Angles and Parallel Lines

This article explains the basics of plane geometry for 8th-grade students. Learn about the relationship between angles and parallel lines, and understand the concepts of vertical angles, corresponding angles, and alternate angles. Use interactive diagrams to visually learn and confirm the properties and conditions of parallel lines. This content provides important foundational knowledge for solving geometric problems.

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Angles

Here, we'll explain the important concepts of plane geometry: angles and parallel lines . Understanding this chapter will provide you with the foundation to solve more complex geometric problems.

Vertical Angles

First, let's draw two lines on a plane so that they intersect. This creates four angles, right? Among these four angles, the two angles that face each other, like $\angle a$ and $\angle c$ in the figure below, are called vertical angles .

Try moving the point in the figure below to change the angles formed by the intersection of the two lines.

No matter how you change the intersection of the two lines, the vertical angles are always equal.

$\angle a$ and $\angle d$ form a straight line, so $\angle a + \angle d = 180^\circ$ . Similarly, $\angle c$ and $\angle b$ form a straight line, so $\angle c + \angle b = 180^\circ$ .

$$\begin{align*} \angle a + \angle d = 180^\circ \\ \angle c + \angle d = 180^\circ \end{align*}$$

Therefore,

$$\begin{align*} \angle a &= 180^\circ - \angle d \\ \angle c &= 180^\circ - \angle d \\ \angle a &= \angle c \end{align*}$$

This means that vertical angles are always equal .

Corresponding Angles and Alternate Angles

In the figure below, when a line $n$ intersects two lines $l$ and $m$ , angles in the same position, like $\angle a$ and $\angle c$ , are called corresponding angles . Angles on opposite sides of line $n$ , like $\angle d$ and $\angle f$ , are called alternate angles . Unlike vertical angles, corresponding angles and alternate angles are not necessarily equal.

Parallel Lines

Properties of Parallel Lines

We'll confirm the properties of parallel lines and the conditions for lines to be parallel by examining what happens to corresponding angles and alternate angles when two lines are parallel.

Corresponding Angles on Parallel Lines

So, what happens to corresponding angles when two lines are parallel? In the figure below, line $l$ and line $m$ are parallel. Try moving the point to change the size of the corresponding angles $\angle a$ and $\angle b$ .

When two lines are parallel, corresponding angles are equal .

Alternate Angles on Parallel Lines

Next, let's check what happens to alternate angles when two lines are parallel. In the figure below, the two lines $l$ and $m$ are parallel. The corresponding angle to $\angle t$ is shown in green, and the alternate angle is shown in red.

We know that corresponding angles are always equal when two lines are parallel. The alternate angle is the vertical angle to the corresponding angle. This means that when two lines are parallel, not only are the corresponding angles equal, but the alternate angles are also always equal .