Practice Maths

L25 — Congruence and Similarity

Key Terms

Congruent
Identical in shape and size — all corresponding sides and angles are equal.
Congruence tests
SSS, SAS, AAS, RHS — the minimum conditions that guarantee two triangles are congruent.
Similar
Same shape, different size — corresponding angles equal, sides in the same ratio.
Scale factor k
The ratio of corresponding sides in similar figures; if k > 1 the figure is enlarged, if k < 1 it is reduced.
Similarity tests
AA, SSS (ratio), SAS (ratio + included angle) — the minimum conditions for similar triangles.
Dilation
A transformation that enlarges or reduces a figure from a centre point by a scale factor, producing a similar figure.

Congruence (same shape, same size)

Two figures are congruent if one can be mapped onto the other by a combination of reflections, rotations, and translations (rigid motions). All corresponding sides and angles are equal.

Congruence tests for triangles

TestConditionsNotes
SSS3 sides equalMost direct
SAS2 sides + included angleAngle between the two sides
AAS2 angles + corresponding sideAny side once angles known
RHSRight angle + hypotenuse + sideRight-angled triangles only

Note: SSA is NOT a valid congruence test (ambiguous case).

Similarity (same shape, different size)

Two figures are similar if one can be obtained from the other by a dilation (enlargement/reduction). Corresponding angles are equal; corresponding sides are in the same ratio (the scale factor k).

PropertyRule
Corresponding sidesAll in ratio k : 1
Corresponding anglesAll equal
Perimeter ratiok : 1
Area ratiok² : 1
Volume ratiok³ : 1

Similarity tests for triangles

TestConditions
AA2 pairs of equal angles (third follows)
SSSAll 3 pairs of sides in same ratio
SAS2 pairs of sides in same ratio + included angle equal
A B C 4 5 3 D E F 8 10 6 k = 2 ▲ABC ~ ▲DEF
Similar triangles — sides in ratio 2:1, angles equal
Hot Tip: SSA is not a valid congruence test — two sides and a non-included angle can produce two different triangles. The included angle (SAS) must sit between the two known sides.

Worked Example 1 — Identifying congruent triangles

In triangles PQR and XYZ: PQ = XY = 7 cm, QR = YZ = 5 cm, PR = XZ = 9 cm. Are they congruent?

Step 1: Check conditions. PQ=XY, QR=YZ, PR=XZ — all three pairs of sides are equal.

Step 2: Apply SSS. ∴ ▵PQR ≅ ▵XYZ (SSS).

Step 3: State corresponding angles: ∠P = ∠X, ∠Q = ∠Y, ∠R = ∠Z.

Worked Example 2 — SAS congruence

In triangles ABC and DEF: AB = DE = 6 cm, BC = EF = 8 cm, ∠ABC = ∠DEF = 55°. Are they congruent?

Step 1: AB = DE (given), BC = EF (given), ∠ABC = ∠DEF (included angle, between the two sides).

Step 2: Apply SAS. ∴ ▵ABC ≅ ▵DEF (SAS).

Worked Example 3 — Finding a scale factor

Triangles ABC and PQR are similar. AB = 10 cm, PQ = 15 cm. Find: (a) the scale factor, (b) BC if QR = 12 cm.

(a) Scale factor k = PQ/AB = 15/10 = 1.5.

(b) BC = QR/k = 12/1.5 = 8 cm.

Worked Example 4 — Area and volume ratios

Two similar cylinders have radii 4 cm and 6 cm. (a) Find the scale factor. (b) Find the ratio of their surface areas. (c) Find the ratio of their volumes.

(a) k = 6/4 = 1.5.

(b) Area ratio = k² = 1.5² = 2.25 : 1.

(c) Volume ratio = k³ = 1.5³ = 3.375 : 1.

Worked Example 5 — Proving similarity (AA test)

In the diagram, DE is parallel to BC in triangle ABC, with D on AB and E on AC. Prove ▵ADE ~ ▵ABC.

Step 1: ∠DAE = ∠BAC (common angle).

Step 2: ∠ADE = ∠ABC (corresponding angles, DE ∥ BC).

Step 3: Two pairs of equal angles. ∴ ▵ADE ~ ▵ABC (AA).

See Answers ➔

  1. Congruence tests. Fluency

    • (a) Two triangles have all three pairs of sides equal. Which test applies?
    • (b) Two triangles have two sides and the included angle equal. Which test applies?
    • (c) Two right-angled triangles have equal hypotenuses and one other equal side. Which test applies?
    • (d) Can SSA be used to prove congruence? Explain why or why not.

