CHAPTER 6: TRIANGLES
Question Bank
A. Multiple Choice Questions (MCQs)
1.
Two figures having the same shape but possibly different sizes are called:
A. Congruent figures
B. Similar figures
C. Equal figures
D. Symmetric figures
Answer: B
2.
Which statement is always true?
A. All similar figures are congruent
B. All congruent figures are similar
C. Similar figures always have the same size
D. Congruent figures have different shapes
Answer: B
3.
All circles are:
A. Congruent
B. Similar
C. Neither similar nor congruent
D. Equal in size
Answer: B
4.
For two similar polygons, corresponding sides are:
A. Always equal
B. Always perpendicular
C. Proportional
D. Unrelated
Answer: C
5.
If DE is parallel to BC in triangle ABC, then which relation follows from BPT?
A. AD/AB = DB/BC
B. AD/DB = AE/EC
C. AB/AC = DB/EC
D. AD/AE = DB/EC
Answer: B
6.
The converse of the Basic Proportionality Theorem is used when:
A. Two angles are equal
B. Two sides are equal
C. Two sides are divided in the same ratio
D. All angles are right angles
Answer: C
7.
If two angles of one triangle are respectively equal to two angles of another triangle, the triangles are:
A. Congruent
B. Similar
C. Isosceles
D. Equilateral
Answer: B
8.
Which is a valid similarity criterion?
A. AAA
B. SAS
C. SSS
D. All of these
Answer: D
9.
If AB/DE = BC/EF = CA/FD, then the triangles ABC and DEF are similar by:
A. AA
B. SAS
C. SSS
D. BPT
Answer: C
10.
In SAS similarity, the equal angle must be:
A. Any angle of either triangle
B. An exterior angle
C. The angle between the two proportional sides
D. A right angle only
Answer: C
11.
If triangle ABC ~ triangle PQR, then the side corresponding to AB is:
A. QR
B. PR
C. PQ
D. RP only
Answer: C
12.
The symbol used to denote similarity is:
A. =
B. ~
C. ≅
D. ∥
Answer: B
13.
If corresponding sides of two triangles are proportional, their corresponding angles are:
A. Unequal
B. Supplementary
C. Equal
D. Always right angles
Answer: C
14.
A line joining the midpoints of two sides of a triangle is:
A. Perpendicular to the third side
B. Equal to the third side
C. Parallel to the third side
D. Twice the third side
Answer: C
15.
Which theorem directly connects a parallel line inside a triangle with proportional division of its sides?
A. Pythagoras theorem
B. Basic Proportionality Theorem
C. Angle Sum Theorem
D. SSS criterion
Answer: B
B. Fill in the Blanks
1.
Two figures having the same shape but not necessarily the same size are called __________ figures.
Answer: similar
2.
All congruent figures are __________, but similar figures need not be congruent.
Answer: similar
3.
The ratio between corresponding sides of similar polygons is called the __________ factor.
Answer: scale
4.
Two triangles with equal corresponding angles are called __________ triangles.
Answer: equiangular
5.
BPT is also known as the __________ Theorem.
Answer: Thales
6.
If DE || BC in triangle ABC, then AD/DB = __________.
Answer: AE/EC
7.
The converse of BPT states that equal division ratios imply that the line is __________ to the third side.
Answer: parallel
8.
The symbol ~ means __________.
Answer: similar to
9.
The similarity criterion based on three proportional sides is called __________.
Answer: SSS
10.
The similarity criterion based on two equal angles is called __________.
Answer: AA
11.
The similarity criterion involving two proportional sides and the included equal angle is called __________.
Answer: SAS
12.
The angles of a triangle have a total measure of __________.
Answer: 180°
13.
A line through the midpoint of one side of a triangle and parallel to another side __________ the third side.
Answer: bisects
14.
The line joining the midpoints of two sides of a triangle is __________ to the third side.
Answer: parallel
15.
For similar triangles, corresponding sides are in the same __________.
Answer: ratio
C. True or False
1.
All similar figures are congruent.
Answer: False
2.
All congruent figures are similar.
