Parallel and Intersecting Lines

Given: Two lines intersect at a point forming four angles: $\angle a$, $\angle b$, $\angle c$, $\angle d$ (going around the intersection).

Concept:
- A linear pair consists of two adjacent angles whose non-common arms form a straight line; they add up to $180°$.
- Vertically opposite angles are the angles across the intersection from each other; they are always equal.

Linear Pairs (each pair sums to $180°$):
$$\angle a \text{ and } \angle b, \quad \angle b \text{ and } \angle c, \quad \angle c \text{ and } \angle d, \quad \angle d \text{ and } \angle a$$

Pairs of Vertically Opposite Angles (each pair is equal):
$$\angle a = \angle c \quad \text{and} \quad \angle b = \angle d$$

Summary Table:

| Linear Pairs | $\angle a$ and $\angle b$, $\angle b$ and $\angle c$, $\angle c$ and $\angle d$, $\angle d$ and $\angle a$ |
|---|---|
| Pairs of Vertically Opposite Angles | $\angle a$ and $\angle c$; $\angle b$ and $\angle d$ |

Given: Two lines intersect forming four angles. We want all four angles to be equal.

Concept: The four angles around a point sum to $360°$. If all four angles are equal, each angle $= \dfrac{360°}{4} = 90°$.

Reasoning: When two lines intersect, vertically opposite angles are equal, so we already have two pairs of equal angles. For all four to be equal, both pairs must be equal to each other. This happens when each angle is $90°$, i.e., the two lines are perpendicular to each other.

Answer: Yes, such a pair of lines can be drawn — they are perpendicular lines. Each of the four angles measures $\mathbf{90°}$.

Method (Activity):

Step 1: Identify each given line segment on the dot paper.

Step 2: For a line segment going horizontally (left–right), draw a line segment going vertically (up–down) through any point on it. These two directions are perpendicular on a rectangular dot grid.

Step 3: For a line segment inclined at $45°$ (going diagonally), draw a line segment inclined at $135°$ (the other diagonal direction) through any point on it — these are perpendicular to each other.

Step 4: Mark the right-angle symbol ($\square$) at the point of intersection to indicate perpendicularity.

Note: On a rectangular dot paper, two line segments are perpendicular if one goes $m$ dots across and $n$ dots up, while the other goes $n$ dots across and $m$ dots down (i.e., their direction vectors are $(m, n)$ and $(n, -m)$, whose dot product is $mn - mn = 0$).

Activity: (Based on Fig. 5.11 — a figure with several line segments)

- Mark lines that go in the same direction (same slope/orientation) with matching arrow symbols (→ → for one pair, ⇒ ⇒ for another pair, etc.).
- Mark the point where two lines meet at a right angle with a small square symbol $\square$.

(a) How to spot perpendicular lines:
Two lines are perpendicular if they meet and form a $90°$ angle. On dot paper, if one line goes $a$ steps right and $b$ steps up, a perpendicular line goes $b$ steps right and $a$ steps down (or vice versa). Visually, they form a perfect 'L' or '+' shape.

(b) How to spot parallel lines:
Two lines are parallel if they have the same direction/slope and never meet. On dot paper, if one line segment goes $a$ steps right and $b$ steps up for each unit, a parallel line also goes $a$ steps right and $b$ steps up. Visually, they look like they are always the same distance apart and point in the same direction.

Activity (Construction):

Set 1 — Horizontal parallel lines:
Choose two dots in the same row and connect them (horizontal segment). Then choose two dots in another row at the same horizontal positions and connect them. Both segments are horizontal, so they are parallel.

Set 2 — Vertical parallel lines:
Similarly, draw two vertical segments in different columns.

Set 3 — Diagonal parallel lines (slope 1):
Draw a segment from dot $(0,0)$ to dot $(2,2)$. Draw another from dot $(0,1)$ to dot $(2,3)$. Both go 2 right and 2 up — they are parallel.

**Set 4 — Diagonal parallel lines (slope $\frac{1}{2}$):**
Draw a segment from $(0,0)$ to $(4,2)$. Draw another from $(0,1)$ to $(4,3)$. Both go 4 right and 2 up — they are parallel.

Key idea: Two segments on dot paper are parallel if, for each segment, the number of dots moved horizontally and vertically (the 'rise over run') is the same.

Activity:

For each given line segment, identify its direction by counting how many dots it moves horizontally ($\Delta x$) and vertically ($\Delta y$). Then, starting from a different dot, draw a new segment that moves the same $\Delta x$ horizontally and $\Delta y$ vertically.

(a) Yes, some line segments are more challenging to draw parallel lines to.

(b) Line segments that are neither horizontal, nor vertical, nor at exactly $45°$ are harder. For example, a segment going 3 dots right and 1 dot up has a less obvious direction, and it is harder to replicate accurately by eye.

(c) Method used:
- Count the horizontal and vertical steps of the given segment (e.g., 3 right, 2 up).
- Start at a new dot and move the same number of steps (3 right, 2 up) to find the endpoint.
- Connect the new starting dot to the new endpoint.
- This ensures the new segment has the same slope, making it parallel to the original.

Given: Three lines $a$, $b$, $c$ are shown in Fig. 5.13. We must decide which of $b$ or $c$ is parallel to $a$.

Concept: Two lines are parallel if they have the same direction (same slope) and never intersect.

Method 1 — Visual/Slope comparison:
Compare the direction of each line with line $a$. The line ($b$ or $c$) that goes in exactly the same direction (same rise over run on the dot grid) as $a$ is parallel to $a$.

Method 2 — Using a transversal:
Draw a transversal (a line crossing all three lines). Measure the corresponding angles formed at the intersections. The line for which the corresponding angle equals the corresponding angle with line $a$ is parallel to $a$.

Answer: The line that has the same slope (same $\Delta x$ and $\Delta y$ per unit) as line $a$ is parallel to it. Based on typical figures of this type, **line $c$** is parallel to line $a$ (since $b$ has a different slope). The decision is made by checking that corresponding angles are equal or that the direction vectors match.