InterMath Dictionary

Angles

 Acute Angle Inscribed Angle Right Angle Adjacent Angle Interior Angle Straight Angle Alternate Exterior Angle Linear Pair Supplementary Angles Alternate Interior Angle Obtuse Angle Transversal Angle Opposite Angles TVertex Angle Bisector Protractor Vertical Angles Complementary Angles Ray Corresponding Angles Degree of an Angle Reflex Angle Exterior Angle

Parts of an Angle
 Angle - The union of two line segments or two rays that have a common endpoint. Vertex - The point where two rays or two line segments meet. Ray - Part of a line that starts at one endpoint and continues in one direction.

The arrows on the sides of the angle tell you that you can extend the sides (These are called rays). The space between the two rays (the two sides) is what we call an angle. The point where the two rays meet is called the vertex.

An angle separates a plane into three parts: the interior of the angle, the exterior of the angle, and the angle itself. The exterior of the angle includes all the points in the plane not on the angle or in its interior. Neither the interior nor the exterior of an angle contains points on the angle.

Check for Understanding

 What letter represents the vertex of the angle shown to the right? Enter your answer here and press tab

External Links to Extend Basic Understanding

Angles at Hamilton's "Math to Build On": Read more about parts of angles and naming them.

Classifying Angles at Math.com: Read about and practice classifying and naming angles.

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Carpenters use many different tools to make sure that all the pieces of wood they are working with fit together. One tool they use is a miter box. A miter box guides the saw so that the correct angle is cut. You can see a corner that was cut with a miter box probably in your own home or at your school. For instance, look at the top of a wooden door frame (the wood around the opening of the door). A carpenter used a miter box to cut the wood so that the pieces fit together to form the corner of the frame.

Building picture frames is another area where using a miter box to cut angles is useful. Frame shop personnel also cut mats at angles to accent pictures and paintings.

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Naming an Angle

An angle can be named by its vertex, or it can be named using a point from each side and the vertex. So the angle below can be called angle B, angle ABC or angle CBA.

We can use the symbol to mean "angle." So, instead of saying angle B, we could say B.

Check for Understanding

What are three different ways to name the angle to the right?

Use the less than sign, <, to represent an angle symbol when you input your answer.

Enter each answer in the left column followed by pressing tab

External Links to Extend Basic Understanding

Angles at Hamilton's "Math to Build On": Read more about parts of angles and naming them.

Classifying Angles at Math.com: Read about and practice classifying and naming angles.

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 Angle Bisector - A ray dividing an angle into two equal parts.

A ray (BD) is an angle bisector of ABC if and only if D is in the interior of the angle and ABD is congruent to CBD.

Check for Understanding

Ray MA is an angle bisector of <TMH. If the measure of <TMA is 26 degrees, then state the measure of:

a) <AMH, and
b) <TMH

Enter each answer in the left column followed by pressing tab
 a)  degrees b)  degrees

External Links to Extend Basic Understanding

Harcourt's Animated Math Glossary: Read another definition of an angle bisector.

Related Investigations on InterMath to Challenge Teachers

Relationships: Explore various relationships in a given construction involving a parallelogram.
Angle bisector
: Explore relationships of parallelograms and angle bisectors.
: Do the angle bisectors of the opposite sides of an inscribed quadrilateral meet at right angles?
Angle Bisector
: Given a triangle and an angle bisector, determine proportions comparing the lengths of the various segments that form the triangle.
Moving Walls
: Determine the ideal position to stand when you are surrounded.

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Types of Angles
 Degree of an Angle - The unit for measuring angles. Protractor - A measuring device that can be used to approximate the measure of an angle.

An angle is classified by its measure (the number of degrees or amount space between the two rays forming the angle). Various classifications of angles are in the following table.

 Acute Angle Right Angle An acute angle is an angle whose measure is between 0and 90. As you can see an acute angle is bigger than 0and less than 90. A right angle forms a square corner and measures exactly 90. Obtuse Angle Straight Angle An obtuse angle is greater than 90but less than 180. A straight angle is formed by a straight line and measures exactly 180 Reflex Angle A reflex angle measures more than 180, but less than 360.

