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Notes on Angle Relationships Angle: Definition: A shape formed by two lines or rays diverging from a common point (the vertex). In easier terms: is a figure that is formed from two rays that extend from a common point called the vertex. Vertex Angle B is the name of the angle Angle ABC is also the name of the angle B is the vertex The vertex is the common point at which the two lines or rays are joined. Point B in the figure above is the vertex of the angle ∠ABC. Legs The legs (sides) of an angle are the two lines that make it up. In the figure above, the lines AB and BC are the legs of the angle ∠ABC. Interior The interior of an angle is the space in the 'jaws' of the angle extending out to infinity. Exterior All the space on the plane that is not the interior. The symbol m∠ is used to denote the measure of an angle. The unit of measure for angles is in degrees. There are 360 degrees around a circle. Type of angles Acute angle Right angle Obtuse angle Straight angle Reflex angle description an angle that is less than 90° an angle that is 90° (exactly) an angle that is greater than 90° but less than 180° an angle that is 180° (exactly) an angle that is greater than 180° In One Diagram This diagram might make it easier to remember: Also: Acute, Obtuse and Reflex are in alphabetical order. Be Careful What You Measure This is an Obtuse Angle.And this is a Reflex Angle. But the lines are the same ... so when naming the angles make sure that you know which angle is being asked for! The measure of an angle is the smallest amount of rotation about the vertex from one ray to the other, measured in degrees. According to this definition, the measure of an angle can be any value between 0° and 180°. The largest amount of rotation less than 360° between the two rays is called the reflex measure of an angle. Angle Relationships: Supplementary angles: These two angles (140° and 40°) are Supplementary Angles, because they add up to 180°. Notice that together they make a straight angle. But the angles don't have to be together. These two are supplementary because 60° + 120° = 180° Complementary: Two Angles are Complementary if they add up to 90 degrees (a Right Angle). These two angles (40° and 50°) are Complementary Angles, because they add up to 90°. Notice that together they make a right angle. But the angles don't have to be together. These two are complementary because 27° + 63° = 90° Right Angled Triangle In a right angled triangle, the two nonright angles are complementary, because in a triangle the three angles add to 180°, and 90° has already been taken by the right angle. Complementary vs Supplementary. "C" of Complementary stands for "Corner" (a Right Angle), and "S" of Supplementary stands for "Straight" (180 degrees is a straight line) Adjacent angles Definition: two coplanar angles with a common side, a common vertex, and no common interior points. In other words:Two angles that share a common side and a common vertex, but do not overlap Adjacent angles: angle 1 and 2 are adjacent They may or may not be supplementary Linear Pairs Definition: Two angles that are adjacent (share a leg) and supplementary (add up to 180°) Angle 3 and 4 are linear pairs Vertical Angles Definition: A pair of non-adjacent angles formed by the intersection of two straight lines. They are congruent. In other words: Vertical angles are the angles that are opposite each other when two lines intersect. (Technically, these two lines need to be on the same plane) In the picture on the left, 1 and 2 are vertical angles. Likewise, A and B are vertical. Vertical angles are always congruent (equal). Angle bisector Definition: it is a line or ray passing through the vertex of the angle that cuts it into two equal smaller angles. In other words: A line which cuts an angle into two equal halves In general 'to bisect' something means to cut it into two equal parts. The 'bisector' is the thing doing the cutting.