5.1 Classifying Angles
Acute angles measure less than 90 degrees.
Right angles measure exactly 90 degrees.
Obtuse angles measure greater than 90 degrees, but less than 180 degrees.
Straight angles measure exactly 180 degrees.
Reflex angles measure more than 180 degrees, but less than 360 degrees.
Complementary angles are two angles whose measures total 90 degrees.
Supplementary angles are two angles whose measures total 180 degrees.
Congruent angles have the same angle measure.
Vertical angles are opposite (and congruent) angles created when two lines intersect.
5.2 Angles and Sides of Triangles/Types of Triangles
Equilateral triangles are triangles with three congruent sides.
Isosceles triangles have at least two congruent sides. Therefore, all equilateral triangles can also be called isosceles, but only some isosceles triangles can be called equilateral.
Scalene triangles have no sides that are congruent.
Acute triangles contain three acute angles. Acute angles measure less than 90 degrees. All triangles have two acute angles, so it is the third angle that categorizes the triangle based on the angles.
Right triangles contain a right angle. Right angles measure exactly 90 degrees.
Obtuse triangles contain an obtuse angle. Obtuse angles measure more than 90 degrees, but less than 180 degrees.
The sum of the angle measures of any triangle is always 180 degrees.
5.3 Angles of Polygons
Polygons are closed figures made up of three or more line segments that intersect only at their endpoints.
Convex polygons are polygons that, when connecting all of the vertices, the lines created either fall on or in the polygon.
Concave polygons have at least one line connecting the vertices that falls outside of the polygon’s perimeter.
Congruent lines are equal in length.
Regular polygons have all congruent sides and all congruent angles.
Irregular polygons do not have all sides congruent and all angles congruent.
Equiangular means all interior angles of a polygon are congruent. Regular polygons are equiangular. However, not all equiangular polygons are regular polygons. Regular polygons must also have congruent sides.
Formula for Sum of the Angle Measures of a Polygon S=n-2×180° S is the sum of the interior angles in the polygon, and n is the number of sides in the polygon.
Triangles are 3-sided polygons.
Quadrilaterals are 4-sided polygons. As long as the figure has four sides and is a polygon, it is a quadrilateral.
Pentagons are 5-sided polygons.
Hexagons are 6-sided polygons.
Heptagons are 7-sided polygons.
Octagons are 8-sided polygons.
Nonagons are 9-sided polygons.
Decagons are 10-sided polygons.
5.4 Similar Triangles
Similar Triangles have the same shape, but are not necessarily the same size.
Triangles that are similar will always contain the same three angle measures. For example, every triangle with angles measuring 45, 45, and 90 degrees are similar to each other. The size of the triangle doesn’t matter. A triangle with angles measuring 15, 22.5, and 142.5 degrees is similar to all other triangles with angles measuring 15, 22.5, and 142.5 degrees.
Triangle AEF is similar to triangle ADG. We know that because they both contain angle A and a 90 degree angle. That means their third angles, angle F and angle G, are congruent. Since both triangles have the same three angles, they are similar.
Triangles are similar if the ratios of the corresponding sides are equal. For example, if a triangle has sides measuring 2, 3, and 4 inches, it will be similar to a triangle with sides measuring 4, 6, and 8 inches. This is because 24 , 36, and 48 are all the same ratio after they have been simplified.
5.5 Parallel Lines and Transversals
Parallel lines occur when two or more lines are always an equal distance apart.
Perpendicular lines occur when two lines intersect and create four right angles at their intersection.
A transversal is a line that intersects two or more lines.
If the two lines a transversal intersects happen to be parallel to each other, several pairs of congruent angles are formed.
Interior angles are the four angles created that are in between (or inside) the parallel lines that are intersected by the transversal. (Angles b, c, e, and h are interior angles.)
Exterior angles are the four angles created that are outside the parallel lines that are intersected by the transversal. (Angles a, d, f, and g are exterior angles.)
Each exterior angle has a corresponding interior angle. Each interior angle has a corresponding exterior angle. Corresponding angles are congruent. (Angle a corresponds to angle e. Angle b corresponds to angle f. Angle c corresponds to angle g. Angle d corresponds to angle h.)
Alternate interior angles are congruent. (Angles b and h are alternate interior angles. Angles c and e are alternate interior angles.)
Alternate exterior angles are congruent. (Angles a and g are alternate exterior angles. Angles d and f are alternate exterior angles.)
Transformation Rules on the Coordinate Plane
Translation: Each point moves a units in the x-direction and b units in the y-direction.
(x, y) → (x + a, y + b)
Reflection across the x-axis: Each x-value stays the same and each y-value becomes opposite of what it was.
(x, y) → (x, −y)
Reflection across the y-axis: Each y-value stays the same and each x-value becomes opposite of what it was.
(x, y) → (−x, y)
Reflection across the line y=x: The x and y values switch places. (x, y) → (y, x)
Rotation 90° clockwise about the origin: Each x value becomes opposite of what it was. The x and y values switch places.
(x, y) → (y, -x)
Rotation 180° clockwise about the origin: Each x and y value becomes opposite of what it was.
(x, y) → (−x, −y)
Rotation 270° clockwise about the origin: Each y value becomes opposite of what it was. The x and y values switch places.
(x, y) → (-y, x)
Dilation with respect to the origin and scale factor of k: (x, y) → (kx, ky)
Distortion by altering height only with respect to the origin and scale factor of k: (x, y) → (x, ky)
Distortion by altering width only with respect to the origin and scale factor of k: (x, y) → (kx, y)