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PLANE AND SOLID GEOMETRY
[Plane Geometry] Topic 1: Three Undefined Terms in Geometry In geometry, formal definitions are formed using there defined words or terms. There are, however, three words in geometry that are not formally defines. These words are point, line, and plane, and are referred to as the “three undefined terms of geometry”. Point − A point indicates a location (or position) in space. − A points has no dimension (actual size). − A point has no length, no width, and no height. − A point is usually named with a capital letter. − In the coordinate plane, a point is named by an ordered pair, (x,y). Line (straight line) − A line has no thickness − A line’s length extends in one dimension. − A line goes on forever on both directions. − A line has infinite length, zero width, and zero height. − A line is assumed to be straight. − A line is drawn with arrowheads on both ends. − A line is named by a single lower case script letter, or by any two (or more) points which lie on the line. Plane − A plane has two dimensions. − A plane forms a flat surface extending indefinitely in all directions. − A plane has infinite length, infinite width, and zero height (thickness). − A plane is drawn as a four-sided figure resembling a tabletop or a parallelogram. − A plane is named by a single letter (plane m) or by three coplanar, but non-collinear, points (plane ABC). Collinear Points Are points that lie on the same straight line. Coplanar Points Are points that lie on the same plane.
Subsets of Line Line Segment If you mark two points A and B on it and pick this segment separately, it becomes a line segment. This line segment has two endpoints A and B whose length is fixed. A line segment is named by its endpoints, but other points along its length can be named, too. The length of this line segment is the distance between its endpoints A and B. Ray Is a line with a single endpoint (or points of origin) that extends infinitely in one direction. We can name a ray using its starting points and one other that is on the ray. Ray AB is different from Ray BA. Angle A figure which is formed by two rays or lines that shares a common endpoint. The two rays are called the sides of an angle, and the common endpoints is called the vertex. To name an angle, we use three points listing the vertex in the middle, sin number, Greek letter. Angle is denoted by ∠. Type of Angles Acute Angle 0 ° < x < 90 ° Right Angle x = 90 ° Obtuse Angle^90 °^ <^ x^ <^180 ° Straight Angle x = 180° Reflex Angle 180 ° < x < 360 ° Full Rotation x < 360 ° Lines Parallel Lines Are coplanar lines that never intersect; they travel similar paths at a constant distance from one another. Skew Lines Are noncoplanar lines that never intersect; they travel dissimilar paths on separate planes. Two-Lined Intersection Intersection of two lines is a points at which both lines meet. When two lines share a common points, they are called intersecting lines. When the measures of those four angles are added, the sum equal the rotation of a complete circle, or 360 degrees. How big is the angle? It does not matter how long the side of the angle are. The size of the angle is ONLY determine by how much it has opened as compared to the whole circle.
Exterior Angle Theorem - the measurement of an exterior angle is equal to the sum of the measurement of the two non-adjacent interior angles. Base Angle Theorem - if two side of a triangle are congruent, the angles opposite these angles are congruent. Base Angle Converse Theorem - if two angle of a triangle are congruent, the sides opposite these sides are congruent. Alternate Exterior Angles Theorem - if a transversal line intersects two parallel lines then alternate exterior angles are congruent. Converse of the Alternate Exterior Angles Theorem - if two lines and a transversal line form alternate exterior angles that are congruent, then the lines are parallel. Corresponding Angles Theorem - if a transversal intersects two parallel lines, and then corresponding angles are congruent. Converse of the Corresponding Angles Theorem - if two lines and a transversal line form corresponding angles that are congruent then the lines are parallel. Alternate Interior Angles Theorem - if a transversal line intersects two parallel lines then alternate interior angles are congruent. Converse of the Alternate Interior Angles - if two lines and a transversal line form alternate interior angles that are congruent, then the lines are parallel. Postulate and Corollary Postulate Is a statement that is accepted without proof. The postulates together with undefined terms in geometry are used to prove theorems. Postulate: Through any two points, there is exactly one line containing them. Basic Postulates & Theorems of Geometry Euclid’s Postulates
- Two points determine a line segment.
- A line segment can be extended indefinitely along a line.
- A circle can be draw in with a center and any radius.
