Unformatted text preview: Angles An angle as defined in Euclidean (standard) Geometry is a general measure of the space between two intersecting lines. More explicitly, the larger the angle, the wider the opening two lines create. Angles are usually measured in° for geometry, but in radians for trigonometry later on. There are five general classifications of angles, namely: acute (measure between 0 and 90°), right (90° exactly), obtuse (between 90 and 180°), reflex (greater than 180°). Two other specific cases for angles are those that form a straight line (180°) and go full
circle (360°). Moreover, we say that angles are congruent if their measures, or values, are equal to one another. For example, the angles in a rectangle are all congruent to one another, each measuring 90° (right angle). Vertical angle theorem When two lines intersect (and thereby create 4 regions of intersection), the states that opposite angles are congruent. In the figure below, angles a and b have equal measure. Retrieved from: Definition of an Angle Bisector An angle bisector is a line originating from the vertex (intersection of the two lines creating the angle) that cuts the angle into two equal halves. In the figure below, line BD is an angle bisector and cuts angle CBA into two equal parts (angles CBD and DBA) of measure 65°. Retrieved from: Supplementary and Complementary Angles Finally, we say that angles are supplementary if their sum is 180° and complementary if their sum is 90°. Note that it has to be strictly two angles only, and therefore the angles in a triangle are not supplementary, despite summing to 180°. If two supplementary angles are found adjacent to one another, they are called a linear pair, as a straight line is indeed formed. Supplementary Angle: Complementary Angle: Retrieved from: Retrieved from: Oftentimes, questions involving supplementary and complementary angles are algebraic in nature and require one to solve for the value of x. Angles in Parallel Lines There are three main properties about angles in parallel lines cut by a transversal, which is any line intersecting both parallel lines. They are as follows: Image taken from Corresponding Angles are Congruent Corresponding angles are splitting images of one another without rotation or reflection. In the figure above, angle pairs (1,5), (2,6), (4,8), and (3,7) are corresponding. The members of each pair are congruent to one another. Alternate Angles are Congruent These are the pairs of angles that are “opposite” one another. On the interior above, these are angle pairs (4,6) and (3,5), while on the exterior, these are angle pairs (2,8) and (1,7). The members of each pair are congruent to one another. Consecutive Angles are Supplementary These are the pair of angles on the same side. On the interior, these are angle pairs (4,5) and (3,6), while on the exterior, these are angle pairs (2,7) and (1,8). The members of each
pair sum to 180°. Questions involving parallel lines cut by a transversal usually ask students to determine the measure of a particular angle given another one. Polygons A polygon is defined to be a closed, 2D figure with an integral number of sides. A polygon with just one side is simply a line, while that with two sides is an angle. Thereafter, the polygons become more tangible. The one with 3 sides is the infamous triangle, and that with 4 is known as a quadrilateral. The following table shows the names of higher polygons up to the one with ten sides. Number of sides Name 5 Pentagon 6 Hexagon 7 Heptagon 8 Octagon 9 Nonagon 10 Decagon A regular polygon is one where all sides and angles are equal to one another. Convex polygons are those whose angles have measures all less than 180°, and conversely, concave polygons are those with at least one reflex angle. We now present the following general formulas on polygons: Sum of interior angles in a polygon This is given by the formula 180 * (n − 2)° , where n is the number of sides in a polygon. By applying this formula, we have that the sum of angles in a triangle is indeed 180 * (3 − 2) = 180° , which most know to be true. Measure of an interior angle in a regular polygon As a regular polygon has all angles equal to one another, we can simply divide the sum by the number of sides. We therefore have this to be 180 * (n − 2) / n . It is for this reason that an equilateral triangle (a triangle with all sides and angles congruent) has each angle measuring 60°, while a square (quadrilateral with all sides and angles congruent) has each angle measuring 90°. Sum of exterior angles We define an exterior angle to be the linear pair counterpart of an interior angle. FOr example, if an angle in a triangle has measure 80°, we can extend one of the lines subtending this angle and get its corresponding linear pair to have measure 180 - 80 = 100°. We will show now that the sum of exterior angles, regardless of the nature of the polygon, is always 360°. This is easy to prove. We simply need to extend all lines of the polygon. Then the sum of all exterior and interior angles is simply 180 (measure of a line) * n (number of sides) = 180n. Subtracting this by the sum of all interior angles given by 180 * (n − 2) above, we have 180n − 180n + 360 = 360 degrees as the sum of all exterior angles for any given polygon. Likewise, in a regular polygon, the measure of each exterior angle is 360/n °. Triangles The sum of the interior angles of any triangle in regular geometry is 180°, and triangles always have 3 sides. The following are special variations of triangles: Types of Triangles Properties Right Angle Triangle - - As the name implies, one of its angles is a right angle (90°). The legs are the shorter side of the triangle, while the hypotenuse is the single longest side. The infamous Pythagorean theorem applies to this triangle, and this form of triangle only. Special right triangles are a more specific subset of this type, such as the isosceles right triangle ( 45-45-90) where the two legs are equal. A triangle that is not right is called oblique. Isosceles Triangle Has AT LEAST two equal sides. The angles opposite these two sides are also equal. Equilateral Triangle - Is an isoceles triangle. Has all three sides equal. Hence, has all three angles equal (60° each.) - Has three unequal sides. Hence, three unequal angles. The angles may be either right, obtuse, or acute. It is very much possible for a right triangle to be a scalene one. Scalene Triangles Quadrilaterals The sum of interior angles in any quadrilateral is 360°, and quadrilaterals always have 4 sides. The following are special variations of quadrilaterals: Types of Quadrilaterals Properties Trapezoid - Parallelogram - Rhombus Trapezoid (US) has exactly one pair of parallel sides Trapezium (US) has no parallel sides Opposite sides are parallel and equal in length to one another Opposite angles have the same measure (coloured the same in the figure to the left). Diagonals bisect one another. - All properties of the parallelogram are preserved, additionally: All sides are equal to one another. The diagonals create right angles at the intersection. Rectangle - All properties of the parallelogram are preserved, additionally: All angles are right angles, diagonals are equal in length Square - All properties of the rectangle are preserved, additionally: All sides are equal to one another Kite - The lengths colored red are equal to one another, as with those colored blue. Diagonals form a right angle. An angle of opposite pairing are equal to each other, indicated by the yellow angle. If the other pair was equal, then it can be classed as a rhombus. Image taken from: ...
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- Spring '18