Congruent triangles Quiz (Grade 7)

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Score: 0 / 10

Congruent triangles — Notes & Worked Examples Show / Hide

Explanation:

Two triangles are congruent if they are identical in both shape and size. You must prove congruence by stating three matching pairs of sides/angles with geometric reasons, then concluding with one of the four valid conditions: SSS (3 sides), SAS (2 sides and the included angle), ASA / AAS (2 angles and a corresponding side), or RHS (Right-angle, Hypotenuse, and another Side).

Note that AAA proves similarity only (not congruence), and ASS is not sufficient unless the angle is 90° (RHS). Look out for shared/common sides, vertically opposite angles, alternate angles, and radii of a circle.

Example 1: ABCD is a parallelogram. Prove that ΔABC is congruent to ΔCDA.

Solution:

• AB = CD (opposite sides of parallelogram)

• BC = DA (opposite sides of parallelogram)

• AC is a common side to both triangles

Answer: ΔABC ≅ ΔCDA by SSS

Example 2: AB is a chord of a circle with centre O, and M is the midpoint of AB. Prove ΔOAM ≅ ΔOBM.

Solution:

• OA = OB (both are radii of the circle)

• AM = BM (M is the midpoint of chord AB)

• OM is a common side to both triangles

Answer: ΔOAM ≅ ΔOBM by SSS

One Page Guide (PDF) Congruent triangles Worksheets

Question 1 AQA Jun 2024 1F Q15

[1 mark]

These two triangles are congruent.

Write down the value of x.

x = cm

Worked Solution & Answer

Corresponding sides of congruent triangles are equal, so x = 6 cm. B1

Answer: x = 6 cm

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Question 2 AQA Jun 2024 2H Q12

[1 mark]

Two triangles each have sides 9 cm, 12 cm, 17 cm. Circle the reason why these triangles are congruent: ASA, RHS, SAS, SSS. [1 mark]

Circle the reason why these triangles are congruent.
ASA RHS SAS SSS

Worked Solution & Answer

All three corresponding sides (9 cm, 12 cm, 17 cm) are equal → SSS. B1

Answer: SSS

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Question 3

[1 mark]

The diagram shows four triangles.

Two of these triangles are congruent.
Write down the letters of these two triangles.

Worked Solution & Answer

In Triangle A, angles are 50°, 45°, 85° with 12 cm opposite 85°
In Triangle D, angles are 85°, 45°, 50° with 12 cm opposite 85° (AAS/ASA). B1

Answer: A, D

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Question 4 AQA Jun 2025 3F Q15

[3 marks]

A and B are triangles.

For each statement, tick the correct box to describe A and B.

Worked Solution & Answer

1. If all 3 side lengths are identical (SSS condition), the triangles are always congruent. M1
2. If angles are identical (AAA condition), triangles are similar, but only congruent if scale factor is 1 (sometimes). M1
3. An enlargement by scale factor 2 doubles all side lengths, so they are never congruent. A1

Answer: Sides same: Always; Angles same: Sometimes; Enlargement sf 2: Never

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Question 5

[3 marks]

The diagram shows a triangle ADE.

AE = DE
AB : BC : CD = 1 : 2 : 1
Prove that triangle ACE is congruent to triangle DBE.

Worked Solution & Answer

AE = DE (given), ∠EAC = ∠EDB (base angles of isosceles △ADE), AC = DB (since AB = CD and AC = AB + BC, DB = CD + BC) M2
⇒ congruent by SAS C1

Answer: AE = DE (given); angle EAC = angle EDB (base angles of an isosceles triangle are equal); AC = AB + BC = 1 + 2 = 3 parts and DB = BC + CD = 2 + 1 = 3 parts so AC = DB; therefore triangle ACE is congruent to triangle DBE by SAS.

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Question 6 AQA Nov 2023 3H Q19

[4 marks]

AC is a diameter of a circle, centre E.
E is the midpoint of BD.

Prove that triangle ABE is congruent to triangle CDE.

Worked Solution & Answer

AE = CE because both are radii of the circle M1
BE = DE because E is the midpoint of BD M1
Angle AEB = Angle CED because vertically opposite angles are equal M1
Hence triangle ABE is congruent to triangle CDE by SAS. C1

Answer: AE = CE (both radii of the circle) and BE = DE (E is the midpoint of BD)
Angle AEB = Angle CED (vertically opposite angles)
Therefore triangle ABE is congruent to triangle CDE by SAS

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Question 7

[4 marks]

ABCD is a quadrilateral.

