Similar Triangles & Basic Proportionality Theorem explained through a friendly whiteboard conversation, examples, visuals, and practice for school students.
Step 1
Notes
Step 2
Formulae
Step 3
Examples
Step 4
Practice
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Similar triangles have the same shape but may have different sizes.
Their matching angles are equal and their matching sides are in proportion.
Similarity is used in maps, scale drawings, shadows, heights, and geometry proofs.
Two triangles are similar when their corresponding angles are equal and corresponding sides are proportional.
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AA similarity is often the easiest test because two equal angles are enough.
Scale factor compares matching sides.
The Basic Proportionality Theorem says a line parallel to one side of a triangle divides the other two sides in the same ratio.
Similarity is different from congruency because similar shapes may have different sizes.
Meaning: Same shape, not necessarily same size.
Definition: Two triangles are similar if corresponding angles are equal and corresponding sides are proportional.
A small triangle and its enlarged copy are similar.
Meaning: Tests prove triangles are similar.
Definition: AA, SSS, and SAS are common similarity tests.
If two angles of one triangle equal two angles of another triangle, the triangles are similar by AA.
Meaning: Scale factor compares corresponding side lengths.
Definition: Scale factomage sidriginal side for matching sides.
If matching sides are 6 cm and 3 cm, scale factor is 2.
Meaning: A parallel line divides two sides proportionally.
Definition: If a line is parallel to one side of a triangle and meets the other two sides, it divides those sides in the same ratio.
If DE is parallel to BC in triangle ABC, then AC.
Easy
Question: Two triangles have angles 40 degrees and 70 degrees matching. Are they similar?
Medium
Question: Two similar triangles have matching sides 4 cm and 12 cm. Find the scale factor from smaller to larger.
Exam-level
Question: If AC, A, D, and A, find EC.
Mixing up similar and congruent
Why it happens: Both compare triangles.
Correct approach: Similar means same shape. Congruent means same shape and size.
Comparing wrong sides
Why it happens: Correspondence is not marked.
Correct approach: Match vertices before writing ratios.
Using BPT without parallel lines
Why it happens: The theorem is recognised by diagram shape only.
Correct approach: Check that the line is parallel first.
Are all congruent triangles similar?
Yes. Congruent triangles are similar with scale factor 1.
Can similar triangles have different areas?
Yes. They may be different sizes.
Why does BPT need parallel lines?
The equal angle relationships come from parallel lines.
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