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CBSE · Class 10 · Mathematics

Triangles 6.4: Criteria for Similarity of Triangles

5 concepts · 40 practice questions with answers · Free to read

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Identifying AA, SSS and SAS

From daily life

To check two photo frames have the same shape you do not measure everything: a couple of matching angles or ratios is enough.

Simple definition

Triangles are similar if: AA two angles are equal (the third then matches), SSS all three pairs of sides are in the same ratio, or SAS one angle is equal and the two sides including it are in the same ratio.

50°60°70°50°60°70°AA

In the figure: 50° and 60° match in both triangles, so the third angles (70°) match and the triangles are similar.

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Identifying AA, SSS and SAS

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Rule 1: Which criterion fits? — Look at what is given: angles, three sides, or two sides and an angle.

  • Easy: Angles 50°, 60° and 60°, 70°: both sets are 50°, 60°, 70°: AA.
  • Medium: Sides 3, 4, 5 and 6, 8, 10: SSS.
  • Exam-type: Sides 2, 3 with included angle 40° and 4, 6 with 40°: SAS.

Rule 2: What does not work — Fewer matches than a criterion needs, or the wrong angle, prove nothing.

  • Easy: Only one pair of angles equal: not enough.
  • Medium: Two sides proportional but the angle is not between them: not a test.
  • Exam-type: Only two side ratios equal, no angle given: not enough.
Remember

AA needs 2 angles, SSS needs 3 side ratios, SAS needs an angle between 2 sides.

Common mistake

Using SAS when the equal angle is not between the two proportional sides.

Memory clue

"AA: angles. SSS: sides. SAS: angle in the middle."

Practice questions

Easy
  1. 1In △DRG and △AYB: DR = 7 cm, RG = 9 cm, GD = 12 cm, AY = 14 cm, YB = 18 cm and BA = 24 cm. Which criterion shows △DRG ~ △AYB?
  2. 2In △UYG and △VNR: ∠U = 70°, ∠Y = 55°, ∠V = 70° and ∠N = 55°. Which criterion shows △UYG ~ △VNR?
  3. 3In △RLF and △VQC: RL = 7 cm, LF = 8 cm, FR = 9 cm, VQ = 14 cm, QC = 16 cm and CV = 18 cm. Which criterion shows △RLF ~ △VQC?
Medium
  1. 4In △QZD and △SKF: ∠Q = 45°, ∠Z = 110°, ∠S = 45° and ∠K = 110°. Which criterion shows △QZD ~ △SKF?
  2. 5In △DFA and △HVR: DF = 5 cm, FA = 12 cm, AD = 13 cm, HV = 10 cm, VR = 24 cm and RH = 26 cm. Which criterion shows △DFA ~ △HVR?
  3. 6In △CEY and △UXV: CE = 2 cm, CY = 6 cm, ∠E = 110°, UX = 6 cm, UV = 18 cm and ∠X = 110°. Which criterion shows △CEY ~ △UXV?
Challenging
  1. 7Which one condition is enough to prove △XEB ~ △YDA?
  2. 8Which one condition is enough to prove △WCZ ~ △SHT?
Show answers (try them yourself first)
  1. 1SSS
  2. 2AA
  3. 3SSS
  4. 4AA
  5. 5SSS
  6. 6None of AA, SSS, SAS applies
  7. 7∠X = ∠Y and ∠E = ∠D
  8. 8WC/SH = CZ/HT = ZW/TS

SSS similarity: checking and finding sides

From daily life

Two triangular flags with sides 4, 6, 8 cm and 6, 9, 12 cm are copies of the same design at different sizes.

Simple definition

If AB/PQ = BC/QR = CA/RP, then ΔABC ~ ΔPQR (SSS). Pair the shortest with shortest, middle with middle and longest with longest.

Formula: AB/PQ = BC/QR = CA/RP

84612692 : 3

In the figure: 4 : 6, 6 : 9 and 8 : 12 are all 2 : 3, so the triangles are similar.