- All right angles are congruent.
- If two lines are cut by a transversal, and the interior angles on the same side of the transversal have a total measure of less than 180 degrees, then the lines will intersect on that side of the transversal. Postulate: if a line contains two distinct points of a plane, it is contained in the plane. The measure of any line segment is a unique positive number. Postulate: For any three points, there is at least one plane containing them. Any three non collinear points determine a plane. If X is a points on AB and X is between A and B, then AX + XB = AB. Postulate: if two distinct planes intersect, then their intersection is a line. If two lines intersect, then they intersect in exactly one point. Postulate: space contains at least four non coplanar points. Through any three non collinear points, there is exactly one plane. Postulate (The Ruler Postulate): the points on a line can be placed in one-to-one correspondence with the real numbers so that for every points on the line there corresponds exactly one real number called the coordinate of the point and vice versa. The distance between points A and B, written as AB, is the absolute value of the difference between the coordinate A and B.
Postulate (Angle Measurement Postulate): for every angle, there is unique positive number between 0 and 180 called the degree measure of the angle. If two points lie in a plane, then the line joining them lies in that plane. The protractor A protractor is used to find the degree measure of a given angle. Postulate: the measure of an angle is a unique positive number. Postulate: if a point D lies in the interior of angle ABC, then mABD + mDBC = mABC. Postulate (Parallel Postulate): through a point not on a line, exactly one line is parallel to the given line. Linear Pair Postulate: if two angles form a linear pair, then the measures of the angles add up to 180 degrees. Vertical Angles Postulate: if two angles are vertical angles, then they are congruent. Corresponding Angles Postulate: if two parallel lines are cut by a transversal, then corresponding angles are congruent. Corollary Is a statement that is proven true by another statement or considered to be sequence of an statement’s truth. Corollaries are believed to be true without additional proof beside the initial true statement. An equilateral triangle is always equiangular. If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent. Each angle of an equilateral triangle has measure 60 degrees. In a triangle, there can be at most one right angle or obtuse angle. The acute angle of a right triangle are complementary. Topic 2: Triangles Triangles Is a 3-sided polygon. A closed figure consisting of three lines segments joining three non collinear points. The three angles of a triangle always add to 180 degrees. A closed plane figure having three sides and three angles. Properties of a Triangle The vertex is a corner of the triangle. Every triangle has three vertices. The base of a triangle can be any one of the three sides, usually the one drawn at the bottom. You can pick any side you like to be the base. Commonly used as a reference side for calculating the area of the triangle. In an isosceles triangle, the base is usually taken to be the unequal side. The altitude of a triangle is the perpendicular from the base to the opposite vertex. Since there are three possible bases, there are also three possible altitude. The three altitudes intersect at a single points, called the orthocenter of the triangle. The median of a triangle is a line from a vertex to the midpoint of the opposite side. The three medians intersect at a single point, called the centroid of the triangle. Interior angles are the three angles on the inside of the triangle at each vertex. Exterior angles are angle between a side of a triangle and the extension of an adjacent side. Types of Triangles According to Sides Equilateral Triangle - a triangle with all three sides equal in measure.
Regular Polygon A regular polygon in which all the sides are equal and all the angles are equal. A segment whose endpoint is two non-consecutive vertices of a polygon is called diagonal. Naming Polygons Number of Sides Name of Polygon Figure 3 Triangle 4 Quadrilateral 5 Pentagon 6 Hexagon 7 Heptagon 8 Octagon 9 Nonagon 10 Decagon n n-gon Angles of Polygon Interior Angle Consider each convex polygon below with all possible diagonals drawn from one vertex. Notice that each polygon is separated into triangles. Since the sum of the measures of the interior angles in a triangle is 180, it easy to find the sum of the measures of the interior of the interior angles of each polygon. The sum of the angle measurement is (n-2)180. Convex Polygon Number of Sides Number of Triangles Sum of Interior Angles Triangles 3 1 1\8180) = 180 Quadrilateral 4 2 2(180) = 360 Pentagon 5 3 3(180) = 540 Hexagon 6 4 4(180) = 720 Heptagon 7 5 5(180) = 900 Octagon 8 6 6(180) = 1080 Theorem: If a convex polygon has n sides and S is the sum of the measures of its interior angles, then S = (n - 2) Exterior Angle an exterior angle of a polygon is an angle that forms a linear pair with one of the angles of the polygon.
