AA, SAS, SSS || Similarity Rules || Chapter 6 || Triangles || Class 10 Maths NCERT

Exercise 6.3 | Chapter 6 Triangles | Class 10 Maths NCERT | New Syllabus New Course English Medium Class 6 Maths :    • Class 6 Maths | All Exercise of All Chapters   Class 7 Maths :    • Class 7 Maths | All Exercise of All Chapters   Class 8 Maths :    • Class 8 Maths | New Syllabus 2026 | 2025 |...   Class 9 Maths :    • Class 9 Maths NCERT | New Syllabus 2026 | ...   Class 10 Math :    • Class 10 Maths | New Syllabus 2026 | 2025 ...   New Course In Hindi Medium Class 8 Maths :    • In Hindi | Class 8 Maths | All Exercise of...   Class 9 Maths :    • In Hindi | Class 9 Maths | New Syllabus 20...   Class 10 Math :    • In Hindi | Class 10 Maths | All Exercise o...   Theorem 6.4 : If in two triangles, sides of one triangle are proportional to (i.e., in the same ratio of ) the sides of the other triangle, then their corresponding angles are equal and hence the two triangles are similiar. Theorem 6.5 : If one angle of a triangle is equal to one angle of the other triangle and the sides including these angles are proportional, then the two triangles are similar. This criterion is referred to as the SAS (Side–Angle–Side) similarity criterion for two triangles. Example 4 : In Fig. 6.29, if PQ || RS, prove that Δ POQ ~ Δ SOR. Example 5 : Observe Fig. 6.30 and then find ∠ P. Example 6 : In Fig. 6.31, OA . OB = OC . OD. Show that ∠A = ∠C and ∠B = ∠D. Example 7 : A girl of height 90 cm is walking away from the base of a lamp-post at a speed of 1.2 m/s. If the lamp is 3.6 m above the ground, find the length of her shadow after 4 seconds. Example 8 : In Fig. 6.33, CM and RN are respectively the medians of Δ ABC and Δ PQR. If Δ ABC ~ Δ PQR, prove that : (i) Δ AMC ~ Δ PNR (ii)CM AB=RN PQ (iii) Δ CMB ~ Δ RNQ EXERCISE 6.3 1. State which pairs of triangles in Fig. 6.34 are similar. Write the similarity criterion used by you for answering the question and also write the pairs of similar triangles in the symbolic 2. In Fig. 6.35, Δ ODC ~ Δ OBA, ∠ BOC = 125° and ∠ CDO = 70°. Find ∠ DOC, ∠ DCO and ∠ OAB. 3. Diagonals AC and BD of a trapezium ABCD with AB || DC intersect each other at the point O. Using a similarity criterion for two triangles, show that OA OB=OC OD. 4. In Fig. 6.36, QR QT=QS PR= and ∠ 1 = ∠ 2. Show that Δ PQS ~ Δ TQR. 5. S and T are points on sides PR and QR of Δ PQR such that ∠ P = ∠ RTS. Show that Δ RPQ ~ Δ RTS. 6. In Fig. 6.37, if Δ ABE ≅ Δ ACD, show that Δ ADE ~ Δ ABC. 7. In Fig. 6.38, altitudes AD and CE of Δ ABC intersect each other at the point P. Show that: (i) Δ AEP ~ Δ CDP (ii) Δ ABD ~ Δ CBE (iii) Δ AEP ~ Δ ADB (iv) Δ PDC ~ Δ BEC 8. E is a point on the side AD produced of a parallelogram ABCD and BE intersects CD at F. Show that Δ ABE ~ Δ CFB. 9. In Fig. 6.39, ABC and AMP are two right triangles, right angled at B and M respectively. Prove that: (i) Δ ABC ~ Δ AMP (ii) CA BC=PA MP 10. CD and GH are respectively the bisectors of ∠ ACB and ∠ EGF such that D and H lie on sides AB and FE of Δ ABC and Δ EFG respectively. If Δ ABC ~ Δ FEG, show that: (i) CD AC= GH FG (ii) Δ DCB ~ Δ HGE (iii) Δ DCA ~ Δ HGF 11. In Fig. 6.40, E is a point on side CB produced of an isosceles triangle ABC with AB = AC. If AD ⊥ BC and EF ⊥ AC, prove that Δ ABD ~ Δ ECF. 12. Sides AB and BC and median AD of a triangle ABC are respectively proportional to sides PQ and QR and median PM of Δ PQR (see Fig. 6.41). Show that Δ ABC ~ Δ PQR. Fig. 6.40 13. D is a point on the side BC of a triangle ABC such that ∠ ADC = ∠ BAC. Show that CA2 = CB.CD. 14. Sides AB and AC and median AD of a triangle ABC are respectively proportional to sides PQ and PR and median PM of another triangle PQR. Show that Δ ABC ~ Δ PQR. Fig. 6.41 15. A vertical pole of length 6 m casts a shadow 4 m long on the ground and at the same time a tower casts a shadow 28 m long. Find the height of the tower. 16. If AD and PM are medians of triangles ABC and PQR, respectively where Δ ABC ~ Δ PQR, prove that AB AD=PQ PM⋅