Angle

What Is an Angle?

An angle is a figure formed by two rays that share a common endpoint. This common endpoint is called the vertex, and the two rays are called the sides or arms of the angle. Think of the hands on a clock; they form an angle with the vertex at the center of the clock.

We measure angles to describe the amount of turn or rotation between the two arms. The most common unit for measuring angles is the degree, and we use the symbol ${}^{\circ }$ to represent it. Imagine one ray is fixed and the other rotates around the vertex. If it rotates all the way around to where it started, it has completed a full circle, which we define as ${360}^{\circ }$.

• A quarter turn creates a ${90}^{\circ }$ angle.
• A half turn creates a ${180}^{\circ }$ angle.
• A three-quarter turn creates a ${270}^{\circ }$ angle.

An important thing to remember is that the length of the rays does not affect the measure of the angle. An angle is about the openness between the rays, not how long they are drawn. An angle with short arms can have the exact same degree measure as an angle with very long arms.

How Do We Name Angles?

To talk about angles, we need a clear way to name them. There are three common methods, and the one you use often depends on how complex the diagram is.

1. Using Three Points: This is the most specific and reliable method. You pick one point on each ray and use the vertex as the middle point. For an angle with points A on one ray, B as the vertex, and C on the other ray, you could name it $\mathrm{\angle }ABC$ or $\mathrm{\angle }CBA$. The middle letter must be the vertex.
2. Using Only the Vertex: If there is only one angle at a particular vertex, you can name the angle using just that single letter. For the same angle above, if no other angles share vertex B, you could simply call it $\mathrm{\angle }B$. Be careful with this method; if multiple angles share the same vertex, this name would be ambiguous.
3. Using a Number or Symbol: Sometimes, an angle is labeled with a number or a letter inside its arc. You might see an angle labeled as $\mathrm{\angle }1$ or $\mathrm{\angle }x$. This is a simple way to refer to angles in diagrams with many angles, saving you from writing out three letters every time.

Being precise with your naming is a key skill in geometry. Using three points is almost always the safest and clearest option.

What Are the Main Types of Angles?

Angles are categorized based on their degree measure. Understanding these types is the first step to mastering geometry problems. Each category tells you something important about the angle's size and shape.

Most of the time in 7th and 8th grade, you will be working with acute, right, obtuse, and straight angles. But it's good to know all the classifications!

Understanding Special Angle Pairs

Angles often come in pairs that have special relationships. Identifying these pairs is the key to solving many geometry problems.

Adjacent Angles

Two angles are adjacent if they share a common vertex and a common side, but do not overlap. Think of them as being 'next to' each other.

Complementary Angles

Two angles are complementary if their measures add up to ${90}^{\circ }$. They don't have to be adjacent. If one angle is ${30}^{\circ }$, its complement is ${60}^{\circ }$ because ${30}^{\circ }+{60}^{\circ }={90}^{\circ }$.

Supplementary Angles

Two angles are supplementary if their measures add up to ${180}^{\circ }$. A common example is two angles that form a straight line. If one angle is ${110}^{\circ }$, its supplement is ${70}^{\circ }$ because ${110}^{\circ }+{70}^{\circ }={180}^{\circ }$.

Vertical Angles

When two lines intersect, they form four angles. The angles that are directly opposite each other are called vertical angles. A key property of vertical angles is that they are always equal, or congruent.

What Happens When a Line Crosses Parallel Lines?

Things get really interesting when we have parallel lines—two lines on a plane that never intersect—and a third line, called a transversal, that crosses them. This setup creates eight angles, and they are related in very predictable ways. If you know the measure of just one angle, you can find all the others!

Let's look at the angle pairs formed:

• Corresponding Angles: These are angles in the same relative position at each intersection. For example, the top-left angle at the first intersection corresponds to the top-left angle at the second intersection. Corresponding angles are equal.
• Alternate Interior Angles: These are on opposite (alternate) sides of the transversal and are in between (interior) the parallel lines. Alternate interior angles are equal.
• Alternate Exterior Angles: These are on opposite sides of the transversal and are outside (exterior) the parallel lines. Alternate exterior angles are equal.
• Consecutive Interior Angles: Also called Same-Side Interior Angles. They are on the same side of the transversal and are between the parallel lines. Consecutive interior angles are supplementary (they add up to ${180}^{\circ }$).