Theorem: The sum of the measures of the exterior angles of a convex polygon is 360 degrees. Quadrilaterals Quadrilaterals Is a closed figure consisting of four line segments or sides. Theses sides may or may not be congruent and parallel. Can be named by their vertices. Kinds of Quadrilaterals Trapezoid - with exactly one pair of parallel side. Kite - has 2 pairs of equal adjacent sides. Parallelogram - two pairs of opposite sides are both parallel and congruent. Rhombus - all sides are congruent. Rectangle - all angles are congruent and two pairs of opposite sides are parallel and congruent. Square - all angles are right angles and all sides are congruent. Properties of Trapezoid Only one pair of the opposite side of a trapezium is parallel to each other The two adjacent sides of a trapezium are supplementary (180 degrees) The diagonals of a trapezium bisect each other in the same ratio Properties of Kite The pair of adjacent sides of a kite are of the same length The largest diagonal of a kite bisect the smallest diagonal Only one pair of opposite angles are of the same measure. Properties of Parallelogram The opposite side of the parallelogram are of the same length The opposite sides are parallel to each other The diagonals of a parallelogram bisect each other The opposite angles are of equal measure The sum of two adjacent angles of a parallelogram is equal to 180 degrees Properties of Rhombus All the four sides of a rhombus are of the same measure The opposite sides of the rhombus are parallel to each other The opposite angles are of the same measure The sum of any two adjacent angles of a rhombus is equal to 180 degrees The diagonals perpendicularly bisect each other Properties of Rectangle The opposite sides of a rectangle are of equal length The opposite sides are parallel to each other All the interior angles of a rectangle are 90 degrees. The diagonals of a rectangle bisect each other. Properties of Square All the sides of the square are of equal measure The sides are parallel to each other All the interior angles of a square are at 90 degrees (i.e., right angle) The diagonals of a square perpendicular bisect each other
Topic 5: Triangle Congruent Two triangles are congruent if their corresponding sides are equal in length and their corresponding interior angles are equal in measure. We use the symbol ≅ to show congruence. Triangle Congruence Theorem Side-Angle-Side (SAS) Theorem Two triangles are congruent if two sides and one included angle in a given triangle are equal to the corresponding two side and one included angle in another triangle. Example: Side-Angle-Angle (SAA) Theorem States that two triangles are congruent if their corresponding two angles and one non-included side are equal. Example: Side-Side-Side (SSS) Theorem States that two triangles are congruent if their corresponding three sides lengths are equal. Example: Angle-Side-Angle (ASA) Theorem States that two triangles are congruent if their corresponding two angles and one included side are equal. Example: Topic 6: Circles Circle is a closed two-dimensional figure in which the set of all points in the plane is equidistant from a given points called “centre”.
Parts of a Circle Circumference - the distance around the boundary of a circle. Radius - is a segment, one of its endpoint is in the center of the circle and the other endpoint is on the circle. Chord - is a segment whose endpoints are any two points on the circle. Diameter - is a chord which passes through the center of the circle. Secant - is a line which intersects the circle at two distinct points. Tangent - is a line in the plane of a circle that intersects the circle at exactly one point. Central Angle - is an angle formed by two radii of a circle with its vertex is in the center of the circle. Arc - is a part of a circle. It is a part or segment of the circumference of a circle. If it is half a circle, it is called a semicircle. It it less than half of a circle, it is called a minor arc. If it is more than half a circle, it is called major arc. An arc is intercepted by a given angle or the angle intercepts the arc if the endpoints of the arc are points of the angle and all other points of the arc are in the interior of the angle. An angle is subtended by an arc if two points are endpoint of the arc. Arcs are measured by their corresponding angles. Note: The degree measure of a minor arc is the degree measure of its central angle. The degree measure of a major arc is 360 minus the degree measure of the minor arc. Theorems on Central Angles and Arcs
1. In a circle or in congruent circles, two minor arcs are congruent if and only if their
corresponding central angles are congruent.
