What is an angle?

What is an angle?

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Definition of Angles

An angle is formed by two rays that share an endpoint called the vertex. Angles are measured in degrees. On ACT-style geometry questions, identify the angle relationship before doing arithmetic.

Angles

Different Types of Angles

  • An acute angle is between \(0^\circ\) and \(90^\circ\).
  • A right angle measures exactly \(90^\circ\).
  • An obtuse angle is between \(90^\circ\) and \(180^\circ\).
  • A straight angle measures exactly \(180^\circ\).

Angle Relationships

Complementary angles add to \(90^\circ\). Supplementary angles add to \(180^\circ\). Vertical angles are congruent. A linear pair is supplementary. Triangle angles add to \(180^\circ\), and a polygon with \(n\) sides has interior angle sum \((n - 2)180^\circ\).

Angles2

Example

If two angles form a linear pair and one angle is \(68^\circ\), the other is \(180^\circ - 68^\circ = 112^\circ\).

Angles

Think of this lesson as more than a rule to memorize. Angles is about shape relationships, formulas, and units. A strong student does not rush to the first formula on the page; they pause, identify the structure of the problem, and then choose the tool that matches that structure. That pause is what prevents most mistakes.

Geometry formulas work because they measure a specific feature: length around, space inside, or space enclosed by a solid. Match the question to the measurement first.

Here is the teacher way to approach the topic. First, read the problem slowly and underline the information that is actually given. Next, name the target: are you finding a value, simplifying an expression, comparing two quantities, solving for a variable, or interpreting a graph? Once the target is clear, the calculation becomes much less mysterious because every step has a job.

  • Draw the angle or triangle when possible.
  • Choose the ratio or conversion that matches the information.
  • Use exact special-angle values when they apply.
  • Check the quadrant or unit before finalizing.

A helpful habit is to say what each number represents before using it. For example, if a number is a denominator, a radius, a slope, a common difference, or a coefficient, it should not be treated like an ordinary loose number. Its role tells you where it belongs in the formula. This is especially important on ACT-style questions because many wrong answer choices come from using the right numbers in the wrong places.

Another good habit is to keep the work organized vertically. Write one clean line for substitution, one line for simplifying, and one line for the final answer. If the problem has units, keep the units attached. If the problem has signs, exponents, or parentheses, copy them carefully from one line to the next. Most errors in this topic are not caused by a hard idea; they are caused by dropping a negative sign, combining unlike terms, using the wrong denominator, or skipping a check.

When you finish, ask a quick reasonableness question. Should the answer be positive or negative? Should it be larger or smaller than the original number? Does it fit the graph, table, shape, or equation? Can you plug it back into the original problem? This final check turns the lesson from a procedure into understanding.

On a test, the goal is not to write the longest solution. The goal is to write enough clear work that you can see the structure, avoid traps, and recover quickly if one line goes wrong. Practice the examples below with that mindset: identify the type of problem, choose the matching rule, show the substitution, simplify carefully, and check the answer before moving on.

Free printable Worksheets

Exercises for Angles

1)  Classify an angle measuring \(42^\circ\).

2)  What is the complement of \(37^\circ\)?

3)  What is the supplement of \(112^\circ\)?

4)  Two angles form a linear pair. One is \(68^\circ\). Find the other.

5)  One vertical angle is \(124^\circ\). Find the opposite vertical angle.

6)  A triangle has angles \(42^\circ\) and \(67^\circ\). Find the third angle.

7)  On a straight line, adjacent angles are \(x^\circ\), \(35^\circ\), and \(80^\circ\). Find \(x\).

8)  Corresponding angles are \(75^\circ\) and \((3x + 15)^\circ\). Find \(x\).

9)  Alternate interior angles are \((2x + 10)^\circ\) and \((5x - 35)^\circ\). Find the angle measure.

10)  Same-side interior angles are \((4x + 8)^\circ\) and \((6x - 18)^\circ\). Find the larger angle.

11)  A quadrilateral has angles \(85^\circ\), \(92^\circ\), and \(104^\circ\). Find the fourth angle.

12)  Each exterior angle of a regular polygon is \(24^\circ\). How many sides does it have?

13)  What is each interior angle of a regular decagon?

14)  Two complementary angles are in the ratio \(2:3\). Find the larger angle.

