Congruent & Similar Triangles: The Four Conditions — Worksheet & Memo

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Grade 9 · Geometry of 2D Shapes

Congruent & Similar Triangles: The Four Conditions

MARKING GUIDELINE

Marks

Duration

50 minutes

Questions

Name:

Class:

Date:

Mark

/ 31

Instructions and Information

Answer ALL the questions in this question paper.
Answer QUESTION 1 by circling the letter (A–D) in the answer grid at the end of that section.
Show ALL calculations clearly.
Show all units where applicable.
Number the answers correctly according to the numbering system used in this question paper.
A non-programmable calculator may be used, unless stated otherwise.
Write neatly and legibly.

Question 1

[10 MARKS]

Four options are given as possible answers to the following questions. Choose the correct answer and circle the letter (A–D) in the grid at the end of this section. If you want to change your choice, put a cross through the wrong letter and circle your new choice.

1.1
Which one of the following is NOT a condition for congruency?
(1)
A)SSS
B)AAA✓
C)SAS
D)RHS
Answer: B — AAA fixes the SHAPE but not the size, so it proves similarity only.
- A — three equal sides fix the triangle
- C — two sides and the included angle fix it
- D — right angle, hypotenuse and one side fix it
1.2
△ABC ≡ △DEF means the two triangles are …
(1)
A)the same shape only
B)the same shape AND the same size✓
C)similar only
D)equal in area only
Answer: B — The symbol ≡ means congruent.
- A — that is similarity
- C — similar allows different sizes
- D — equal areas do not force the same shape
1.3
If △ABC ||| △PQR with a scale factor of 3, then the AREA of △PQR is … times that of △ABC.
(1)
A)3
B)9✓
C)6
D)27
Answer: B — Area scales by the SQUARE of the length factor.
- A — that is the LENGTH factor
- C — that is 3 doubled
- D — that is the VOLUME factor
1.4
In △ABC ≡ △PQR, the angle equal to B̂ is …
(1)
A)P̂
B)Q̂✓
C)R̂
D)Â
Answer: B — The letters correspond in order: A↔P, B↔Q, C↔R.
- A — P matches A
- C — R matches C
- D — A is in the first triangle
1.5
Two triangles have equal hypotenuses and one other pair of equal sides, and both are right-angled. They are congruent by …
(1)
A)SAS
B)RHS✓
C)SSS
D)AAA
Answer: B — Right angle, Hypotenuse, Side.
- A — the right angle is not INCLUDED between the two known sides
- C — only two sides are known
- D — no angles other than the right angle are given
1.6
Equiangular triangles are always …
(1)
A)congruent
B)similar✓
C)the same size
D)right-angled
Answer: B — Equal angles fix the shape but not the size.
- A — congruency needs equal sides as well
- C — the sizes may differ
- D — nothing forces a right angle
1.7
△ABC ||| △DEF with AB = 8, DE = 4 and BC = 10. Then EF = …
(1)
A)20
B)5✓
C)10
D)2,5
Answer: B — The factor from ABC to DEF is 1/2, so EF = 10 ÷ 2.
- A — multiplied instead of dividing
- C — similar triangles have equal sides only if the factor is 1
- D — divided by 4 instead of 2
1.8
In similar triangles, corresponding SIDES are …
(1)
A)equal
B)in the same ratio✓
C)supplementary
D)perpendicular
Answer: B — That constant ratio is the scale factor.
- A — equal sides would make them congruent
- C — sides are not angles
- D — sides need not meet at right angles
1.9
Which pair of triangles must be congruent?
(1)
A)both with angles 30°, 60°, 90°
B)both with sides 3, 4, 5✓
C)both with the same area
D)both right-angled
Answer: B — Three equal sides fix the triangle completely (SSS).
- A — equal angles fix the shape, not the size
- C — different shapes can have the same area
- D — a right angle alone fixes nothing
1.10
A photograph is enlarged by a factor of 4. Its PERIMETER is multiplied by …
(1)
A)16
B)4✓
C)8
D)2
Answer: B — Perimeter is a LENGTH, so it scales by the length factor.
- A — that is the AREA factor
- C — 4 doubled is not the scale factor
- D — that is the square root of the factor

Answer grid — marking guideline

1.1	A	B	C	D
1.2	A	B	C	D
1.3	A	B	C	D
1.4	A	B	C	D
1.5	A	B	C	D
1.6	A	B	C	D
1.7	A	B	C	D
1.8	A	B	C	D
1.9	A	B	C	D
1.10	A	B	C	D

Question 2

[11 MARKS]

Study the two triangles in the diagram below. The diagrams are not drawn to scale.

2.1
Write down the condition of congruency that proves △ABC ≡ △DEF, and explain what each letter of that condition stands for.
(4)
SSS (2)
All THREE pairs of corresponding sides are equal (1)
and equal sides force the angles between them to be equal too (1)
2.2
Explain why the ORDER of the letters matters when you write △ABC ≡ △DEF.
(3)
The order shows which vertex corresponds to which (1)
A matches D, B matches E and C matches F (1)
so writing △ABC ≡ △EDF would claim the wrong pairs are equal (1)
2.3
A learner claims that AAA proves two triangles congruent. Explain why the learner is wrong, and state what AAA DOES prove.
(4)
Equal angles fix only the SHAPE of a triangle, not its size (2)
An enlargement has the same three angles but longer sides (1)
AAA proves the triangles are similar, not congruent (1)

Question 3

[10 MARKS]

△PQR and △XYZ are similar. PQ = 8 cm, QR = 6 cm and XY = 20 cm. △PQR is right-angled at Q.

3.1
Calculate the scale factor from △PQR to △XYZ.
(3)
k = 208 (2)
= 2,5 (1)
3.2
Hence calculate the length of YZ.
(3)
YZ = QR × k = 6 × 2,5 (2)
= 15 cm (1)
3.3
Calculate the length of PR, and hence the length of XZ.
(4)
PR² = 8² + 6² = 100 (Pythagoras) (1)
PR = 10 cm (1)
XZ = 10 × 2,5 (1)
= 25 cm (1)

TOTAL: 31 marks

This question paper consists of 3 questions.