Example 3

In the diagram below, line $m$ is parallel to line $n$, and they are cut by transversal line $t$. If $\mathrm{\angle }1={125}^{\circ }$, find the measures of all other angles.

Solution:

We are given $\mathrm{\angle }1={125}^{\circ }$. Let's find the other angles step-by-step using the relationships we just learned.

1. Find $\mathrm{\angle }2$: $\mathrm{\angle }1$ and $\mathrm{\angle }2$ form a straight line, so they are supplementary.
   $$\mathrm{\angle }1+\mathrm{\angle }2={180}^{\circ }$$ $${125}^{\circ }+\mathrm{\angle }2={180}^{\circ }$$ $$\mathrm{\angle }2={180}^{\circ }-{125}^{\circ }={55}^{\circ }$$
2. Find $\mathrm{\angle }3$: $\mathrm{\angle }1$ and $\mathrm{\angle }3$ are vertical angles.
   $$\mathrm{\angle }3=\mathrm{\angle }1={125}^{\circ }$$
3. Find $\mathrm{\angle }4$: $\mathrm{\angle }2$ and $\mathrm{\angle }4$ are vertical angles.
   $$\mathrm{\angle }4=\mathrm{\angle }2={55}^{\circ }$$
4. Find $\mathrm{\angle }5$: $\mathrm{\angle }1$ and $\mathrm{\angle }5$ are corresponding angles.
   $$\mathrm{\angle }5=\mathrm{\angle }1={125}^{\circ }$$
5. Find $\mathrm{\angle }6$: $\mathrm{\angle }2$ and $\mathrm{\angle }6$ are corresponding angles.
   $$\mathrm{\angle }6=\mathrm{\angle }2={55}^{\circ }$$
6. Find $\mathrm{\angle }7$: $\mathrm{\angle }5$ and $\mathrm{\angle }7$ are vertical angles (or $\mathrm{\angle }1$ and $\mathrm{\angle }7$ are alternate exterior angles).
   $$\mathrm{\angle }7=\mathrm{\angle }5={125}^{\circ }$$
7. Find $\mathrm{\angle }8$: $\mathrm{\angle }6$ and $\mathrm{\angle }8$ are vertical angles (or $\mathrm{\angle }2$ and $\mathrm{\angle }8$ are alternate exterior angles).
   $$\mathrm{\angle }8=\mathrm{\angle }6={55}^{\circ }$$

So, the angles are: $\mathrm{\angle }1={125}^{\circ }$, $\mathrm{\angle }2={55}^{\circ }$, $\mathrm{\angle }3={125}^{\circ }$, $\mathrm{\angle }4={55}^{\circ }$, $\mathrm{\angle }5={125}^{\circ }$, $\mathrm{\angle }6={55}^{\circ }$, $\mathrm{\angle }7={125}^{\circ }$, and $\mathrm{\angle }8={55}^{\circ }$. Notice there are only two different angle measures in the entire diagram: ${125}^{\circ }$ and ${55}^{\circ }$.

Common Mistakes to Avoid with Angles

When working with angles, a few common slip-ups can lead to the wrong answer. Being aware of them is the best way to avoid making them!

• Confusing Complementary and Supplementary: This is a classic mix-up. Remember, C comes before S in the alphabet, just like 90 comes before 180. Complementary angles add to ${90}^{\circ }$, and Supplementary angles add to ${180}^{\circ }$.
• Naming Angles Incorrectly: When using the three-letter naming system (like $\mathrm{\angle }ABC$), the vertex must be the middle letter. Writing $\mathrm{\angle }BAC$ would name a completely different angle.
• Assuming Lines are Parallel: Never assume two lines are parallel just because they look parallel in a diagram. You can only use the special angle relationships (like corresponding and alternate interior angles) if the problem states the lines are parallel or if they are marked with parallel symbols (like arrows on the lines).
• Thinking Angle Size Depends on Ray Length: The measure of an angle is about the rotation between the rays, not their length. A ${30}^{\circ }$ angle is a ${30}^{\circ }$ angle whether its sides are an inch long or a mile long.
• Mixing Up Vertical and Adjacent Angles: Vertical angles are opposite each other and are equal. Adjacent angles are next to each other and share a side. They are not necessarily equal, but if they form a straight line, they are supplementary.

Quick Summary and Reference

Here is a quick reference guide to the most important concepts about angles. Use this to review before a test or when you're stuck on a problem.

Key Angle Types

Key Angle Pair Relationships