If you think that you are the master of angles then try this
Some of you may have run into those problems where you have to figure out the angle between the two hands of a clock at weird times. Well, I took that problem as a starting point and added a twist to it.
Consider a 12-hour analog clock with two hands and a round face. Consider the angle between the two hands at any given time and, when the angle between the hands is not 180 degrees, take the smaller of the two angles. Thus, at 12:00 the angle between the two hands is 0 degrees. At 3:00 and at 9:00 it’s 90 degrees.
If we me measure the angle between the two hands at each of the 61 consecutive minutes between 12:00 and 1:00 inclusively, what is the sum of those 61 angles?
Tuesday, March 31, 2009
What is an Angle?
Two rays that share the same endpoint form an angle. The point where the rays intersect is called the vertex of the angle. The two rays are called the sides of the angle.
Example: Here are some examples of angles.
We can specify an angle by using a point on each ray and the vertex. The angle below may be specified as angle ABC or as angle CBA; you may also see this written as ABC or as CBA. Note how the vertex point is always given in the middle.
Example: Many different names exist for the same angle. For the angle below, PBC, PBW, CBP, and WBA are all names for the same angle.
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Degrees: Measuring Angles
We measure the size of an angle using degrees.
Example: Here are some examples of angles and their degree measurements.
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Acute Angles
An acute angle is an angle measuring between 0 and 90 degrees.
Example:
The following angles are all acute angles.
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Obtuse Angles
An obtuse angle is an angle measuring between 90 and 180 degrees.
Example:
The following angles are all obtuse.
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Right Angles
A right angle is an angle measuring 90 degrees. Two lines or line segments that meet at a right angle are said to be perpendicular. Note that any two right angles are supplementary angles (a right angle is its own angle supplement).
Example:
The following angles are both right angles.
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Complementary Angles
Two angles are called complementary angles if the sum of their degree measurements equals 90 degrees. One of the complementary angles is said to be the complement of the other.
Example:
These two angles are complementary.
Note that these two angles can be "pasted" together to form a right angle!
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Supplementary Angles
Two angles are called supplementary angles if the sum of their degree measurements equals 180 degrees. One of the supplementary angles is said to be the supplement of the other.
Example:
These two angles are supplementary.
Note that these two angles can be "pasted" together to form a straight line!
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Vertical Angles
For any two lines that meet, such as in the diagram below, angle AEB and angle DEC are called vertical angles. Vertical angles have the same degree measurement. Angle BEC and angle AED are also vertical angles.
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Alternate Interior Angles
For any pair of parallel lines 1 and 2, that are both intersected by a third line, such as line 3 in the diagram below, angle A and angle D are called alternate interior angles. Alternate interior angles have the same degree measurement. Angle B and angle C are also alternate interior angles.
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Alternate Exterior Angles
For any pair of parallel lines 1 and 2, that are both intersected by a third line, such as line 3 in the diagram below, angle A and angle D are called alternate exterior angles. Alternate exterior angles have the same degree measurement. Angle B and angle C are also alternate exterior angles.
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Corresponding Angles
For any pair of parallel lines 1 and 2, that are both intersected by a third line, such as line 3 in the diagram below, angle A and angle C are called corresponding angles. Corresponding angles have the same degree measurement. Angle B and angle D are also corresponding angles.
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Angle Bisector
An angle bisector is a ray that divides an angle into two equal angles.
Example:
The blue ray on the right is the angle bisector of the angle on the left.
The red ray on the right is the angle bisector of the angle on the left.
________________________________________
Perpendicular Lines
Two lines that meet at a right angle are perpendicular.
Two rays that share the same endpoint form an angle. The point where the rays intersect is called the vertex of the angle. The two rays are called the sides of the angle.
Example: Here are some examples of angles.
We can specify an angle by using a point on each ray and the vertex. The angle below may be specified as angle ABC or as angle CBA; you may also see this written as ABC or as CBA. Note how the vertex point is always given in the middle.
Example: Many different names exist for the same angle. For the angle below, PBC, PBW, CBP, and WBA are all names for the same angle.
________________________________________
Degrees: Measuring Angles
We measure the size of an angle using degrees.
Example: Here are some examples of angles and their degree measurements.
________________________________________
Acute Angles
An acute angle is an angle measuring between 0 and 90 degrees.
Example:
The following angles are all acute angles.
________________________________________
Obtuse Angles
An obtuse angle is an angle measuring between 90 and 180 degrees.
Example:
The following angles are all obtuse.
________________________________________
Right Angles
A right angle is an angle measuring 90 degrees. Two lines or line segments that meet at a right angle are said to be perpendicular. Note that any two right angles are supplementary angles (a right angle is its own angle supplement).
Example:
The following angles are both right angles.
________________________________________
Complementary Angles
Two angles are called complementary angles if the sum of their degree measurements equals 90 degrees. One of the complementary angles is said to be the complement of the other.
Example:
These two angles are complementary.
Note that these two angles can be "pasted" together to form a right angle!
________________________________________
Supplementary Angles
Two angles are called supplementary angles if the sum of their degree measurements equals 180 degrees. One of the supplementary angles is said to be the supplement of the other.
Example:
These two angles are supplementary.
Note that these two angles can be "pasted" together to form a straight line!
________________________________________
Vertical Angles
For any two lines that meet, such as in the diagram below, angle AEB and angle DEC are called vertical angles. Vertical angles have the same degree measurement. Angle BEC and angle AED are also vertical angles.
________________________________________
Alternate Interior Angles
For any pair of parallel lines 1 and 2, that are both intersected by a third line, such as line 3 in the diagram below, angle A and angle D are called alternate interior angles. Alternate interior angles have the same degree measurement. Angle B and angle C are also alternate interior angles.
________________________________________
Alternate Exterior Angles
For any pair of parallel lines 1 and 2, that are both intersected by a third line, such as line 3 in the diagram below, angle A and angle D are called alternate exterior angles. Alternate exterior angles have the same degree measurement. Angle B and angle C are also alternate exterior angles.
________________________________________
Corresponding Angles
For any pair of parallel lines 1 and 2, that are both intersected by a third line, such as line 3 in the diagram below, angle A and angle C are called corresponding angles. Corresponding angles have the same degree measurement. Angle B and angle D are also corresponding angles.
________________________________________
Angle Bisector
An angle bisector is a ray that divides an angle into two equal angles.
Example:
The blue ray on the right is the angle bisector of the angle on the left.
The red ray on the right is the angle bisector of the angle on the left.
________________________________________
Perpendicular Lines
Two lines that meet at a right angle are perpendicular.
Tuesday, March 17, 2009
Tuesday, March 10, 2009
Tuesday, February 24, 2009
Welcome to our blog
Hello our names are Alisha and Verity and we are in class 6. We are doing a project on angles.We will be updating everything we know onto this blog. We hope you find this useful and interesting.
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