Rule 1: Check SSS — Sort the sides, divide matching pairs and compare.

  • Easy: 4, 6, 8 and 6, 9, 12: all 2/3: similar.
  • Medium: 3, 5, 7 and 6, 10, 15: 1/2, 1/2, 7/15: not similar.
  • Exam-type: 5, 12, 13 and 10, 24, 26: all 1/2: similar.

Rule 2: Find the missing side — Find k from two known sides, then apply it.

  • Easy: 5, 12, 13 ~ 10, 24, x: x = 26.
  • Medium: 6, 8, 10 ~ 9, 12, x: k = 3/2, x = 15.
  • Exam-type: 4, 5, 6 ~ 12, 15, x: k = 3, x = 18.
Remember

Always sort both triangles’ sides from shortest to longest before comparing.

Common mistake

Dividing the shortest side of one by the longest of the other.

Memory clue

"Short with short, long with long."

Practice questions

Easy
  1. 1△QZF has sides QZ = 6 cm, ZF = 8 cm, FQ = 11 cm and △XBH has sides XB = 24 cm, BH = 32 cm, HX = 44 cm. Are the triangles similar?
  2. 2△NEY has sides NE = 5 cm, EY = 6 cm, YN = 7 cm and △QGT has sides QG = 10 cm, GT = 12 cm, TQ = 14 cm. Are the triangles similar?
  3. 3△SEM has sides SE = 5 cm, EM = 6 cm, MS = 7 cm and △XLT has sides XL = 15 cm, LT = 18 cm, TX = 21 cm. Are the triangles similar?
Medium
  1. 4△ADW has sides AD = 5 cm, DW = 7 cm, WA = 9 cm and △FYM has sides FY = 14 cm, YM = 21 cm, MF = 27 cm. Are the triangles similar?
  2. 5△SUG has sides SU = 8 cm, UG = 9 cm, GS = 10 cm and △XZB has sides XZ = 16 cm, ZB = 18 cm, BX = 20 cm. Are the triangles similar?
  3. 6In △FAP, FA = 6 cm, AP = 8 cm, PF = 11 cm. △SWQ is similar to △FAP (corresponding order) with SW = 24 cm, WQ = 32 cm. Find QS (in cm).
Challenging
  1. 7In △TDQ, TD = 16 cm, DQ = 18 cm, QT = 20 cm. △EYP is similar to △TDQ (corresponding order) with YP = 36 cm, PE = 40 cm. Find EY (in cm).
  2. 8In △YEF, YE = 16 cm, EF = 20 cm, FY = 24 cm. △GWZ is similar to △YEF (corresponding order) with WZ = 15 cm, ZG = 18 cm. Find GW (in cm).
Show answers (try them yourself first)
  1. 1Yes
  2. 2Yes
  3. 3Yes
  4. 4No
  5. 5Yes
  6. 644
  7. 732
  8. 812

Corresponding angles, sides and perimeters

From daily life

Once two triangles are known to be similar, one fact about one triangle tells you the matching fact about the other.

Simple definition

If ΔABC ~ ΔPQR, then corresponding angles are equal, corresponding sides have the same ratio k, and perimeter ABC / perimeter PQR = k as well.

equalanglessameratio kperimeterratio k

In the figure: AB = 6, PQ = 9, perimeter ABC = 20: perimeter PQR = 20 × 9/6 = 30.

Rule 1: Ratios and perimeters — The perimeter ratio equals the side ratio.

  • Easy: AB = 6, PQ = 9, perimeter ABC = 20: PQR = 30.
  • Medium: k = 3/5, perimeter of the first = 24: second = 40.
  • Exam-type: Perimeters 20 and 30, side 8: matching side 12.

Rule 2: Matching angles — Equal angles pair by the order of letters.

