GEOMETRY is the study of points in space

CHAPTER 5 Logic and Geometry SECTION 5-1 Elements of Geometry GEOMETRY is the study of points in space POINT indicates a specific location and is represented by a dot and a letter R S T LINE is a set of points that extends without end in two opposite directions PLANE is a set of points that extends in all directions along a flat surface R S line RS Y W 1

COLLINEAR POINTS points that lie on the same line C D E F NONCOLLINEAR POINTS points that do not lie on the same line C D E F COPLANAR POINTS are points that lie in the same plane E A B C D NONCOPLANAR POINTS are points that do not lie in the same plane E A B C D INTERSECTION set of all points common to two geometric figures P LINE SEGMENT - part of a line that begins at one endpoint and ends at another F G 2

RAY part of a line that begins at one endpoint and continues in the other direction, without ending CONGRUENT LINE SEGMENTS - have the same measure means congruent to A B MIDPOINT - the point that divides the segment into two congruent segments. BISECTOR - is any line, segment, ray or plane that intersects the segment at its midpoint POSTULATE - 1 Through any two points, there is exactly one line. POSTULATE - 2 Through any three noncollinear points, there is exactly one plane. 3

POSTULATE - 3 If two points lie in a plane, then the line joining them lies in that plane POSTULATE - 4 If two planes intersect, then their intersection is a line SECTION 5-2 ANGLES AND PERPENDICULAR LINES ANGLE the union of two rays with a common endpoint. VERTEX endpoint of an angle A B C Right Angle measures exactly 90º Acute Angle measure is greater than 0 º and less than 90º 4

Obtuse Angle measure is greater than 90º and less than 180º Straight Angle measures exactly 180º COMPLEMENTARY angles whose sum measures 90º SUPPLEMENTARY angles whose sum measures 180º J 39º J 59 51º K 121º K ADJACENT ANGLES two angles in the same plane that share a common side and a common vertex O ADJACENT ANGLES A B C AOB and BOC 5

CONGRUENT ANGLES angles having the same measure PERPENDICULAR LINES - two lines that intersect to form right angles VERTICAL ANGLES two angles whose sides form two pairs of opposite rays. Vertical angles are congruent. ANGLE BISECTOR - ray that divides the angle into two congruent adjacent angles O A B C 6

EXAMPLE 1 Find the measure of angle JXF SOLUTION Read the Protractor Angle JXF = 90º right EXAMPLE 2 Find the measure of angle HXL SOLUTION Read the Protractor Angle HXL = 120º obtuse 7

EXAMPLE 3 Find the measure of angle KXG SOLUTION Read the Protractor Angle KXG = 120º obtuse EXAMPLE 4 Find the measure of angle GXJ SOLUTION Read the Protractor Angle GXJ = 65º acute 8

SECTION 5-3 PARALLEL LINES and TRANSVERSALS PARALLEL LINES - coplanar lines that do not intersect Parallel Lines m n PARALLEL PLANES - planes that do not intersect Parallel Planes m n Skew Lines - noncoplanar lines that do not intersect and are not parallel 9

Skew Lines n m n Transversal - is a line that intersects each of two other coplanar lines in different points to produce interior and exterior angles l Transversal 6 5 2 1 4 8 3 7 ALTERNATE INTERIOR ANGLES - two nonadjacent interior angles on opposite sides of a transversal Alternate Interior Angles 2 1 3 4 SAME SIDE INTERIOR ANGLES - interior angles on the same side of a transversal 10

Same Side Interior Angles 2 1 3 4 ALTERNATE EXTERIOR ANGLES - two nonadjacent exterior angles on opposite sides of the transversal Alternate Exterior Angles 6 5 8 7 Corresponding Angles - two angles in corresponding positions relative to two lines cut by a transversal Corresponding Angles 6 5 2 1 3 4 8 7 PARALLEL LINE POSTULATES 11

POSTULATE - 5 2 8, 6 4, 5 3, 1 7 If two parallel lines are cut by a transversal, then corresponding angles are congruent 6 5 2 1 4 8 3 7 STATEMENT - 5A 2 3, 1 4 If two parallel lines are cut by a transversal, then alternate interior angles are congruent 2 1 4 3 STATEMENT - 5B If two parallel lines are cut by a transversal, then alternate exterior angles are congruent 6 7, 5 8 6 5 8 7 12