2. In a circle or in congruent circles, two minor arcs are congruent if and only if their
corresponding chords are congruent.
Arc Length
Inscribed Angles and Arcs An inscribed angle is an angle whose vertex lies on the circle and whose sides contain the endpoint of an arc of the same circle. Theorems on Inscribed Angles Tangent of a Circle A line tangent to a circle is a line on the same plane which intersects the circle in one plane and only one point. This point of intersection is called the points of tangency or point of contact.
If two tangent circles are coplanar and their centers are on the opposite side of their common tangent, then there are externally tangent. Secant of a Circle Secant is a line that intersects a circle at two different points. Theorems on Angles Formed by Tangents and Secants
Topic 7: Area and Perimeter of a Plane Shapes Definition Perimeter , generally denoted by P, is the measure of the contour of a figure. In the case of the circle, the perimeter is called circumference and is denoted by C. Area, generally denoted by A, is the surface occupied by an object. Plane Figure Perimeter Area Triangle P = a + b + c (^) A = b × h 2 Square (^) P = s + s + s + s P = 4s A = s × s A = s^2 Rectangle P = b + b + h + h P = 2b + 2b P = 2(b + h) A = bh Parallelogram P = a + a + b + b P = 2a + 2b P = 2(a + b) A = bh Rhombus P = s + s + s + s P = 4s
A =
Dd 2 Trapezoid P = b + a + B + c (^) A = (b + B) × h 2 Regular Polygon P = n × c A =^ san 2 Circle C = 2πr A = πr^2
- It has two identical ends that are either circular or oval in shape. Surface Area Curved Surface Are = 2πrh square units Total Surface Area = 2πr(h + r) square units Where: r = radius h = height Volume V = πr^2 h^ cubic^ units Cuboid A cuboid is a solid that has 6 rectangular faces, 8 vertices, and 12 edges. Properties of a Cuboid
- It has all the faces in the shape of a rectangle.
- All the faces or sides of a cube have different dimensions.
- The angle of planes of the cube are the right angle.
- Each of the faces of a cube meet the other four faces.
- Each of the vertices of a cube meet the three faces and three edges.
- The edges that are opposite to each other are parallel. Surface Area Curved Surface Area = 2h(l + b) square units Total Surface Area = 2(lb + bh + h) square units Where: l = length b = breadth h = height Volume V = lbh cubic units Cube A cube is a solid that has 6 square faces, 8 vertices, and 12 edges. Properties of a Cube
- It has all the faces in the shape of a square.
- All the faces or sides of a cube have equal dimensions.
- The angles of planes of the cube are the right angle.
- Each of the faces of a cube meet the other four faces.
- Each of the vertices of a cube meet the three faces and three edges.
- The edges that are opposite to each other are parallel. Surface Area Curved Surface Area = 4a^2 square units Total Surface Area = 6a^2 square units Where: a = edge of a cube Volume V = a^3 cubic units
Cone A cone is a solid shape that has a flat surface and a curved surface, points towards the top. It is formed by a set of line segments connected from the circular base to a common point, which is known as the apex or vertex. Properties of a Cone
- It has a circular or oval base with an apex or a vertex.
- It has one curved surface and one flat surface.
- A cone is a rotated triangle. Surface Area Curved Surface Area = πrl square units Total Surface Area = πr(l + r) square units Where: r = radius l = slant height Volume V =
πr^2 h cubic units Pyramid A pyramid is a polyhedron with a polygon base and all its lateral faces are triangular in shape. Pyramids are typically categorized by the shape of their bases. A pyramid with: A triangular base is known as tetrahedron A quadrilateral base is known as square pyramid A pentagon base is known as pentagonal pyramid A regular hexagon is known as hexagonal pyramid Properties of a Pyramid
- A pyramid is a polyhedron with a polygon base and a vertex with straight lines.
- It has one curved surface and one flat surface.
- A cone is a rotated triangle. Surface Area Surface Area = B +
× P × l Lateral Surface Are =
Pl Where: B = area of the base P = perimeter of the base l = slant height of the lateral sides Volume V =
× B × h Where: h = height of the pyramid