Answer: True
3.
All circles are similar.
Answer: True
4.
All squares are similar.
Answer: True
5.
Equal corresponding angles alone are not useful for proving triangle similarity.
Answer: False
6.
Two triangles having two pairs of equal corresponding angles are similar.
Answer: True
7.
SSS similarity requires all three corresponding sides to be proportional.
Answer: True
8.
In SAS similarity, the equal angle can be any unrelated angle.
Answer: False
9.
If a line divides two sides of a triangle in the same ratio, it is parallel to the third side.
Answer: True
10.
The ratio of corresponding sides of similar triangles is always 1.
Answer: False
11.
Similar triangles always have equal corresponding angles.
Answer: True
12.
Similar triangles must have the same size.
Answer: False
D. Match the Following
| Column A | Column B |
|---|---|
| 1. AA | a. Three proportional sides |
| 2. SSS | b. Two equal corresponding angles |
| 3. SAS | c. Parallel line and proportional division |
| 4. BPT | d. Two proportional sides and included equal angle |
5. ~ | e. Similarity |
Answers:
1-b
2-a
3-d
4-c
5-e
E. Very Short Answer Questions
1.
What are similar figures?
Answer: Figures having the same shape but not necessarily the same size.
2.
Are all congruent figures similar?
Answer: Yes.
3.
Are all similar figures congruent?
Answer: No.
4.
What is the scale factor?
Answer: The common ratio of corresponding sides of similar figures.
5.
State BPT.
Answer: A line parallel to one side of a triangle divides the other two sides in the same ratio.
6.
State the converse of BPT.
Answer: If a line divides two sides of a triangle in the same ratio, it is parallel to the third side.
7.
Name the three main similarity criteria for triangles.
Answer: AA, SSS and SAS.
8.
What does SSS similarity mean?
Answer: Three corresponding sides of two triangles are proportional.
9.
What does SAS similarity mean?
Answer: Two corresponding sides are proportional and the included angles are equal.
10.
Why is AA sufficient for proving triangle similarity?
Answer: Once two corresponding angles are equal, the third angles are also equal because the angle sum of a triangle is 180°.
F. Short Answer Questions
1.
Differentiate between similar and congruent figures.
Answer:
Similar figures have the same shape but may have different sizes. Congruent figures have both the same shape and the same size.
2.
Why are two squares always similar?
Answer: Every square has four equal right angles, and the ratios of corresponding sides of any two squares are equal.
3.
Why are two rectangles not necessarily similar?
Answer: Although their corresponding angles are equal, their corresponding sides may not be in the same ratio.
4.
In triangle ABC, DE || BC. If AD = 4 cm, DB = 6 cm and AE = 5 cm, find EC.
Answer:
AD/DB = AE/EC
4/6 = 5/EC
EC = 7.5 cm
5.
In triangle PQR, S lies on PQ and T lies on PR. If ST || QR, what theorem can be used to relate PS/SQ and PT/TR?
Answer: Basic Proportionality Theorem.
6.
If two triangles have sides in the ratios 2:3, 4:6 and 6:9, are they similar?
Answer: Yes, because all three corresponding side ratios are equal. The SSS criterion applies.
7.
If two angles of triangle ABC are equal to two angles of triangle DEF respectively, what can you conclude?
Answer: Triangle ABC is similar to triangle DEF by the AA criterion.
8.
Why is the order important in writing △ABC ~ △DEF?
Answer: The order shows the correct correspondence: A ↔ D, B ↔ E and C ↔ F.
G. Numerical/Application Questions
1.
In triangle ABC, DE || BC. If AD = 3 cm, DB = 5 cm and AE = 6 cm, find EC.
Answer:
AD/DB = AE/EC
3/5 = 6/EC
EC = 10 cm
2.
In triangle ABC, DE || BC. If AD = 4 cm, AB = 10 cm and AC = 15 cm, find AE.
Answer:
AD/AB = AE/AC
4/10 = AE/15
AE = 6 cm
3.