SECTION 5-4 Properties of Triangles Triangle is a figure formed by the segments that join three noncollinear points Vertex point of a triangle Side segment of a triangle Congruent Segments segments with the same length Congruent Angles angles with the same measure Scalene Triangle is a triangle with all three sides of different length. Isosceles Triangle is a triangle with two sides (legs) of equal length and a third side called the base and 13

Angles at the base are called base angles and the third angle is the vertex angle. Equilateral Triangle is a triangle with three sides of equal length Acute Triangle is a triangle with three acute angle (<90 ) Obtuse Triangle is a triangle with one obtuse angle (>90 ) Right Triangle is a triangle with one right angle (90 ) Equiangular Triangle is a triangle with three angles of equal measure. 14

Interior angles angles determined by the sides of a triangle Exterior angle an angle that is both adjacent and supplementary to an interior angle Base angle angles opposite congruent sides PROPERTIES of TRIANGLES The sum of the measures of the angles of a triangle is 180 The sum of the lengths of any two sides is greater than the length of the third side. The longest side is opposite the largest angle, and the shortest side is opposite the smallest angle. 15

If one side of a triangle is extended, then the exterior angle formed is equal to the sum of the two remote interior angles of the triangle. If two sides of a triangle are congruent, then the angles opposite those sides are congruent. SECTION 5-5 CONGRUENT TRIANGLES Congruent Triangles - the vertices can be matched so that corresponding parts fit exactly over each other, and Corresponding angles lie opposite corresponding sides, and vice versa. Corresponding Sides - corresponding sides of congruent triangles are congruent. 16

Corresponding Angles - corresponding angles of congruent triangles are congruent. POSTULATES If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent (SSS) If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent (SAS) If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent (ASA) SECTION 5-6 QUADRILATERALS and PARALLELOGRAMS 17

Quadrilateral - a closed plane figure that has four sides Parallelogram - is a quadrilateral with both pairs of opposite sides parallel. Trapezoid - is a quadrilateral with exactly one pair of sides parallel. Rectangle - is a quadrilateral with four right angles. Rhombus - is a quadrilateral with four sides of equal length. Square - is a quadrilateral with four right angles and four sides of equal length. 18

Opposite angles - two angles that do not share a common side Consecutive angles - two angles that share a common side Opposite sides - two sides that do not share a common endpoint Consecutive sides - two sides that share a common endpoint. PROPERTIES of PARALLELOGRAMS The opposite sides of a parallelogram are congruent. 19

The opposite angles of a parallelogram are congruent. The consecutive angles of a parallelogram are supplementary. The sum of the angle measures of a parallelogram is 360. The diagonals of a parallelogram bisect each other. The diagonals of a rectangle are congruent. The diagonals of a rhombus are perpendicular. 20

SECTION 5-7 DIAGONALS and ANGLES of POLYGONS Polygon is a closed plane figure that is formed by joining three or more coplanar segments at their endpoints, and Each segment of the polygon is called a side, and the point where two sides meet is called a vertex Convex A polygon is Convex if each line containing a side contains no points in the interior of the polygon. Concave A polygon is Concave if a line containing a side contains a point in the interior of the polygon. Regular Polygon - a polygon that has all sides congruent and all angles congruent. 21

Diagonal - a segment of a polygon that joins two vertices but is not a side. THEOREMS The sum of the measures of the angles of a polygon with n sides is (n-2)180 The measure of each interior angle of a regular polygon with n sides is (n-2)180 n SECTION 5-8 PROPERTIES of CIRCLES Circle - the set of all points in a plane that are a given distance from a fixed point in the plane, and 22

The fixed point is called the CENTER. The given distance is the RADIUS. Radius - is a segment that has one endpoint at the center and one on the circle. Chord - is a segment with both endpoints on the circle. Diameter - is a chord that passes through the center of the circle. Circumference - is the distance around a circle. Arc - is a section of the circumference of a circle. 23

Semicircle - is a arc with endpoints that are the endpoints of a diameter. Minor Arc - is an arc that is smaller than a semicircle. Major Arc - is an arc that is larger than a semicircle. Central Angle - is an angle with its vertex at the center of a circle. *The measure of a central angle is equal to the measure of the arc it intercepts. Inscribed Angle - is an angle whose vertex lies on the circle and whose sides contain chords of the circle, and The measure of an inscribed angle is ½ the measure of the arc it intercepts. 24

SECTION 5-9 CIRCLE GRAPHS Circle Graph - is a way to display data to make comparisons. THE END! 25