  2. Scale factors. Fluency

    • (a) Similar triangles have sides 6, 8, 10 and 9, 12, 15. Find the scale factor.
    • (b) Similar shapes have perimeters 20 cm and 35 cm. Find the scale factor.
    • (c) If k = 4, find the area ratio.
    • (d) If the volume ratio of two similar solids is 27 : 8, find the scale factor and the length ratio.

  3. Finding missing lengths. Fluency

    • (a) ▵ABC ~ ▵DEF, AB = 5 cm, DE = 8 cm, BC = 7 cm. Find EF.
    • (b) ▵PQR ~ ▵XYZ, PQ = 12, XY = 9, QR = 16. Find YZ.
    • (c) Two similar rectangles: first has dimensions 6 cm × 9 cm. Scale factor k = 1.5. Find the dimensions of the second rectangle.
    • (d) A photograph 15 cm × 10 cm is enlarged to 24 cm × 16 cm. Find the scale factor.

  4. Identify the similarity test. Fluency

    • (a) ▵ABC has angles 50°, 70°, 60°. ▵PQR has angles 50°, 70°, 60°. Which test proves similarity?
    • (b) In ▵ABC and ▵DEF: AB/DE = BC/EF = CA/FD = 2. Which test?
    • (c) ▵XYZ and ▵LMN: XY/LM = YZ/MN = 3 and ∠XYZ = ∠LMN. Which test?
    • (d) ▵RST ~ ▵UVW by AA. If ∠R = 48° and ∠S = 75°, find ∠W.

  5. Similar triangles in a diagram. Understanding

    In the diagram, DE is parallel to BC. AD = 6, DB = 4, DE = 9.

    A B C D E 6 4 9
    • (a) Explain why ▵ADE ~ ▵ABC, stating the similarity test used.
    • (b) Find the scale factor k = AB/AD.
    • (c) Find BC.
    • (d) Find AE if AC = 15.

  6. Congruence proof. Understanding

    ABCD is a parallelogram. AC is a diagonal.

    • (a) State why AB ∥ DC and AB = DC.
    • (b) State why ∠BAC = ∠DCA (alternate angles).
    • (c) State why ∠BCA = ∠DAC (alternate angles).
    • (d) Hence prove ▵ABC ≅ ▵CDA.

  7. Area and volume ratios. Understanding

    • (a) Two similar triangles have sides in ratio 3:5. Find the ratio of their areas.
    • (b) Two similar prisms have surface areas 72 cm² and 128 cm². Find the scale factor.
    • (c) A model car is built at 1:24 scale. The real car has volume 3 m³. Find the model's volume in cm³.
    • (d) Two similar spheres have volumes 250 cm³ and 432 cm³. Find the ratio of their radii.

  8. Similar triangles in context. Understanding

    • (a) A 2-m vertical pole casts a 3-m shadow. At the same time, a tree casts a 12-m shadow. Using similar triangles, find the height of the tree.
    • (b) On a map with scale 1:50 000, two towns are 4.6 cm apart. Find the real distance in km.
    • (c) A ramp rises 1.2 m over a horizontal run of 4.8 m. A similar ramp rises 2.1 m. Find its horizontal run.
    • (d) In ▵ABC, D and E are midpoints of AB and AC respectively. Find DE if BC = 18 cm.

  9. Height of a building. Problem Solving

    A surveyor uses similar triangles to find the height of a building. She places a 1.5-m measuring rod 6 m from the building. Standing 20 m from the building, she aligns the top of the rod with the top of the building. Her eye is at ground level (simplified).

    • (a) Draw and label the two similar triangles formed.
    • (b) Identify the relevant lengths in each triangle.
    • (c) Set up a proportion and solve for the building height h.
    • (d) If the surveyor’s eye is actually 1.6 m above the ground (not at ground level), how does this change the height calculated? Find the adjusted height.

  10. Nested similar triangles. Problem Solving

    In ▵ABC, point D is on AB with AD = 4 and DB = 6. A line through D parallel to BC meets AC at E. Let DE = x and BC = y.

    • (a) Prove ▵ADE ~ ▵ABC.
    • (b) Find the scale factor in terms of AD and AB.
    • (c) Express x in terms of y.
    • (d) The area of ▵ADE is 16 cm². Find the area of trapezium BCED.