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CBSE Class 10 · Mathematics

Triangles

Official NCERT chapter from Mathematics (book code jemh1). ExamMaster notes are original teaching at CBSE Class 10 depth.

This lesson follows the official chapter “Triangles” in Mathematics. The words below are ExamMaster’s teaching, not a paste from the book. Use the NCERT chapter for the classroom sequence; use these notes to hold the idea without copying exercises or figures.

  • CBSE Class 10
  • Medium level
  • 8 concepts

1Similar Figures

Similar figures have the same shape: matching angles equal, sides in one scale-factor. A square and a larger square are similar; a square and a 2-by-3 rectangle are not. Similar is a scale-story, not “both look pointy”.

A doodle that is “almost the same” is not similar.

Figure. Similar figures keep the same shape. The large rectangle is the small one scaled by 2: every side doubles, and the 2:3 ratio stays.

How it works

  1. Match vertices in orderThe correspondence.
  2. Check angles equal and sides proportionalThe shape-test.
  3. Name the scale-factor if sides match a ratioSimilar figures.

2Similarity of Triangles

Similar triangles are the same shape-test on three sides: corresponding angles equal, corresponding sides proportional. Writing △ABC ~ △DEF names the order — A with D, B with E, C with F. A shuffled correspondence is a different claim.

Two triangles that share a look but not a named order are not this write.

Figure. Similarity names matching vertices in order. ABC ~ DEF means A matches D, B matches E, C matches F — that order is the correspondence.

How it works

  1. Name the correspondenceWhich vertex with which.
  2. Check AAA or the side-ratios as taughtThe similarity.
  3. Keep the order in the ~ writeTriangles, named.

3Criteria for Similarity of Triangles

Criteria: AAA (two equal angles suffice, the third follows), SAS similarity (an included angle and the sides about it proportional), SSS similarity (three side-ratios equal), as taught. A criterion is a shortest honest test, not “they look alike”. AA is the school short form of AAA.

SAS congruence (same lengths) is not SAS similarity (same ratios).

Figure. AA is enough: if two angles of one triangle equal two of another, the third pair is forced and the triangles are similar.

How it works

  1. Name which criterion you claimAAA, SAS~, SSS~.
  2. Show the equal angles and/or the ratiosThe evidence.
  3. Then write ~ with the matching orderThe criterion ran.

4A definition is a test you can run

Similar is a test: a correspondence plus equal angles or proportional sides. If you only say “same shape”, you have a heading.

A pretty pair of sketches is not the test.

Figure. Similarity is a test you run: matching angles, or three side ratios, or SAS. Looking roughly the same is not the test.

How it works

  1. Name the vertex-orderThe object.
  2. Run a criterionThe test.
  3. Then the word has contentThe definition ran.

5Name the given before the unknown

The given are the marked angles or sides. The unknown is “similar or not” or a missing side via the scale. Copy the correspondence before you cross-multiply.

Matching the longest sides because they “look like” the pair is a silent given.

Figure. Name the given before the unknown. Mark the two known angles first; only then chase the unmarked side x.

How it works

  1. Copy marks and the named orderThe given.
  2. Name ~ or the missing lengthThe unknown.
  3. Then run the criterion or the ratioGiven first.

6One worked case is enough at this class

One similar pair is enough. A page of the same AA-clone does not add a new SSS~.

Ten twins of “same shape” are still one idea.

Figure. One worked pair is enough: sides 6, 8, 10 sit on 3, 4, 5 with scale 2. Area would need 2 squared, not 2.

How it works

  1. Run one criterion or one scaleThe case.
  2. A second only as a checkOptional.
  3. StopThis class.

7A miss: swapping the school name for the picture

A shadow-photo is a setting. The maths is a correspondence and a criterion. A shadow is not AAA until angles are argued.

A tourist leaning-tower still is not ~.

Figure. ABC ~ DEF matches A to D, not A to E. Swapping the school names for a pretty picture breaks every later ratio.

How it works

  1. Keep the photo as a storyA setting.
  2. Name vertices and a criterionThe maths.
  3. Then the scale if askedDo not swap.

8Check by the opposite action or the opposite test

Check similarity by a second criterion, or check a side by the inverse scale. If AA held, a third angle must match; if a ratio fails, the ~ fails.

Measuring one side only is not the opposite test for SSS~.

Figure. The opposite test undoes the scale: 6 to 3 is 2, and 3 to 6 is 1/2. Both must name the same pairing.

How it works

  1. State ~ or a lengthThe claim.
  2. Rebuild angles or invert the scaleThe check.
  3. Mend if they disagreeHonest.
AAA similarity needs
  1. Corresponding angles equal (two suffice as AA)
  2. All three sides equal in length
  3. A shared side only

Shape, not congruence lengths.

Notes

  • The official chapter title is “Triangles”. Teach the school test for that title, not a contest shortcut.
  • If a step needs a later class, stop. The next official chapter will pick it up.

Formulas

  • ~ : corresponding angles equal, sides proportional
  • criteria: AAA/AA, SAS~, SSS~ (as taught)

Recap

Hold these pegs from the official chapter “Triangles”. The wording is ExamMaster’s teaching, not a textbook recap.

Similar Figures
Similar figures have the same shape: matching angles equal, sides in one scale-factor.
Similarity of Triangles
Similar triangles are the same shape-test on three sides: corresponding angles equal, corresponding sides proportional.
Criteria for Similarity of Triangles
Criteria: AAA (two equal angles suffice, the third follows), SAS similarity (an included angle and the sides about it proportional), SSS similarity (three side-ratios equal), as taught.
A definition is a test you can run
Similar is a test: a correspondence plus equal angles or proportional sides.
Name the given before the unknown
The given are the marked angles or sides. The unknown is “similar or not” or a missing side via the scale. Copy the correspondence before you cross-multiply.
One worked case is enough at this class
One similar pair is enough. A page of the same AA-clone does not add a new SSS~.

Practise Triangles

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