Triangles • Topic 1 of 3

Similar figures and criteria of similarity

What is a similar figure? In geometry, similar figures are shapes that have the exact same shape, but not necessarily the same size. Think of a photograph of yourself: whether it is printed as a small passport-size photo or a large wall poster, your features look exactly the same because the proportions are preserved. Only the size changes!

In contrast, congruent figures are identical twins: they have both the same shape and the same size. Therefore, all congruent figures are similar, but all similar figures are not congruent.

For two polygons to be similar, they must satisfy two strict conditions:

  • Their corresponding angles must be equal.
  • Their corresponding sides must be in the same ratio (proportional).

Criteria for Similarity of Triangles Instead of checking all three angles and all three sides every time, we can use short-cut rules to prove that two triangles are similar:

1. AAA (Angle-Angle-Angle) Criteria: If all three corresponding angles of two triangles are equal, their corresponding sides will automatically be proportional, making the triangles similar. (Even if just two angles are equal, the third must be equal due to the angle sum property. This is often called the AA criteria). 2. SSS (Side-Side-Side) Criteria: If the three corresponding sides of two triangles are in the same ratio, then their corresponding angles will automatically be equal, making them similar. 3. SAS (Side-Angle-Side) Criteria: If one angle of a triangle is equal to one angle of another triangle, and the sides including these angles are proportional, the triangles are similar.

FeatureCongruent TrianglesSimilar Triangles
ShapeExactly the sameExactly the same
SizeExactly the sameCan be different
Corresponding AnglesEqualEqual
Corresponding SidesEqual (Ratio is 1:1)Proportional (Ratio is equal)
Symbol≅~

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DIAGRAM 1: CONGRUENT VS SIMILAR TRIANGLES

    Congruent Triangles (Same Shape & Size)
         /\                 /\
        /  \               /  \
     5 /____\ 5         5 /____\ 5
         6                  6
  
    Similar Triangles (Same Shape, Scaled Size)
         /\
        /  \                   /\
     3 /____\ 3               /  \
         4                 6 /____\ 6
                                8

DIAGRAM 2: AA SIMILARITY CRITERION

     Triangle ABC               Triangle DEF
          /\                         /\
         /  \                       /  \
        / 60 \                     / 60 \
       /      \                   /      \
      /40    80\                 /40    80\
     /__________\               /__________\
     
     Since Angle A = Angle D, Angle B = Angle E, and Angle C = Angle F,
     Triangle ABC ~ Triangle DEF by AAA Similarity.

DIAGRAM 3: REAL-LIFE APPLICATION (SHADOW PUPPET SCALE)

       Light Source
           O
          / \
         /   \     Object (Hand)
        /     \_______|_
       /       \      |
      /         \     v
     /___________\________ [Wall Shadow: Scaled Up & Similar]
Solved Examples
1
Worked Example
In triangle ABC and triangle PQR, Angle A = 50 degrees, Angle B = 70 degrees, Angle P = 50 degrees, and Angle Q = 70 degrees. Are these triangles similar? If yes, state the criterion.
Solution
  1. Step 1: Identify the given information.*
  2. In triangle ABC, Angle A = 50 degrees and Angle B = 70 degrees.*
  3. In triangle PQR, Angle P = 50 degrees and Angle Q = 70 degrees.*
  4. Step 2: Match the corresponding angles.*
  5. Angle A = Angle P = 50 degrees.*
  6. Angle B = Angle Q = 70 degrees.*
  7. Step 3: Apply the triangle similarity rule.*
  8. Since two pairs of corresponding angles are equal, the third pair must also be equal (180 - 50 - 70 = 60 degrees for both).*
  9. Therefore, by the AA (Angle-Angle) similarity criterion, Triangle ABC is similar to Triangle PQR.*
  10. Answer: Yes, the triangles are similar by the AA similarity criterion.
2
Worked Example
A 6-foot tall school student stands next to a vertical flagpole. The student casts a shadow that is 4 feet long on the ground, while the flagpole casts a shadow that is 20 feet long. Find the height of the flagpole.
Solution
  1. Step 1: Set up the scenario as similar triangles.*
  2. The sun's rays hit the ground at the same angle for both the student and the flagpole. Both the student and flagpole stand perpendicular (90 degrees) to the ground.*
  3. Therefore, the triangle formed by the student and their shadow is similar to the triangle formed by the flagpole and its shadow by AA similarity.*
  4. Step 2: Set up the ratio of corresponding sides.*
  5. (Height of Flagpole) / (Height of Student) = (Shadow of Flagpole) / (Shadow of Student)*
  6. Let the height of the flagpole be H.*
  7. H / 6 = 20 / 4*
  8. Step 3: Solve for H.*
  9. H / 6 = 5*
  10. H = 5 6 = 30 feet.
  11. Answer: The height of the flagpole is 30 feet.
3
Worked Example
In a triangle ABC, a line segment DE is drawn parallel to base BC such that D lies on AB and E lies on AC. If AD = 2 cm, DB = 4 cm, and AC = 9 cm, find the length of AE.
Solution
  1. Step 1: Analyze the triangles formed.*
  2. We have a small triangle ADE inside a large triangle ABC. Since DE is parallel to BC, Angle ADE = Angle ABC (corresponding angles) and Angle AED = Angle ACB (corresponding angles).*
  3. Therefore, Triangle ADE ~ Triangle ABC by AA similarity.*
  4. Step 2: Use the property of similar triangles.*
  5. The ratio of corresponding sides must be equal: AD / AB = AE / AC.*
  6. Step 3: Calculate the missing values.*
  7. Total length of AB = AD + DB = 2 cm + 4 cm = 6 cm.*
  8. Substitute the known values into the ratio: 2 / 6 = AE / 9.*
  9. Step 4: Solve for AE.*
  10. 1 / 3 = AE / 9*
  11. AE = 9 / 3 = 3 cm.*
  12. Answer: The length of AE is 3 cm.
  13. --

Key Points

  • Similar figures share identical shapes but can have different physical dimensions.
  • Two triangles are similar if their corresponding angles are equal and their corresponding sides are proportional.
  • The AA criterion states that if two angles of one triangle match two angles of another, the triangles are similar.
  • The SSS criterion confirms similarity when all three pairs of corresponding sides share a common ratio.
  • The SAS criterion requires one matching angle pair and two adjacent matching side ratios.
Quick Check — 4 MCQs
Tap an option to check your answer0 / 4
Q1.Two triangles are similar if their corresponding angles are equal and corresponding sides are:
Explanation: Similarity requires proportional sides.
Q2.The AAA condition is a criterion for:
Explanation: Angle-Angle-Angle similarity.
Q3.All circles are:
Explanation: Any two circles are similar.
Q4.If $\triangle ABC\sim\triangle DEF$ then $\tfrac{AB}{DE}=\tfrac{BC}{EF}=$
Explanation: All ratios equal.
Practice more MCQs → 📝 Assignment · 35 marks