15)  Two supplementary angles differ by \(34^\circ\). Find the larger angle.

16)  An exterior angle of a triangle is \(128^\circ\). One remote interior angle is \(47^\circ\). Find the other.

17)  An isosceles triangle has vertex angle \(38^\circ\). Find each base angle.

18)  What is the smaller angle between the hands of a clock at exactly 4:00?

19)  The sum of the interior angles of a polygon is \(1260^\circ\). How many sides does it have?

20)  Same-side interior angles are \((7x - 5)^\circ\) and \((5x + 17)^\circ\). Find the obtuse angle.

 
1) Compare \(42^\circ\) to \(90^\circ\).
It is greater than zero and less than a right angle.
Answer: \(\color{red}{\text{acute}}\)
2) Complements add to \(90^\circ\).
Compute \(90^\circ - 37^\circ = 53^\circ\).
Answer: \(\color{red}{53^\circ}\)
3) Supplements add to \(180^\circ\).
Compute \(180^\circ - 112^\circ = 68^\circ\).
Answer: \(\color{red}{68^\circ}\)
4) A linear pair is supplementary.
Compute \(180^\circ - 68^\circ = 112^\circ\).
Answer: \(\color{red}{112^\circ}\)
5) Vertical angles are congruent.
The opposite angle has the same measure.
Answer: \(\color{red}{124^\circ}\)
6) Triangle angles add to \(180^\circ\).
Compute \(180 - 42 - 67 = 71\).
Answer: \(\color{red}{71^\circ}\)
7) Angles on a line add to \(180^\circ\).
Set up \(x + 35 + 80 = 180\).
Solve \(x = 65\).
Answer: \(\color{red}{65}\)
8) Corresponding angles are congruent.
Set up \(3x + 15 = 75\).
Solve \(x = 20\).
Answer: \(\color{red}{20}\)
9) Alternate interior angles are congruent.
Solve \(2x + 10 = 5x - 35\) to get \(x = 15\).
Substitute: \(2(15) + 10 = 40\).
Answer: \(\color{red}{40^\circ}\)
10) Same-side interior angles are supplementary.
Set up \(4x + 8 + 6x - 18 = 180\), so \(x = 19\).
The angles are \(84^\circ\) and \(96^\circ\).
Answer: \(\color{red}{96^\circ}\)
11) Quadrilateral angles add to \(360^\circ\).
Compute \(360 - 85 - 92 - 104 = 79\).
Answer: \(\color{red}{79^\circ}\)
12) Exterior angles add to \(360^\circ\).
Compute \(360 \div 24 = 15\).
Answer: \(\color{red}{15}\)
13) A decagon has 10 sides.
Use \(\frac{(10 - 2)180}{10} = 144\).
Answer: \(\color{red}{144^\circ}\)
14) The total is \(90^\circ\) across \(5\) parts.
One part is \(18^\circ\).
The larger angle is \(3\cdot18^\circ = 54^\circ\).
Answer: \(\color{red}{54^\circ}\)
15) Let the smaller angle be \(x\); the larger is \(x + 34\).
Solve \(x + x + 34 = 180\) to get \(x = 73\).
The larger angle is \(107^\circ\).
Answer: \(\color{red}{107^\circ}\)
16) An exterior angle equals the sum of the two remote interior angles.
Solve \(47 + x = 128\).
So \(x = 81\).
Answer: \(\color{red}{81^\circ}\)
17) The base angles are congruent.
Remaining sum is \(180 - 38 = 142\).
Divide by 2: \(71^\circ\).
Answer: \(\color{red}{71^\circ}\)
18) Each hour mark is \(360^\circ \div 12 = 30^\circ\).
The hands are 4 marks apart: \(4\cdot30^\circ = 120^\circ\).
Answer: \(\color{red}{120^\circ}\)
19) Use \((n - 2)180 = 1260\).
Then \(n - 2 = 7\).
So \(n = 9\).
Answer: \(\color{red}{9}\)
20) Set up \(7x - 5 + 5x + 17 = 180\).
Solve \(x = 14\).
The angles are \(93^\circ\) and \(87^\circ\); the obtuse angle is \(93^\circ\).
Answer: \(\color{red}{93^\circ}\)

Angles Practice Quiz