Two similar triangles have corresponding sides 6 cm and 9 cm. If another side of the smaller triangle is 8 cm, find the corresponding side of the larger triangle.
Answer:
6/9 = 8/x
x = 12 cm
4.
Two triangles have corresponding sides:
Triangle 1: 5 cm, 7 cm, 9 cm
Triangle 2: 10 cm, 14 cm, 18 cm
Are they similar?
Answer: Yes. Each corresponding side of Triangle 2 is twice the corresponding side of Triangle 1.
5.
A pole 3 m high casts a shadow 2 m long. At the same time, a tower casts a shadow 14 m long. Find the height of the tower.
Answer:
3/2 = h/14
h = 21 m
6.
A student 1.2 m tall casts a shadow 0.8 m long. A tree casts a shadow 6 m long at the same time. Find the tree’s height.
Answer:
1.2/0.8 = h/6
h = 9 m
H. Identify the Correct Similarity Criterion
1.
∠A = ∠D and ∠B = ∠E.
Answer: AA
2.
AB/DE = BC/EF = CA/FD.
Answer: SSS
3.
AB/DE = AC/DF and ∠A = ∠D.
Answer: SAS
4.
Two right triangles have proportional hypotenuses and one corresponding side.
Answer: RHS similarity
5.
Three pairs of corresponding angles are equal.
Answer: AAA, equivalently AA for triangles.
I. Assertion–Reason Questions
For each question, choose:
A. Both Assertion and Reason are true, and Reason correctly explains Assertion.
B. Both are true, but Reason does not correctly explain Assertion.
C. Assertion is true, Reason is false.
D. Assertion is false, Reason is true.
1.
Assertion: Two similar triangles have equal corresponding angles.
Reason: Similar triangles have the same shape.
Answer: A
2.
Assertion: Two triangles are similar if their three corresponding sides are proportional.
Reason: This is the SSS similarity criterion.
Answer: A
3.
Assertion: All similar figures are congruent.
Reason: Similar figures can have different sizes.
Answer: D
4.
Assertion: If DE || BC in triangle ABC, then AD/DB = AE/EC.
Reason: A line parallel to one side of a triangle divides the other two sides in the same ratio.
Answer: A
5.
Assertion: In SAS similarity, the equal angle must be included between the proportional sides.
Reason: SAS compares two sides and the angle between them.
Answer: A
J. Case-Based Questions
Case 1: Parallel Line in a Triangle
In triangle ABC, a line DE is drawn parallel to BC, meeting AB at D and AC at E.
Questions
- Which theorem relates AD/DB and AE/EC?
- If AD = 6 cm, DB = 4 cm and AE = 9 cm, find EC.
- If AD/DB = AE/EC, what can be concluded about DE?
Answers:
- Basic Proportionality Theorem
- EC = 6 cm
- DE || BC by the converse of BPT
Case 2: Similar Triangles
Two triangles ABC and PQR satisfy:
AB/PQ = BC/QR = CA/RP = 3/5.
Questions
- Are the triangles similar?
- Which criterion is used?
- If AB = 9 cm, find PQ.
- If BC = 12 cm, find QR.
Answers:
- Yes
- SSS similarity criterion
- PQ = 15 cm
- QR = 20 cm
Case 3: Indirect Measurement
A vertical stick and a tower cast shadows at the same time. The stick is 2 m high and its shadow is 1.5 m long. The tower’s shadow is 12 m long.
Questions
- Why can similar triangles be used?
- Which similarity criterion is most naturally used?
- Find the height of the tower.
Answers:
- The vertical objects and their shadows form triangles with equal corresponding angles, including right angles.
- AA similarity
- Tower height = 16 m
K. Find the Error
1.
A student says:
“Since two quadrilaterals have equal corresponding angles, they must be similar.”
Is the statement always correct?
Answer: No. Their corresponding sides must also be proportional.
2.
A student writes:
△ABC ~ △DEF
but then pairs AB with EF.
Is this correspondence correct?
Answer: No. From the similarity statement, AB corresponds to DE.
3.
A student uses SAS because two sides are proportional and an unrelated angle is equal.