  • Easy: ΔABC ~ ΔPQR, ∠B = 70°, ∠C = 50°: ∠P = ∠A = 60°.
  • Medium: ∠A = ∠P = 80°, ∠Q = 45°: ∠C = ∠R = 55°.
  • Exam-type: The side opposite ∠B (AC) corresponds to the side opposite ∠Q (PR).
Remember

Similar means all the equalities hold together: angles, side ratios and perimeter ratio.

Common mistake

Applying the ratio of sides to angles; angles are equal, not scaled.

Memory clue

"Angles stay, lengths scale."

Practice questions

Easy
  1. 1△NYR ~ △TSX. If ∠T = 50° and ∠X = 95°, find ∠Y (in degrees).
  2. 2△KRV ~ △XGN. If ∠V = 75° and ∠K = 55°, find ∠G (in degrees).
  3. 3△ELK ~ △ZHF. If ∠E = 80° and ∠L = 40°, find ∠F (in degrees).
Medium
  1. 4△MDY ~ △BPL and MD : BP = 3 : 4. If the perimeter of △MDY is 24 cm, find the perimeter of △BPL (in cm).
  2. 5△BKF ~ △GHV. If BK = 30 cm, KF = 20 cm, HV = 12 cm, find GH (in cm).
  3. 6△GHS ~ △UEX and GH : UE = 7 : 4. If the perimeter of △GHS is 126 cm, find the perimeter of △UEX (in cm).
Challenging
  1. 7△DWY ~ △RNS. If DW = (x + 5) cm, RN = (x + 8) cm, WY = 8 cm and NS = 11 cm, find x.
  2. 8△CTX ~ △UKN. If CT = (x + 4) cm, UK = (x + 6) cm, TX = 3 cm and KN = 4 cm, find x.
Show answers (try them yourself first)
  1. 135
  2. 250
  3. 360
  4. 432
  5. 518
  6. 672
  7. 73
  8. 82

SAS, AA and parallel-line applications

From daily life

A line drawn parallel to one side cuts off a smaller triangle that is a scaled copy of the whole.

Simple definition

If DE ∥ BC, then ∠A is common and ∠ADE = ∠ABC, so ΔADE ~ ΔABC (AA) and AD/AB = AE/AC = DE/BC. For SAS, one equal angle and the two including sides proportional give similarity.

Formula: AD/AB = AE/AC = DE/BC

ABCDE46ΔADE ~ ΔABC

In the figure: AD = 4, AB = 10, BC = 15: DE = 4/10 × 15 = 6.

Rule 1: Lengths from parallel lines — Use AD/AB = AE/AC = DE/BC with the whole sides.

  • Easy: AD = 4, AB = 10, BC = 15: DE = 6.
  • Medium: AD = 3, AB = 9, DE = 4: BC = 12.
  • Exam-type: AE = 6, AC = 15, BC = 20: DE = 8.

Rule 2: SAS test — The angle must lie between the two proportional sides.

  • Easy: AB = 3, AC = 6, PQ = 6, PR = 12, ∠A = ∠P: ratios 1/2, 1/2: similar.
  • Medium: AB = 4, AC = 6, PQ = 6, PR = 8: 2/3 ≠ 3/4: not similar.
  • Exam-type: If ∠A ≠ ∠P, the test fails.
Remember

The common angle at A comes free in every parallel-line triangle.

Common mistake

Using DE/DB or AD/DB for DE/BC; the correct pairs are with whole sides.

Memory clue

"Parallel gives AA; use whole sides AB, AC, BC."

