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?

Angles on a straight line always add up to 180 degrees. Step 1 — Angles on line Let us look at line . Angles and are on this straight line. They form a linear pair. So, their sum is 180 degrees. Step 2 — Angles on line Now, let us look at line . Angles and are on this straight line. They also form a linear pair. So, their sum is 180 degrees. Step 3 — Comparing the sums We have two equations. Both equations equal 180 degrees. So, their left sides must be equal. Step 4 — Finding equal angles We can subtract from both sides. This leaves us with . This shows that angles and are equal. They are vertically opposite angles. > Step 5 — Proving the other pair Let us do the same for angles and . Angles and are on line . So, their sum is 180 degrees. We already know . So, . Subtract from both sides. This gives us . Angles and are also vertically opposite. > Answer > (i) Yes, this is always true for any pair of intersecting 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

Angles on a straight line always add up to 180 degrees.

Step 1 — Angles on line $l$

Let us look at line $l$ . Angles $a$ and $b$ are on this straight line. They form a linear pair. So, their sum is 180 degrees.

$\angle a + \angle b = 180^\circ$

Step 2 — Angles on line $m$

Now, let us look at line $m$ . Angles $b$ and $c$ are on this straight line. They also form a linear pair. So, their sum is 180 degrees.

$\angle b + \angle c = 180^\circ$

Step 3 — Comparing the sums

We have two equations. Both equations equal 180 degrees. So, their left sides must be equal.

$\angle a + \angle b = \angle b + \angle c$

Step 4 — Finding equal angles

We can subtract $\angle b$ from both sides. This leaves us with $\angle a = \angle c$ . This shows that angles $a$ and $c$ are equal. They are vertically opposite angles.

$\boxed{\angle a = \angle c}$

Diagram 1

Step 5 — Proving the other pair

Let us do the same for angles $b$ and $d$ . Angles $c$ and $d$ are on line $l$ . So, their sum is 180 degrees.

$\angle c + \angle d = 180^\circ$

We already know $\angle b + \angle c = 180^\circ$ . So, $\angle c + \angle d = \angle b + \angle c$ . Subtract $\angle c$ from both sides. This gives us $\angle d = \angle b$ . Angles $b$ and $d$ are also vertically opposite.

$\boxed{\angle b = \angle d}$

Answer

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

Question 4

Answer

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