- Each smaller side is exactly half the matching larger side.
Answer: Yes. The matching side ratios are all 1:2, so the triangles are similar by SSS.
Check: 2.5 x 2 = 5, 6 x 2 = 12 and 6.5 x 2 = 13.
Watch for this: Decimal sides are fine; the common ratio, not whole numbers, decides similarity.
Now try it yourself
Sides 7, 9, 11 correspond to 3.5, 4.5, 5.5. Similar?
Check your answer
Yes.
Each smaller side is half the larger side.
Use similar triangles for a shadow problem
Choose the method: Keep height over shadow for both objects and solve the resulting proportion.
EZT practice question
A 6 m pole casts a 4 m shadow. At the same moment a tree casts a 14 m shadow. How tall is the tree?
Show the worked solution
- At the same time of day, the triangles are similar, so pole height/pole shadow = tree height/tree shadow.
- 6/4 = h/14.
Answer: h = 21 m.
Check: 6/4 = 1.5 and 21/14 = 1.5.
Watch for this: Do not invert one ratio and not the other; use height over shadow for both.
Now try it yourself
A 2 m stick casts a 1.5 m shadow while a tower casts a 12 m shadow. Find the tower height.
Check your answer
16 m.
2/1.5 = h/12 gives h = 16.
Scale a perimeter with the side ratio
Choose the method: Use the same ratio for the perimeter as for each corresponding side.
EZT practice question
Two similar triangles have corresponding sides in the ratio 3:5. The smaller triangle's perimeter is 21 cm. Find the larger triangle's perimeter.
Show the worked solution
- The side ratio from smaller to larger is 3:5.
- Multiply the smaller perimeter by 5/3.
Answer: The larger perimeter is 35 cm.
Check: 21 x 5/3 = 35, and 21:35 simplifies to 3:5.
Watch for this: Do not square the ratio; perimeters scale like sides, not areas.
Now try it yourself
The side ratio is 2:7 and the smaller perimeter is 18 cm. Find the larger perimeter.
Check your answer
63 cm.
18 x 7/2 = 63.
Try a mixed set
Choose a method or explanation before opening each answer.
1. Are all equilateral triangles similar?
Show answer and reason
Yes.
Every angle is 60 degrees, and corresponding sides are in one common ratio.
2. Does the Basic Proportionality Theorem need the two triangles to be similar first?
Show answer and reason
No.
The parallel line itself creates the equal side-segment ratio; you do not assume the conclusion.
3. Which similarity criterion is usually quickest when angle pairs are given?
Show answer and reason
AA.
Two matching angle pairs force the third to match.
4. Do perimeters of similar triangles scale like areas?
Show answer and reason
No.
Perimeters use the side ratio; areas use the square of that ratio.
Try it yourself
Work these on paper before checking any notes. Answers are deliberately not shown here.
- AD = 2, DB = 4, AE = 3 and DE is parallel to BC. Find EC.
- Are triangles with sides 3, 4, 5 and 6, 8, 10 similar? Name the criterion.
- Angles 30 degrees and 80 degrees appear in both triangles. Are they similar? Find the third angle.
- A 1.5 m stick casts a 2 m shadow; a building casts a 30 m shadow. How tall is the building?
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Questions about this practice page
- What makes two triangles similar?
- Corresponding angles are equal and corresponding sides are in the same ratio. In practice, the AA, SSS or SAS criteria usually prove it.
- What is the Basic Proportionality Theorem?
- A line parallel to one side of a triangle divides the other two sides in the same ratio: AD/DB = AE/EC. Its converse is also true.
- What is the quickest similarity criterion?
- AA. If two angles of one triangle match two angles of another, the triangles are similar.
- Do perimeters of similar triangles scale?
- Yes. Perimeters are in the same ratio as corresponding sides.
- Are these the NCERT exercise questions for Triangles?
- No. EZT pages are original practice on this chapter - use them to prepare for the textbook exercises, not as a substitute for your book.