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GEOMETRY

GEOMETRY. - A mathematics subject taught in Secondary level for third year students…. Geometry of shape and size.

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GEOMETRY

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  1. GEOMETRY - A mathematics subject taught in Secondary level for third year students…

  2. Geometry of shape and size Everywhere, geometric forms and patterns emerge in nature and in man-made objects. These motivate and facilitate the learning of geometry because their beauty and symmetry appeal to one’s senses.

  3. GEOMETRY Recognition and analysis of their properties and measures not only develop geometric intuition needed in problem solving, but also open the path for logical thinking…

  4. “GEOMETRY as a subject” Geometry comes from the two Greek words “geo” meaning earth and “metri” meaning measurements. Therefore, GEOMETRY means earth measurements.

  5. Geometry of Shape and Size • 1.UNDEFINED TERMS • 1.1 describe the ideas of point, line and plane • 1.2 define, identify and name the subsets of a line • Segment • Ray • 2. ANGLES • 2.1 illustrate, name, identify and define an angle • 2.2 name and identify the parts of an angle • 2.3 read or determine the measure of an angle using a protractor • 2.4 illustrate, name, identify and define different kinds of angles

  6. 3. POLYGONS • 3.1 illustrate, identify and define different kinds of polygons according to the number of sides • illustrate and identify convex and non-convex polygons • identify the parts of a regular polygon (vertex angle, central angle, exterior angle) • 3.2 illustrate, name and identify a triangle and its basic and secondary parts (vertices, sides, angles, median, angle bisector, altitude) • 3.3 illustrate, name and identify different kinds of triangles and their parts (legs, base, hypotenuse) • Classify triangles according to their angles and according to their sides

  7. 3.4 illustrate, name and define a quadrilateral and its parts • 3.5 illustrate, name and identify the different kinds of quadrilaterals • 3.6 determine the sum of the measures of the interior and exterior angles of a polygon • Sum of the measures of the angles of a triangle is 180 • Sum of the measures of the exterior angles of a quadrilateral is 360 • Sum of the measures of the interior angles of a quadrilateral is (n-2) 180 • 4. CIRCLE • 4.1 define a circle • 4.2 illustrate, name, identify and define the terms related to the circle (radius, diameter and chord)

  8. 5. MEASUREMENTS • 5.1 identify the following common solids and their parts: cone, pyramid, sphere, cylinder, rectangular prism) • 5.2 state and apply the formulas for the measurements of plane and solid figures • Perimeter of a triangle, square and rectangle • Circumference of a circle • Area of a triangle, square, parallelogram, trapezoid and circle • Surface area of a cube, rectangular prism, square pyramid, cylinder, cone and a sphere • SA (cube)= 6s ; SA (cube)= 2B+ LA • SA (square pyramid) = B+ 4{(bxs)/2} • SA (cylinder) = 2(3.1415…)rh • SA (cone)= (3.1415…)rs • SA (sphere) = 4(3.1415…) r(squared)

  9. B. GEOMETRIC RELATIONS • 1. Relations Involving Segments and Angles • 1.1 illustrate and define betweeness and collinearity of points • 1.2 illustrate, identify and define congruent segments • 1.3 illustrate, identify and define the midpoint of a segment • 1.4 illustrate, identify and define the bisector of an angle • 1.5 illustrate, identify and define the different kinds of angle pairs • Supplementary • Complementary • Congruent • Adjacent • Linear pair • Vertical angles

  10. 1.6 illustrate, identify and define perpendicularity • 1.7 illustrate and identify the perpendicular bisector of a segment • 2. ANGLES AND SIDES OF A TRIANGLE • 2.1 derive/ apply relationships among the sides and angles of a triangle • Exterior and corresponding remote interior angles of a triangle • Triangle inequality • 3. ANGLES FORMED ANY PARALLEL LINES CUT BY A TRANSVERSAL • 3.1 illustrate and define parallel lines • 3.2 illustrate and define a transversal • 3.3 identify the angles formed by parallel lines cut by a transversal • 3.4 determine the relationship between pairs of angles formed by parallel lines cut by a transversal

