InterMath Dictionary
Angles
Parts of an Angle
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
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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Angles in Your World
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.
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.
Use the less than sign, <, to represent an angle symbol when you input your answer.
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.
a) <AMH, and b) <TMH
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. Inscribed quadrilateral: 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.
Types of Angles
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.
Types of Angles in Your World
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.
a) <CIE, b) <EIA, c) <BIC (region from B to C in counterclockwise direction) d) <CIA, and e) <AIB
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.
Examples of Inscribed Angles
Non-Examples of Inscribed Angles
Circles at Math.com: Scroll down the page to read another definition of an inscribed angle.
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.
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.
a) an interior angle of the quadrilateral, and b) an exterior angle of the quadrilateral.
Exterior Angles Conjectures: Read more about the exterior angles of polygons.
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. Top of Page
Pairs of Angles
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?
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.
a) <LCT and <TCR, b) <LCN and <NCP, and c) <LCT and <PCR
a) alternate interior angles, b) alternate exterior angles, and c) vertical angles
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.
Constructing quadrilaterals 1: Given two parallel lines, form various quadrilaterals.
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