PSM - Reviewer Q1.pdf

Plane and Solid Mensuration Quarter 1 | 1st Semester | 2024 Points, Lines, and Planes Plane - flat surface which extends indefinitely Euclid - has lengths and width -...

Plane and Solid Mensuration Quarter 1 | 1st Semester | 2024 Points, Lines, and Planes Plane - flat surface which extends indefinitely Euclid - has lengths and width - Developed a geometric system - no thickness - Wrote the book “The Elements” using the Coplanar Points axiomatic method o points that lie on the same plane Axioms - Theorems that don’t need to be proven Mensuration - Process of measurement - Based on the use of algebraic equations and geometric calculations Point Parallel Lines - Fixed location in a space - lines in the same planes that have no points - Represented by a dot in common - Has no dimensions - Referred by a capital letter o Pt. A o Point A Skew Lines o A - noncoplanar lines that are neither parallel Line nor intersecting - Has length but no width - Extends indefinitely in two opposite directions - Referred by lower case letters Line Segment o a part of a straight line that is bounded by two distinct end points Parallel Planes - planes that do not intersect Ray o a line with a single end point Theorems Theorem 1 Two parallel lines determine a plane Collinear Points o points that lie on the same straight line Theorem 2 If a line that is not on the plane Noncollinear Points intersects on the o if three or more points do not lie on plane, then the the same line intersection is exactly one point  Angles Between Lines and Planes Theorem 3 Through a given line, any number Angles of planes could be - A set of all points that is the union of two passed rays having the same end point Theorem 4 Can be named by: A line and a point Capital Letters that is not on the o Represent common vertex same line are Greek Letters contained in o Placed inside the angle exactly one plane Three Capital Letters o Represents each point from each ray Theorem 5 Types of Angles Two intersecting Acute Angle Measures 0° and 90° lines are Right Angle Measures exactly 90° contained in Obtuse Angle Between 90° 180° exactly one plane Straight Angle Exactly 180° Reflex Angle Between 180° and 360° Theorem 6 360° Angle One complete rotation If two planes intersect then Geometric Properties they intersect in exactly one line Theorem 7 Two planes perpendicular to the same straight line are parallel Theorem 8 If a plane intersects two Vertical Angles parallel planes - A = D, C = B, E = H, F = G then the Alternate Exterior Angles intersections are - A = H, B = G parallel Alternate Interior Angles - C = F, D = E Corresponding Angles - B = F, D = H, E = A, G = C Postulates Postulate 1: Dihedral Angle - For only two points there is exactly one time - Any angle formed by two containing them intersecting planes Postulate 2: - Three collinear points are contained in exactly one plane Postulate 3: Polyhedron Angle - A plane containing any two points contains - Angle formed by three all of the points on the line determined by or more intersecting those two points points at a common Postulate 4: point or vertex - Space is determined by at least four points which are not all in the same plane Postulate 5: - At a given point on a line, there are infinitely many lines perpendicular to the given line Theorems Classifying Polygons Theorem 9 1. Equilateral A line is perpendicular o a polygon which has all sides of the to a plane if an only if same length it is perpendicular to 2. Equiangular each line in the plane o a polygon whose vertex angles are passing through the equal intersection of the line Note: and the plane  A polygon is regular if it is a convex polygon Theorem 10 that is both equilateral and equiangular Through a given point Interior Angle of a Polygon on a plane, there is - the angle inside only one line the polygon perpendicular to the formed by each given plane pair of adjacent sides and it is located at the Theorem 11 vertices If two planes are perpendicular to each Note: other, one plane  Theorem 10: If a convex polygon has n contains a line sides, then the sum of the measures of its perpendicular to the interior angles is (n − 2) 180° other plane Exterior Angle of a Polygon - the angle between any Theorem 12 side and a line Every point in a plane extended from the next bisecting a dihedral side angle is equally distant from the faces of the Note: dihedral angle  Theorem 11: In any convex polygon, the sum of the measures of the exterior angles, one at each vertex is 360° Generally Accepted Names for Polygons Classifying Polygons Sides Name n n-gon Polygons 3 Triangle - A closed plane figure formed by three or 4 Quadrilateral more lines segments called sides 5 Pentagon - Named by the number of its sides, angles or 6 Hexagon vertices 7 Heptagon Vertex 8 Octagon - Intersection of two sides 10 Decagon Consecutive Sides 12 Dodecagon - the sides that share common vertex of a polygon Formulas: Consecutive Vertices Sum of all interior angles - two endpoints of any side of a polygon (𝑛 − 2)180 Diagonal Each angle of regular