  11. alternate interior angles • alternate exterior angles • corresponding angles • angles on the same side of the transversal • 4. PROBLEM SOLVING INVOLVING THE RELATIONSHIPS BETWEEN SEGMENTS AND BETWEEN ANGLES • 4.1 Solve problems using the definitions and properties involving relationships between segments and between angles • C. TRIANGLE CONGRUENCE • 1. Conditions for Triangle Congruence • 1.1 Define and illustrate congruent triangles • 1.2 State and apply the properties of congruence

  12. Reflexive property (AB= AB ; XY= XY) • Symmetric Property (If AB= CD, then CD= AB) • Transitive Property (If AB= CD and CD= EF, then AB= EF) • 1.3 Use inductive skills to establish the conditions or correspondence sufficient to guarantee congruence between triangles • SSS Congruence • SAS Congruence • ASA Congruence • SAA Congruence • 2. Applying the Conditions for Triangle Congruence • 2.1 Prove congruence and inequality properties in an isosceles triangle using the congruence conditions in 1.3

  13. Congruent sides in a triangle imply that the angles opposite them are congruent • Congruent angles in a triangle imply that the sides opposite them are congruent • Non-congruent sides in a triangle imply that the angles opposite them are not congruent • Non- congruent angles in a triangle imply that the sides opposite them are not congruent • 2.2 Use the definition of congruent triangles and the conditions for triangle congruence to prove congruent segments and congruent angles between two triangles • 2.3 Solve routine and non- routine problems

  14. ENRICHMENT • Apply inductive and deductive skills to derive other conditions for congruence between two right triangles • LL Congruence • LA Congruence • HyL Congruence • HyA Congruence • D. PROPERTIES OF QUADRILATERALS • 1. Different types of Quadrilaterals and their Properties • 1.1 recall previous knowledge on the different kinds of quadrilaterals and their properties (square, rectangle, rhombus, trapezoid, parallelogram) • 1.2 apply inductive and deductive skills to derive certain properties of the trapezoid

  15. median of a trapezoid • base angles and diagonals of an isosceles trapezoid • 1.3 apply inductive and deductive skills to derive the properties of a parallelogram • each diagonal divides a parallelogram into two congruent triangles • opposite angles are congruent • non- opposite angles are supplementary • opposite sides are congruent • diagonals bisect each other • 1.4 apply inductive and deductive skills to derive the properties of the diagonals of special quadrilaterals • diagonals of a rectangle • diagonals of a square • diagonals of a rhombus

  16. 2. Conditions that Guarantee that a Quadrilateral is a Parallelogram 2.1 verify sets of sufficient conditions which guarantee that a quadrilateral is a parallelogram 2.2 apply the conditions to prove that a quadrilateral is a parallelogram 2.3 apply the properties of quadrilaterals and the conditions for a parallelogram to solve problems ENRICHMENT Apply inductive and deductive skills to discover certain properties of the kite E. SIMILARITY 1. Ratio and Proportion 1.1 state and apply the definition of a ratio 1.2 define a proportion and identify its parts

  17. 1.3 state and apply the fundamental law of proportion • Product of the means is equal to the product of the extremes • 1.4 define and identify proportional segments • 1.5 apply the definition of proportional segments to find unknown lengths • 2. PROPORTIONALITY THEOREMS • 2.1 state and verify the Basic Proportionality Theorem and its Converse • 3. SIMILARITY BETWEEN TRIANGLES • 3.6 define similar figures • 3.7 define similar polygons • 3.8 define similar triangles • 3.9 apply the definition of similar triangles

  18. determining if two triangles are similar • finding the length of a side of measure of an angle of a triangle • 3.5 state and verify the Similarity Theorems • 3.6 apply the properties of similar triangles and the proportionality theorems to calculate lengths of certain line segments and to arrive at other properties • 4. SIMILARITIES IN A RIGHT TRIANGLE • 4.1 apply the AA Similarity Theorem to determine similarities in a right triangle • in a right triangle the altitude to the hypotenuse divides it into two right triangles which are similar to each other and to the given right triangle • 4.2 derive the relationships between the sides of an isosceles triangle and between the sides of a 30-60-90 triangle using the Pythagorean Theorem

