Context: When two lines intersect each other and form four angles, labelled , , and , then and are equal, and and are equal. Q. Is this always true for any pair of intersecting lines?

Understand the Question When two straight lines intersect, they form two pairs of vertically opposite angles ( and ). Adjacent angles on a straight line form a linear pair, meaning their sum is always . By equating the sums of adjacent linear pairs and canceling the common angle, we prove that vertically opposite angles are always equal for any pair of intersecting lines. Step 1 · Prove that Consider two straight lines and intersecting to form angles and . Since angles and lie on straight line , they form a linear pair: Similarly, angles and lie on straight line , forming a linear pair: Equating both sums: Subtracting from both sides: Step 2 · Prove that Angles and lie on straight line , forming a linear pair: Since , equating both expressions gives: Subtracting from both sides: Thus, vertically opposite angles are equal for any pair of intersecting lines. Answer > Yes, this is always true for any pair of intersecting lines. Common Mistakes Confusing Linear Pair with Vertically Opposite Angles: Adjacent angles on a line add up to (supplementary), whereas vertically opposite angles directly facing each other are equal. Assuming It Only Holds at Right Angles: Vertically opposite angles are equal for all intersecting lines, not just perpendicular lines ().

Parallel and Intersecting Lines | IT

Question 4

Context: When two lines intersect each other and form four angles, labelled $a$ , $b$ , $c$ and $d$ , then $\angle a$ and $\angle c$ are equal, and $\angle b$ and $\angle d$ are equal.

Q. Is this always true for any pair of intersecting lines?

Solution

Understand the Question

When two straight lines intersect, they form two pairs of vertically opposite angles ( $\angle a, \angle c$ and $\angle b, \angle d$ ).
Adjacent angles on a straight line form a linear pair, meaning their sum is always $180^\circ$ .
By equating the sums of adjacent linear pairs and canceling the common angle, we prove that vertically opposite angles are always equal for any pair of intersecting lines.

Step 1 · Prove that $\angle a = \angle c$

Consider two straight lines $l$ and $m$ intersecting to form angles $a, b, c,$ and $d$ . Diagram 1

Since angles $a$ and $b$ lie on straight line $l$ , they form a linear pair: $\angle a + \angle b = 180^\circ$

Similarly, angles $b$ and $c$ lie on straight line $m$ , forming a linear pair: $\angle b + \angle c = 180^\circ$

Equating both sums: $\angle a + \angle b = \angle b + \angle c$

Subtracting $\angle b$ from both sides: $\angle a = \angle c$

Step 2 · Prove that $\angle b = \angle d$

Angles $c$ and $d$ lie on straight line $l$ , forming a linear pair: $\angle c + \angle d = 180^\circ$

Since $\angle b + \angle c = 180^\circ$ , equating both expressions gives: $\angle c + \angle d = \angle b + \angle c$

Subtracting $\angle c$ from both sides: $\angle d = \angle b \implies \angle b = \angle d$

Thus, vertically opposite angles are equal for any pair of intersecting lines.

Answer

Yes, this is always true for any pair of intersecting lines.

Common Mistakes

Confusing Linear Pair with Vertically Opposite Angles: Adjacent angles on a line add up to $180^\circ$ (supplementary), whereas vertically opposite angles directly facing each other are equal.
Assuming It Only Holds at Right Angles: Vertically opposite angles are equal for all intersecting lines, not just perpendicular lines ( $90^\circ$ ).

Question 4

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