Here are the solutions for each question based on the image you uploaded:

4. A linear equation in two variables represents a:

• A linear equation in two variables has the general form $ax + by + c = 0$, which represents a straight line.
• Answer: b) Line

5. Which of the following is not a postulate of Euclid's geometry?

• Euclid's postulates are well-known:

  1. A straight line may be drawn from any one point to any other point.
  2. A finite line can be extended indefinitely.
  3. A circle can be drawn with any center and radius.
  4. All right angles are equal.
  5. Through a point not on a line, exactly one line can be drawn parallel to the given line (Playfair's axiom).
  • Answer: d) There are infinite lines passing through two distinct points (This is false, as only one straight line can pass through two distinct points).

6. If two parallel lines are intersected by a transversal, then each pair of alternate interior angles is:

• Alternate interior angles formed when a transversal cuts two parallel lines are always equal.
• Answer: c) Equal

7. In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. This is known as:

• This is the well-known Pythagorean Theorem ($c^2 = a^2 + b^2$).
• Answer: b) Pythagoras Theorem

8. Which of the following is a factor of $x^2 - 5x + 6$?

• We need to factor $x^2 - 5x + 6$. It factors as: $x^2 - 5x + 6 = (x - 2)(x - 3)$ So, $x - 2$ and $x - 3$ are factors.
• Answer: b) $x - 2$

Would you like more details on any of the solutions? Here are 5 related questions to expand your understanding:

1. What is the general form of a quadratic equation in two variables, and what does it represent geometrically?
2. Can you state all of Euclid’s five postulates, and how do they form the basis for Euclidean geometry?
3. Why are alternate interior angles always equal when a transversal intersects two parallel lines?
4. How is the Pythagorean Theorem derived from Euclid’s axioms?
5. How do you factor a quadratic equation like $x^2 - 5x + 6$ systematically?

Tip: Always check for common factorization patterns, such as difference of squares or perfect square trinomials, to simplify quadratic equations.