Practice questions

Easy
  1. 1In △DFK and △RCP, DF = 3 cm, DK = 4 cm, RC = 6 cm, RP = 8 cm and ∠D = ∠R = 40°. Are the triangles similar?
  2. 2In △BPD and △RNV, BP = 9 cm, BD = 6 cm, RN = 18 cm, RV = 14 cm and ∠B = ∠R = 100°. Are the triangles similar?
  3. 3In △WFR and △MUK, WF = 10 cm, WR = 9 cm, MU = 40 cm, MK = 36 cm and ∠W = ∠M = 55°. Are the triangles similar?
Medium
  1. 4In △WNX, A lies on WN, L lies on WX and AL ∥ NX. If WA = 4 cm, WN = 18 cm, NX = 36 cm, find AL (in cm).
  2. 5In △NDU, G lies on ND, Z lies on NU and GZ ∥ DU. If ND = 45 cm, GZ = 12 cm, DU = 54 cm, find NG (in cm).
  3. 6In △RLS, A lies on RL, G lies on RS and AG ∥ LS. If RA = 4 cm, AG = 8 cm, LS = 20 cm, find RL (in cm).
Challenging
  1. 7Lines KL and YH meet at B (with B between K and L, and between Y and H) and KY ∥ LH. If BK = 8 cm, KY = 20 cm, LH = 35 cm, find BL (in cm).
  2. 8Lines FW and SY meet at V (with V between F and W, and between S and Y) and FS ∥ WY. If VF = 35 cm, VW = 20 cm, WY = 8 cm, find FS (in cm).
Show answers (try them yourself first)
  1. 1Yes
  2. 2No
  3. 3Yes
  4. 48
  5. 510
  6. 610
  7. 714
  8. 814

Shadows and heights

From daily life

At the same time of day, a tall tree and a short pole make shadows with the same slope, so height can be found from a shadow.

Simple definition

At the same time an object, its shadow and the sun’s ray form a triangle, and two such triangles are similar (AA). So height/shadow is the same for both: h₁/s₁ = h₂/s₂.

Formula: h₁/s₁ = h₂/s₂

6 m4 m15 m10 msame sunsame angle

In the figure: Pole 6 m, shadow 4 m; tree shadow 10 m: tree height = 6/4 × 10 = 15 m.

Rule 1: Find the height — Equal height-to-shadow ratios; solve for the unknown height.

  • Easy: Pole 6 m, shadow 4 m; tree shadow 10 m: 15 m.
  • Medium: Person 1.5 m, shadow 2 m; tower shadow 40 m: 30 m.
  • Exam-type: Stick 2 m, shadow 3 m; building shadow 30 m: 20 m.

Rule 2: Find the shadow — Put the unknown shadow in the ratio instead.

  • Easy: Pole 8 m, shadow 6 m; tree 12 m: shadow 9 m.
  • Medium: Person 1.8 m, shadow 1.2 m; tree 9 m: shadow 6 m.
  • Exam-type: Both measurements must be taken at the same time of day.
Remember

Same time of day is what makes the triangles similar.

Common mistake

Mixing height with shadow in one ratio, such as h₁/s₂.

Memory clue

"Height over shadow, same for both."

Practice questions

Easy
  1. 1At the same time of the day, a pole of height 6 m casts a shadow 42 m long. Find the length of the shadow (in m) of a building of height 1 m.
  2. 2At the same time of the day, a stick of height 45 m casts a shadow 10 m long. A tower casts a shadow 16 m long. Find the height of the tower (in m).
  3. 3At the same time of the day, a stick of height 15 m casts a shadow 35 m long. Find the length of the shadow (in m) of a tree of height 9 m.
Medium
  1. 4At the same time of the day, a tower of height 2 m casts a shadow 5 m long. Find the length of the shadow (in m) of a pole of height 10 m.
  2. 5A man 160 cm tall casts a shadow 120 cm long. At the same time, a flagpole casts a shadow 6 m long. Find the height of the flagpole (in m).
  3. 6A man 180 cm tall casts a shadow 150 cm long. At the same time, a tower casts a shadow 25 m long. Find the height of the tower (in m).
Challenging
  1. 7A girl 180 cm tall casts a shadow 100 cm long. At the same time, a tower casts a shadow 50 m long. Find the height of the tower (in m).
  2. 8A boy 120 cm tall casts a shadow 90 cm long. At the same time, a building casts a shadow 27 m long. Find the height of the building (in m).
Show answers (try them yourself first)
  1. 17
  2. 272
  3. 321
  4. 425
  5. 58
  6. 630
  7. 790
  8. 836