Chapter 6 — TRIANGLES

1. Answer: similar
   
   Step 1: A circle is defined by its radius.
   Step 2: Two circles may have different radii.
   Step 3: Shape remains same but size changes.
   Step 4: Hence, circles are similar but not necessarily congruent.
   
   Conclusion: All circles are similar.
2. Answer: similar
   
   Step 1: In a square, all angles are 90°.
   Step 2: Ratio of corresponding sides between any two squares is constant.
   Step 3: Hence, they satisfy similarity conditions.
   Step 4: They are congruent only if sides are exactly equal.
   
   Conclusion: All squares are similar.

Equilateral triangles → all angles equal → always similar

3. Answer: equilateral
   
   Step 1: Equilateral triangle has all angles = 60°.
   Step 2: Any equilateral triangle will have same angle measure.
   Step 3: Hence, corresponding angles are equal.
   Step 4: Therefore, all equilateral triangles are similar.
   
   Conclusion: All triangles are similar equilateral.
4. Answer: (a) equal, (b) proportional
   
   Step 1: For similarity, angles must match.
   Step 2: So corresponding angles must be equal.
   Step 3: Side lengths should follow same ratio.
   Step 4: Hence, corresponding sides must be proportional.
   
   Conclusion:
   • (a) equal
   • (b) proportional

1. Similar figures:

   Rectangles with proportional sides → Similar

   • Example 1: Two rectangles (6×4) and (9×6)
     
     Step 1: Check angles → All angles = \(90^\circ\) in both rectangles.
     Step 2: Check ratio of corresponding sides:
     \[ \frac{9}{6} = \frac{6}{4} = \frac{3}{2} \] Step 3: Ratios are equal → sides are proportional.
     
     Conclusion: Rectangles are similar.

   Equilateral triangles → all angles equal → Similar

   • Example 2: Two equilateral triangles (side 5 cm and 8 cm)
     
     Step 1: Each angle in both triangles = \(60^\circ\).
     Step 2: Corresponding angles are equal.
     Step 3: Ratio of sides: \[ \frac{8}{5} \] Step 4: Constant ratio → proportional sides.
     
     Conclusion: Triangles are similar.
2. Non-similar figures:
   • Example 1: Square (5 cm) and Rectangle (6×4)
     
     Step 1: Both have angles \(90^\circ\).
     Step 2: Square sides = 5, 5, 5, 5.
     Step 3: Rectangle sides = 6, 4, 6, 4.
     Step 4: Ratios: \[ \frac{6}{5} \neq \frac{4}{5} \] Step 5: Ratios not equal → sides not proportional.
     
     Conclusion: Not similar.
   • Example 2: Right triangle (3,4,5) and Equilateral triangle (side 4)
     
     Step 1: Right triangle has one angle \(90^\circ\).
     Step 2: Equilateral triangle has all angles \(60^\circ\).
     Step 3: Corresponding angles are not equal.
     
     Conclusion: Not similar.

Solution:

Step 1: Identify the figures

• PQRS is a rhombus → all sides equal, but angles are not \(90^\circ\)
• ABCD is a square → all sides equal and all angles \(90^\circ\)

Step 2: Compare corresponding sides

Side of rhombus = \(1.5\) cm
Side of square = \(3\) cm

Ratio of corresponding sides: \[ \frac{1.5}{3} = \frac{1}{2} \]

Observation: All sides are proportional.

Step 3: Compare corresponding angles

• Square → each angle = \(90^\circ\)
• Rhombus → angles are not \(90^\circ\) (only opposite angles equal)

Observation: Corresponding angles are not equal.

Step 4: Apply similarity condition

• Condition 1 (angles equal) ❌ Not satisfied
• Condition 2 (sides proportional) ✔ Satisfied

Since both conditions must be satisfied for similarity, and angle condition fails:

Final Conclusion: The given quadrilaterals are not similar.