Section 9.1 Points, Lines, Planes, and Angles
What You Will Learn Points Lines Planes Angles 9.1-2
Basic Terms A point, line, and plane are three basic terms in geometry that are NOT given a formal definition, yet we recognize them when we see them. 9.1-3
Lines, Rays, Line Segments A line is a set of points. Any two distinct points determine a unique line. Any point on a line separates the line into three parts: the point and two half lines. A ray is a half line including the endpoint. A line segment is part of a line between two points, including the endpoints. 9.1-4
Basic Terms Description Diagram Symbol Line AB A B AB Ray AB A B AB Ray BA A B BA Line segment AB A B AB 9.1-5
Plane We can think of a plane as a twodimensional surface that extends infinitely in both directions. Any three points that are not on the same line (noncollinear points) determine a unique plane. 9.1-6
Plane Two lines in the same plane that do not intersect are called parallel lines. 9.1-7
Plane A line in a plane divides the plane into three parts, the line and two half planes. 9.1-8
Plane Any line and a point not on the line determine a unique plane. The intersection of two distinct, non-parallel planes is a line. 9.1-9
Plane Two planes that do not intersect are said to be parallel planes. 9.1-10
Angles An angle is the union of two rays with a common endpoint; denoted. 9.1-11
Angles The vertex is the point common to both rays. The sides are the rays that make the angle. There are several ways to name an angle: ABC, CBA, B 9.1-12
Angles The measure of an angle is the amount of rotation from its initial to its terminal side. Angles can be measured in degrees, radians, or gradients. 9.1-13
Angles Angles are classified by their degree measurement. Right Angle is 90º Acute Angle is less than 90º Obtuse Angle is greater than 90º but less than 180º Straight Angle is 180º 9.1-14
Angles 9.1-15
Types of Angles Adjacent Angles - angles that have a common vertex and a common side but no common interior points. Complementary Angles - two angles whose sum of their measures is 90 degrees. Supplementary Angles - two angles whose sum of their measures is 180 degrees. 9.1-16
Example 3: Determining Complementary Angles In the figure, we see that ABC = 28 ABC & CBD are complementary angles. Determine m CBD. 9.1-17
Example 3: Determining Complementary Angles Solution m ABC + m CBD = 90 28 + m CBD = 90 m CBD = 90-28 = 62 9.1-18
Example 3: Determining Supplementary Angles In the figure, we see that ABC = 28. ABC & CBE are supplementary angles. Determine m CBE. 9.1-19
Example 3: Determining Supplementary Angles Solution m ABC+m CBE=180 28 +m CBE=180 m CBE=180-28 =152 9.1-20
Definitions When two straight lines intersect, the nonadjacent angles formed are called Vertical angles. Vertical angles have the same measure. 9.1-21
Definitions A line that intersects two different lines, at two different points is called a transversal. 9.1-22
Definitions Special names are given to the angles formed by a transversal crossing two parallel lines. 9.1-23
Special Names Alternate interior angles 3 & 6; 4 & 5 Alternate exterior angles 1 & 8; 2 & 7 Corresponding angles 1 & 5, 2 & 6, 3 & 7, 4 & 8 Interior angles on the opposite side of the transversal have the same measure Exterior angles on the opposite sides of the transversal have the same measure One interior and one exterior angle on the same side of the transversal have the same measure 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 9.1-24
Parallel Lines Cut by a Transversal When two parallel lines are cut by a transversal, 1. alternate interior angles have the same measure. 2. alternate exterior angles have the same measure. 3. corresponding angles have the same measure. 9.1-25
Example 6: Determining Angle Measures The figure shows two parallel lines cut by a transversal. Determine the measure of R1 through R7. 9.1-26