polygons - a line segment whose endpoints are (𝑛 − 2)180 nonconsecutive vertices of polygon Convex 𝑛 Diagonals - all diagonals lie on the interior of the 𝑛(𝑛 − 3) polygon 2 Concave Measure of Each Interior Angle - any part of the diagonals lies outside of the 𝑠 polygon 𝑛 Lesson 2: Perimeter, Triangles Based on the Angles Right Triangle Circumference and Area of - triangle with one right angle Different Plane Figures May be Solved Using: o Pythagorean Theorem o Trigonometric Ratios Polygons Oblique Triangle - Does not contain a right angle Perimeter of a Polygon - three interior angles has a measure greater - Measurement of the distance around the than 90° polygon Acute Triangle Area of a Polygon - All three sides of the triangle are acute - A positive number that represents the Triangles based on the Length of Sides number of square units needed to cover the Scalene Triangle polygon - all side lengths has different measures - The region bounded by the sides of a - No side will be equal in length to any of the polygon other sides Apothem Isosceles Triangle - the line segment from the center of the - two of the three sides are equal polygon perpendicular to the midpoint of a - has two equal sides and two equal angles side of the polygon Equilateral Triangle - all the lengths of the sides are equal - each of the interior - angles will have a measure of 60° - also known as an equiangular triangle Formulas: Perimeter of a Triangle Area of a Triangle 1 𝑃 = 𝑎+𝑏+𝑐 𝐴 = 2 𝑏ℎ Where: b = base Formulas: h = height Perimeter of a Polygon Apothem 𝑠 𝑃 = 𝑛𝑠 𝑎= 180 2 𝑡𝑎𝑛( 𝑛 ) Formulas for Oblique Triangles: Where: Where: Solving Oblique Triangles n = no. of sides a = apothem s = length of sides s = side length Area of a Polygon n = no. of sides 1 𝐴 = 𝑎𝑃 Side – Angle – Side (SAS) 2 Where: P = perimeter a = apothem Triangles Side – Side – Side (SSS) Triangle - a polygon bounded by three sides - integral part in most computations involved in architecture and engineering Architecture Angle – Side – Angle (ASA) o used in creating scale models Engineering o measures the height of inaccessible objects Surveyor o computes distance in navigation Quadrilaterals Perimeters Quadrilateral Formula Quadrilateral Parallelogram and 𝑃 = 2 (𝑙 + 𝑊 ) - A polygon bounded by four sides with four Rectangle Where: L = length vertices and corners W = width Special Quadrilaterals Square and Rhombus 𝑃 = 4𝑠 - Quadrilaterals with specific characteristics Where: Parallelogram s = side - a quadrilateral with both pairs of opposite Trapezoid 𝑃 =𝑎+𝑏+𝑐+𝑑 Where: sides parallel a, b = base Properties: c, d = side opposite sides of a parallelogram are Kite 𝑃 = 2(𝑎 + 𝑏) equal Where: a, b = sides opposite angles of a parallelogram are congruent Areas opposite sides of a parallelogram are Quadrilateral Formula parallel Parallelogram and 𝐴 = 𝐿𝑊 adjacent sides of a parallelogram are Where: Rectangle complementary L = length diagonals of a parallelogram bisect W = width each other Square 𝐴 = 𝑠2 Where: Rectangle s = side - a parallelogram with four right angles Rhombus and Kite 𝑑1 𝑑2 Rhombus 𝐴= 2 - a parallelogram with four congruent sides Where: d₁ = diagonal 1 Square d₂ = diagonal 2 - a parallelogram with four congruent sides Trapezoid 𝑏1 + 𝑏2 and four right angles 𝐴= ℎ 2 - a regular parallelogram Where: - it is a rhombus and a rectangle b₁ = base 1 b₂ = base 2 Trapezoid h = height - a quadrilateral with exactly one pair of parallel sides Circles and Irregular Figures Isosceles Trapezoid a trapezoid with legs (nonparallel Circle sides) that are congruent - a collection of points equidistant from a Properties fixed point called the center with a fixed the bases angles of a isosceles distance called the radius trapezoid are always congruent Chord the two diagonals of an isosceles - a straight line which joins any two points on trapezoid are equal the circle the opposite angles of an isosceles - the longest chord is the diameter of the trapezoid are congruent circle Kite Circumference - quadrilateral with two pairs of consecutive - the length or the distance around the circle congruent sides Area of a Circle - no opposite sides congruent and no parallel - the region bounded by the circle and its sides area Arc length of a Circle - distance between two points on the circle in relation to the central angle Sector of a Circle - portion of the area of the circle bounded by an arc and two radii of the circle Segment - portion of a sector bounded by the arc and the chord passing through the ends of the arc Concentric Circles - are circles having the same center ad with equal or unequal radii Polygon Inscribed in a Circle - A polygon is inscribed in a circle when each vertex of the polygon is a point of the circle Polygon Circumscribed about a Circle - A polygon is circumscribed about a circle when each side of the polygon is tangent to the circle Formulas: Circumference 𝐶 = 2𝜋𝑟 Area of a Circle 𝐴 = 𝜋𝑟 Arc 𝑠 = 𝑟𝜃 Sector 1 𝐴 = 𝑟 2𝜃 2 Segment 1 1 𝐴 = 𝑟 2 𝜃 − 𝑎𝑏 2 2 Okay lang yan guys wala naman itong unit, HAHAHA char, kaya niyo yan! Goodluck!!! - Ate Hannah