Types of Angles in Your World

 Buildings and architectural drawings often display good examples of right angles. There are a lot of right angles in the drawing of the Sears Tower, a skyscaper in Chicago, shown at right.

In the floor plan of an apartment shown below, locate a right angle, obtuse angle, and an acute angle. Why do you think there are fewer acute angles in the plan than right or obtuse angles?

Some buildings don't contain very many right angles, as you can see in the model for a Concert Hall below. The architect, Frank Gehry, is known for using unusual curved shapes in his designs.

Check for Understanding

Identify the types of angles in the diagram to the right:

a) <CIE,
b) <EIA,
c) <BIC (region from B to C in counterclockwise direction)
d) <CIA, and
e) <AIB

 a)  Choose one acute angle obtuse angle reflex angle right angle straight angle b)  Choose one acute angle obtuse angle reflex angle right angle straight angle c)  Choose one acute angle obtuse angle reflex angle right angle straight angle d)  Choose one acute angle obtuse angle reflex angle right angle straight angle e)  Choose one acute angle obtuse angle reflex angle right angle straight angle

External Links to Extend Basic Understanding

Classifying Angles at Math.com: Read about and practice classifying and naming angles.

Angles at Hamilton's "Math to Build On": Read more about parts of angles and naming them.

Related Investigations on InterMath to Challenge Teachers

Slopes and Angles of Elevation: Determine the relationship between the slope of a line and the angle the line makes with a horizontal line.
Tessellation Restrictions: Determine which polygons can tessellate.
Sum of Angles in a Polygon: Explore the relationship between the number of sides of a polygon and the sum of its interior angles.
Twice Reflected over Intersecting Lines: Find an equation which relates the angles formed by intersecting lines and points reflected over these lines.
Angles in a Circle: Investigate how the location of an angle's vertex affects its relationship to the measure(s) of the arc(s) it intercepts.

 Inscribed Angle - An angle is an inscribed angle if and only if its vertex lies on a circle and its sides contain chords of the circle.

Examples of Inscribed Angles

Non-Examples of Inscribed Angles

 This angle is not inscribed because although the angle's vertex lies on a circle, one of its sides does not contain a chord of the circle. This angle is not inscribed because the angle's vertex does not lies on a circle.

Check for Understanding

 Which of the following angles is an inscribed angle? Enter your answer to the left and press tab <

External Links to Extend Basic Understanding

Circles at Math.com: Scroll down the page to read another definition of an inscribed angle.

Related Investigations on InterMath to Challenge Teachers

Angles in a Circle: Investigate how the location of an angle's vertex affects its relationship to the measure(s) of the arc(s) it intercepts.

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Location, Location, Location
 Interior Angle - An interior angle of a polygon is formed by two adjacent sides. Also, when a third line (a transversal) crosses two other lines, the angles formed in between the other two lines are called interior angles. (see alternate interior angles) Exterior Angle - An exterior angle of a polygon is formed when you extend a side of a polygon. Also, when a third line (a transversal) crosses two other lines, the angles formed outside the other two lines are called exterior angles. (see alternate exterior angles)

An angle is an exterior angle of a polygon if and only if it forms a linear pair with one of the angles of the polygon.

An interior angle of a polygon is formed by two consecutive sides of a polygon. You will sometimes see an interior angle called a vertex angle.

Check for Understanding

Use three letters to identify:

a) an interior angle of the quadrilateral, and
b) an exterior angle of the quadrilateral.

Use the less than sign, <, to represent an angle symbol when you input your answer.

Enter each answer in the left column followed by pressing tab
 a) b)

External Links to Extend Basic Understanding

Related Investigations on InterMath to Challenge Teachers

Sum of Angles in a Polygon: Explore the relationship between the number of sides of a polygon and the sum of its interior angles.
Sum of Exterior Angles: Determine the sum of the exterior angles in any polygon.
Interior angles
: Explore the sum of interior angles of a quadrilateral.
Outside Looking In: Identify the sum of the exterior angles in a triangle.
Remotely Interior
: Compare exterior angles with remote interior angles.