Example 6: Determining Angle Measures Solution 6 = 49 5 = 131 7 = 131 8 & 6 vertical angles 8 & 5 supplementary angles 5 & 7 vertical angles 9.1-27
Example 6: Determining Angle Measures Solution 1 = 131 4 = 49 2 = 49 3 = 131 1 & 7 alternate exterior 4 & 6 alternate exterior 6 & 2 corresponding angles 3 & 1 vertical angles 9.1-28
Section 9.2 Polygons
What You Will Learn Polygons Similar Figures Congruent Figures 9.2-30
Polygons A polygon is a closed figure in a plane determined by three or more straight line segments. 9.2-31
Polygons The straight line segments that form the polygon are called its sides, and a point where two sides meet is called a vertex (plural, vertices). The union of the sides of a polygon and its interior is called a polygonal region. A regular polygon is one whose sides are all the same length and whose interior angles all have the same measure. 9.2-32
Polygons Polygons are named according to their number of sides. Number of Sides Name Number of Sides Name 3 Triangle 8 Octagon 4 Quadrilateral 9 Nonagon 5 Pentagon 10 Decagon 6 Hexagon 12 Dodecagon 7 Heptagon 20 Icosagon 9.2-33
Polygons Sides Triangles Sum of the Measures of the Interior Angles 3 1 1(180º) = 180º 4 2 2(180º) = 360º 5 3 3(180º) = 540º 6 4 4(180º) = 720º The sum of the measures of the interior angles of an n-sided polygon is (n 2)180º. 9.2-34
Types of Triangles Acute Triangle All angles are acute. Obtuse Triangle One angle is obtuse. 9.2-35
Types of Triangles (continued) Right Triangle One angle is a right angle. Isosceles Triangle Two equal sides. Two equal angles. 9.2-36
Types of Triangles (continued) Equilateral Triangle Three equal sides. Three equal angles, 60º each. Scalene Triangle No two sides are equal in length. 9.2-37
Similar Figures Two figures are similar if their corresponding angles have the same measure and the lengths of their corresponding sides are in proportion. 6 9 4 4 6 6 3 4.5 9.2-38
Example 3: Using Similar Triangles to Find the Height of a Tree Monique Currie plans to remove a tree from her backyard. She needs to know the height of the tree. Monique is 6 ft tall and determines that when her shadow is 9 ft long, the shadow of the tree is 45 ft long (see Figure). How tall is the tree? 9.2-39
Example 3: Using Similar Triangles to Find the Height of a Tree 9.2-40
Example 3: Using Similar Triangles to Find the Height of a Tree Solution Let x represent the height of the tree height of tree height of Monique x 45 6 9 9x 270 x 30 The tree is 30 ft tall. length of tree's shadow length of Monique's shadow 9.2-41
Congruent Figures If corresponding sides of two similar figures are the same length, the figures are congruent. Corresponding angles of congruent figures have the same measure. 9.2-42
Quadrilaterals Quadrilaterals are four-sided polygons, the sum of whose interior angles is 360º. Quadrilaterals may be classified according to their characteristics. 9.2-43
Quadrilaterals Trapezoid Parallelogram Two sides are parallel. Both pairs of opposite sides are parallel. Both pairs of opposite sides are equal in length. 9.2-44
Quadrilaterals Rhombus Rectangle Both pairs of opposite sides are parallel. The four sides are equal in length. Both pairs of opposite sides are parallel. Both pairs of opposite sides are equal in length. The angles are right angles. 9.2-45
Quadrilaterals Square Both pairs of opposite sides are parallel. The four sides are equal in length. The angles are right angles. 9.2-46
Example 5: Angles of a Trapezoid Trapezoid ABCD is shown. a) Determine the measure of the interior angle, x. b) Determine the measure of the exterior angle, y. 9.2-47
Example 5: Angles of a Trapezoid Solution a) Determine the measure of the interior angle, x. m DAB + m ABC + m ABC + m x = 360 130 + 90 + 90 + m x = 360 310 + m x = 360 m x = 50 9.2-48
Example 5: Angles of a Trapezoid Solution b) Determine the measure of the exterior angle, y. m x + m y = 180 m y = 180 - m x m y = 180-50 m y = 130 9.2-49