AB = CD.
Angle ABC = angle BCD.
Prove that AC = BD.

Worked Solution & Answer

Consider triangles ABC and DCB: 1) AB = CD (given) M1
2) Angle ABC = Angle DCB (given) M1
3) BC = CB (common side) M1
By SAS congruence, triangle ABC is congruent to triangle DCB
Therefore corresponding sides AC = BD. C1

Answer: In triangles ABC and DCB: AB = DC (given), ∠ ABC = ∠ DCB (given), and BC is common. Therefore △ ABC ≡ △ DCB (SAS), so AC = BD.

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Question 8 OCR Nov 2017 5H Q10

[4 marks]

ABCD is a quadrilateral.
AD = AB and CD = CB.

Prove that angle ADC is equal to angle ABC.

Worked Solution & Answer

Draw diagonal AC M1
In triangles ADC and ABC:
1. AD = AB (given) M1
2. CD = CB (given)
3. AC is common to both triangles M1
Therefore triangle ADC is congruent to triangle ABC (SSS)
Hence corresponding angles ADC and ABC are equal. C1

Answer: In triangles ADC and ABC: AD = AB (given), CD = CB (given), AC is common, so triangle ADC is congruent to triangle ABC by SSS. Therefore angle ADC = angle ABC.

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Question 9 OCR Nov 2019 4H Q11

[4 marks]

(a) Are these two triangles definitely congruent?
Give a reason.
(b) Prove that these two triangles are congruent.

(a) Are these two triangles definitely congruent?
Give a reason. [1 mark]

(b) Prove that these two triangles are congruent. [3 marks]

Worked Solution & Answer

(a)

No, having the same angles only proves similarity
the side lengths are unknown and could be different. C1

Answer (a): No, because none of the side lengths are known (the triangles are similar but could be different sizes).

(b)

1. GH = JK = 9 cm (side) M1
2. Angle HGI = angle KJL = 36° (included angle) M1
3. GI = JL = 7 cm (side)
Therefore triangles GHI and JKL are congruent by SAS. C1

Answer (b): GH = JK = 9 cm, GI = JL = 7 cm, and the included angle HGI = angle KJL = 36°. Therefore the triangles are congruent by SAS.

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Question 10

[5 marks]

ABCD is a parallelogram, ABP and QDC are straight lines, angle ADP = angle CBQ = 90°. (a) Prove triangle ADP is congruent to triangle CBQ (3) (b) Explain why AQ is parallel to PC (2)

ABCD is a parallelogram.
ABP and QDC are straight lines.
Angle ADP = angle CBQ = 90°

(a) Prove that triangle ADP is congruent to triangle CBQ. [3 marks]

(b) Explain why AQ is parallel to PC. [2 marks]

Worked Solution & Answer

(a)

AD = BC (opposite sides of parallelogram), ∠ADP = ∠CBQ = 90° (given), ∠PAD = ∠QCB (opposite angles of parallelogram ABCD) M2
⇒ congruent by ASA C1

Answer (a): AD = BC (opposite sides of parallelogram ABCD are equal); angle ADP = angle CBQ = 90° (given); angle PAD = angle QCB (alternate angles as AB is parallel to DC and AD is parallel to BC, or opposite angles of parallelogram ABCD). Therefore triangle ADP is congruent to triangle CBQ by ASA.

(b)

AP = QC from congruence and AP is parallel to QC, so APCQ is a parallelogram and its opposite sides AQ and PC are parallel M1 C1

Answer (b): Since triangles ADP and CBQ are congruent, AP = QC. Also AP is parallel to QC (since ABP and QDC are parallel lines). A quadrilateral APCQ with a pair of opposite sides equal and parallel is a parallelogram, so AQ is parallel to PC.

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Score: 0 / 10 (0 / 30 marks)

Congruent triangles Worksheets Next: 3D pythagoras / trig. →

Grade 7 Quizzes

19 Topics

7A - Surds	7B - Bounds
7C - Proportion (harder)	7D - Algebraic fractions
7E - The quadratic formula	7F - Factorising harder quadratics
7G - Trigonometric & exp. Graphs	7H - Rearranging (harder)
7J - Functions (harder)	7K - Iteration
7L - Area of any triangle	7M - The sine rule
7N - The cosine rule	7P - Congruent triangles
7Q - 3D pythagoras / trig.	7R - Histograms
7S - Conditional probability	7T - Exact trig. values (harder)
7U - Area of a segment