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Here are a few recommended readings before getting started with this lesson.
An angle is a plane figure formed by two rays that have the same starting point. This common point is called the vertex of the angle and the rays are the sides of the angle.
There are different ways to denote an angle and all involve the symbol ∠
in front of the name. One way is to name an angle by its vertex alone. Alternatively, it can be named by using all three points that make up the angle. In this case, the vertex is always in the middle of the name. Additionally, angles within a diagram can be denoted with numbers or lowercase Greek letters.
Using the Vertex | Using the Vertex and One Point on Each Ray | Using a Number | Using Greek Letters |
---|---|---|---|
∠B | ∠ABC or ∠CBA | ∠1 | ∠α or ∠β or ∠θ |
The measure of an angle, denoted by m∠, is the number of degrees between the rays. It is found by applying the Protractor Postulate.
An angle divides the plane into two parts.
interiorof the angle
exteriorof the angle
The measure of an angle can range from 0∘ to 360∘ or from 0 to 2π radians, a unit of measure that will be studied in the future. Angles can be classified according to their measures into six different types.
An acute angle is an angle whose measure is greater than 0∘ but less than 90∘.
As with right angles, the following type of angle involves only those angles whose measure is exactly 180∘.
A straight angle is an angle whose measure is exactly 180∘.
The fifth type of angle includes angles whose measure is greater than 180∘ but less than 360∘. This is the largest range of measures.
A reflexive angle is an angle whose measure is greater than 180∘ but less than 360∘. An alternative name for this type of angle is reflex angle.
As time passes, the hands of a clock form different angles. Classify the indicated angle by estimating its measure.
When a laser is pointed at a mirror, the light beam is reflected in such a way that the angle between the incident beam and the mirror measures the same as the angle between the reflected beam and the mirror.
In the diagram, ∠1 and ∠3 have the same measure. Angles with the same measure have a special name.
Angles can also be classified based on their position relative to other angles.
Going back to the diagram of the laser and the mirror, notice that the point where the beam hits the mirror is the vertex of three angles.
The incident beam is a common side to ∠1 and ∠2, so these angles are adjacent. Similarly, ∠2 and ∠3 are also adjacent angles because they share the side made by the reflected beam. Next, a third way of relating a pair of angles is presented.Two angles are vertical angles if they are opposite angles formed by the intersection of two lines or line segments. In the diagram, vertical angles are marked with the same number of angle markers.
Vertical angles are always congruent.
Based on the characteristics of the diagram, the following relations hold true.
∠1≅∠3
∠2≅∠4
In the diagram, lines AD and CE intersect at point X, and B is a point on the interior of ∠AXC.
Congruent Angles | Adjacent Angles | Vertical Angles |
---|---|---|
∠AXB and ∠BXC ∠CXD and ∠EXA ∠DXE and ∠AXC |
∠AXB and ∠BXC ∠BXC and ∠CXD ∠CXD and ∠DXE ∠DXE and ∠EXA ∠EXA and ∠AXB |
∠CXD and ∠EXA ∠DXE and ∠AXC |
Since ∠AXB and ∠BXC have the same number of markers, the angles are congruent.
Congruent Angles |
---|
∠AXB and ∠BXC |
Next, focus on identifying adjacent angles. Adjacent angles have the same vertex, share one side, and they do not overlap. These three conditions are met by ∠DXE and ∠EXA.
In the diagram there are five pairs of adjacent angles.
Adjacent Angles |
---|
∠AXB and ∠BXC |
∠BXC and ∠CXD |
∠CXD and ∠DXE |
∠DXE and ∠EXA |
∠EXA and ∠AXB |
Finally, look for vertical angles. Vertical angles are opposite angles formed when two lines or line segments intersect. Since lines AD and CE intersect at X, they form two pairs of vertical angles. To make it easier to see, ignore the unnecessary parts of the diagram and focus on just these two lines.
From the diagram, ∠CXD and ∠EXA are vertical angles, as are ∠DXE and ∠AXC.
Vertical Angles |
---|
∠CXD and ∠EXA |
∠DXE and ∠AXC |
Because vertical angles are always congruent, the last pairs of angles are also congruent angles. All the information obtained from the diagram is summarized in the following table.
Congruent Angles | Adjacent Angles | Vertical Angles |
---|---|---|
∠AXB and ∠BXC ∠CXD and ∠EXA ∠DXE and ∠AXC |
∠AXB and ∠BXC ∠BXC and ∠CXD ∠CXD and ∠DXE ∠DXE and ∠EXA ∠EXA and ∠AXB |
∠CXD and ∠EXA ∠DXE and ∠AXC |
The angles have the same vertex but they do not have a common side. Therefore, they are not adjacent angles. Notice that ∠DXE has no angle marker and its measure seems to be greater than the measure of ∠BXC. Therefore, the angles are not congruent.
∠BXC and ∠DXE | ||
---|---|---|
Adjacent | Congruent | Vertical |
× | × | ? |
Lastly, note that E, X, and C lie on the same line but D, X, and B do not. Therefore, ∠BXC and ∠DXE are not vertical angles. As such, there is no relation between these angles.
∠BXC and ∠DXE | ||
---|---|---|
Adjacent | Congruent | Vertical |
× | × | × |
In addition to adjacent, vertical, and congruent angles, pairs of angles can be classified in three more ways based on the sum of their measures.
Notice that if two angles are complementary, they are by necessity acute angles. Also, if two complementary angles are adjacent, the angle formed by the not common sides is a right angle.
When the clock shows 09:00 and 55 seconds, the angle between the minute hand and the second hand is 30∘, while the angle between the minute hand and the hour hand is 90∘.
If the angle between the second hand and the hour hand is 5x∘, what is the value of x?The angle formed by the minute hand and the second hand is complementary to the angle formed by the second hand and the hour hand.
Start by marking a few points on the diagram to make it easy to reference its parts.
Note that ∠JKL and ∠LKM together make a right angle. As a result, these angles are complementary. This means that the sum of their measures is 90∘.