Is the reasoning correct?
Answer: No. The equal angle must be the included angle between the proportional sides.
4.
A student says:
“All similar triangles are congruent.”
Correct or incorrect?
Answer: Incorrect. Similar triangles may have different sizes.
L. Proof-Based Questions
1.
In triangle ABC, DE || BC. Prove that:
AD/DB = AE/EC.
Key theorem: Basic Proportionality Theorem.
2.
In triangle ABC, D lies on AB and E lies on AC. If:
AD/DB = AE/EC,
prove that:
DE || BC.
Key theorem: Converse of BPT.
3.
Prove that if two angles of one triangle are respectively equal to two angles of another triangle, the triangles are similar.
Key criterion: AA similarity.
4.
Prove that if the three corresponding sides of two triangles are proportional, the triangles are similar.
Key criterion: SSS similarity.
5.
Prove that if one angle of each of two triangles is equal and the sides including those angles are proportional, the triangles are similar.
Key criterion: SAS similarity.
6.
Prove that the line joining the midpoints of two sides of a triangle is parallel to the third side.
Hint: Use the converse of BPT.
M. Higher-Order Thinking Questions
1.
Can two triangles have equal corresponding angles but different sizes? Explain.
Answer: Yes. Equal corresponding angles establish similarity, but the corresponding sides may have a common ratio other than 1.
2.
Can two triangles have proportional corresponding sides but unequal corresponding angles?
Answer: No. If all three corresponding sides are proportional, SSS similarity gives equal corresponding angles.
3.
Why is it unnecessary to check all six conditions for triangle similarity?
Answer: For triangles, appropriate similarity criteria such as AA, SSS or SAS are sufficient; one condition establishes the remaining corresponding relationships.
4.
A rectangle and a square have all corresponding angles equal. Does this prove they are similar?
Answer: No. Their corresponding sides need not be proportional.
5.
A line divides two sides of a triangle in the same ratio. What additional conclusion can be made?
Answer: The line is parallel to the third side.
6.
Why is similarity useful in measuring the height of a tower?
Answer: Similar triangles allow an unknown height to be calculated from measurable corresponding lengths, such as a known height and shadows.
N. Rapid Revision Questions
1.
What is similarity?
Answer: Same shape, possibly different size.
2.
What is the symbol for similarity?
Answer: ~
3.
What is BPT?
Answer: Parallel line → proportional division of two sides.
4.
What is the converse of BPT?
Answer: Proportional division → parallel line.
5.
AA stands for?
Answer: Angle-Angle.
6.
SSS stands for?
Answer: Side-Side-Side.
7.
SAS stands for?
Answer: Side-Angle-Side.
8.
What must be true for SAS similarity?
Answer: Two corresponding sides must be proportional and their included angles equal.
9.
What happens to corresponding angles of similar triangles?
Answer: They are equal.
10.
What happens to corresponding sides of similar triangles?
Answer: They are proportional.
11.
What is the angle sum of a triangle?
Answer: 180°.
12.
What is the midpoint-line result?
Answer: The line joining the midpoints of two sides is parallel to the third side.
13.
What is RHS similarity?
Answer: For right triangles, proportional hypotenuse and one corresponding side imply similarity.
14.
Can similar triangles be of different sizes?
Answer: Yes.
15.
Can congruent triangles be similar?
Answer: Yes.
MOST IMPORTANT EXAM CHECKLIST
Before answering a question from this chapter, check:
- Is there a parallel line?
→ Think BPT or Converse BPT. - Are two angles equal?
→ Think AA. - Are three pairs of sides proportional?
→ Think SSS. - Are two sides proportional and the included angle equal?
→ Think SAS. - Are the triangles right-angled with proportional hypotenuse and one side?
→ Think RHS similarity. - If triangles are similar:
→ Corresponding angles are equal.
→ Corresponding sides are proportional. - Always write the correspondence correctly.
- For numerical questions, write the ratio first, then substitute values.
- Never assume similarity merely because two figures look alike.
- Remember the central idea:
Similarity = same shape + proportional corresponding sides.