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Pairs of Angles

 Adjacent angles are angles in the same plane that have a common vertex and a common side but no common interior points. For example, angle HKF and angle FKI are adjacent angles. Can you find some more pairs of adjacent angles in the diagram? How many adjacent angles are in the picture at the left? Vertical angles (or opposite angles) are two nonadjacent angles formed by intersecting lines. Their sides form two lines (EF & GH); the vertical angles are the ones opposite each other (EKG & HKF). Notice that the angles have only one point in common, the vertex (K). How many pairs of vertical angles are in this diagram? What is the relationship between the angles opposite each other? Congruent angles are the same size; they have exactly the same measure. Perhaps you noticed that all vertical angles are also congruent angles. Complementary angles are two angles whose sum is 90. Notice that at all times angle BAT + angle CAT = 90.The diagram to the left is the typical representation of complementary angle but the angles do not have to be adjacent as shown here to be complementary. Supplementary angles are two angles whose sum in 180. Notice that at all times angle ABG + angle CBG = 180. If they are the angles are adjacent (as in this diagram) they may be called a linear pair because they form a straight line. In the figure at the left, angles 1, 2, 3, and 4 are exterior angles. Alternate exterior angles are pairs of angles formed when a third line (a transversal) crosses two other lines. These angles are on opposite sides of the transversal and are outside the other two lines. Angles 1 and 4 are alternate exterior angles. Angles 2 and 3 are also alternate exterior angles. Line t (red) is called a transversal, a line crossing two or more lines. When the transversal cuts through 2 parallel lines, the alternate exterior angles formed have a special relationship. Using GSP, what do you notice about this situation? In the figure at the left, angles 1,2,3, and 4 are interior angles. Alternate interior angles are pairs of angles formed when a third line (a transversal) crosses two other lines. These angles are on opposite sides of the transversal and are in between the other two lines. Angles 1 and 4 are alternate interior angles. Angles 2 and 3 are also alternate interior angles. Line t (red) is called a transversal, a line crossing two or more lines. When the transversal cuts through 2 parallel lines, the alternate interior angles formed have a special relationship. Using GSP, what do you notice in this situation? In the figure at the left, angles 1 and 3 are corresponding angles. Angles 2 and 4 are also corresponding angles. Corresponding angles are pairs of angles formed when a third line (a transversal) crosses two other lines. These angles are on the same side of the transversal and are in the same relative position. For example, both angles 1 and 3 are in the upper left position at each intersection. Line t (red) is called a transversal, a line crossing two or more lines. When the transversal cuts through 2 parallel lines, the alternate interior angles formed have a special relationship. Using GSP, what do you notice in this situation?

Pairs of Angles in Your World

The railing on a set of stairs provides a good example of corresponding angles. Can you spot pairs of corresponding angles in the picture below?

The use of alternate interior angles allows an ironing board to stay parallel to the floor or a clothes rack to stay parallel to a wall.

Check for Understanding

Identify the types of angle pairs in the diagram to the right:

a) <LCT and <TCR,
b) <LCN and <NCP, and
c) <LCT and <PCR

 a)  Choose one adjacent angles vertical angles congruent angles complementary angles supplementary angles AND     Choose one adjacent angles vertical angles congruent angles complementary angles supplementary angles b)  Choose one adjacent angles vertical angles congruent angles complementary angles supplementary angles AND     Choose one adjacent angles vertical angles congruent angles complementary angles supplementary angles c)  Choose one adjacent angles vertical angles congruent angles complementary angles supplementary angles AND     Choose one adjacent angles vertical angles congruent angles complementary angles supplementary angles

Check for Understanding

Identify a pair of:

a) alternate interior angles,
b) alternate exterior angles, and
c) vertical angles

Enter each answer on the left followed by pressing tab
 a) < and< a) < and< a) < and<

External Links to Extend Basic Understanding

Angles and Intersecting Lines at Math.com: Read more about and practice naming angles formed by intersecting lines.

Angles and angle terms: Read explanations of different types of angles.

Related Investigations on InterMath to Challenge Teachers