  19. ENRICHMENT • State, verify and apply the ratio between the perimeters and areas of similar triangle. • Apply the definition of similar triangles to derive the Pythagorean Theorem • If a triangle is a right triangle, then the square of the hypotenuse is equal to the sum of the squares of the legs • 6. Word Problems Involving Similarity • 6.1 apply knowledge and skills related to similar triangles to word problems • F. CIRCLES • 1. The Circle • 1.1 recall the definition of a circle and the terms related to it

  20. radius • diameter • chord • secant • tangent • interior and exterior • 2. Arcs and Angles • 2.1 define and identify a central angle • 2.2 define and identify a minor and major arc of a circle • 2.3 determine the degree measure of an arc of a circle • 2.4 define and identify an inscribed angle • 2.5 determine the measure of an inscribed angle

  21. 3. Tangent Lines and Tangent Circles • 3.1 State and apply the properties of a line tangent to a circle • If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency. • If two segments from the same exterior point are tangent to a circle, then the two segments are congruent. • 4. Angles formed by Tangent and Secant Lines • 4.1 Determine the measure of the angle formed by the following: • two tangent lines • a tangent line and a secant line • two secant lines • ENRICHMENT • Illustrate and identify externally and internally tangent circles • Illustrate and identify a common internal tangent or a common external tangent

  22. GEOMETRIC CONSTRUCTIONS • Duplicate or copy a segment • Duplicate or copy an angle • Construct the perpendicular bisector and the midpoint of a segment • Derive the Perpendicular Bisector Theorem • Construct the perpendicular to a line • From a point on the line • From a point not on the angle • Construct the bisector of an angle • Construct parallel lines • Perform construction exercises using the constructions in 4.1 to 4.6

  23. Math I: Linear Equations in Two Variables Competency E1. describe the Cartesian coordinate Plane (x-axis, y-axis, quadrant, origin) Time Frame: 2 Sessions Objectives: At the end of the sessions, the students must be able to: 1. describe the Cartesian coordinate plane 2. given a point, describe its distance from the x or y axis 3. given a point on the coordinate plane, give its coordinates 4. given a pair of coordinates, plot the points 5. given the coordinates of a point, determine the quadrant where it is located

  24. Development of the Lesson: • A. Introduce the Cartesian coordinate plane using the number line. State that the rectangular coordinate plane are also called Cartesian plane can be constructed by drawing a pair of perpendicular number lines to intersect at zero on each line. • B. Ask the students to describe the two lines and their point of intersection to develop the following ideas: • Use construction to derive some other geometric properties (e.g.. Shortest distance from an external point to a line, points on the angle bisector are equidistant from the sides of the angle) • G. PLANE COORDINATE GEOMETRY • 1. Review the Cartesian Coordinate System, Linear Equations and Systems of Linear Equations in 2 Variables

  25. 1.1 name the parts of a Cartesian Plane 1.2 represent ordered pairs on the Cartesian plane and denote points on the Cartesian plane 1.3 define the slope of a line and compute for the slope given the graph of a line 1.4 define a linear equation 1.5 define the y-intercept 1.6 derive the equation of a line given two points of the line 1.7 determine algebraically the point of intersection of two lines 1.8 state and apply the definitions of parallel and perpendicular lines 2. COORDINATE GEOMETRY 2.1 Derive and state the Distance Formula using the Pythagorean Theorem 2.2 Derive and state the Midpoint Formula 2.3 Apply the Distance and Midpoint Formulas to find or verify the lengths of segments and find unknown vertices or points

  26. 2.4 Verify properties of triangles and quadrilaterals using coordinate proof • 3. CIRCLES IN THE COORDINATE PLANE • 3.1 derive/ state the standard form of the equation of a circle with radius r and center at (0,0) and at (h, k) • 3.2 given the equation of a circle, find its center and radius • 3.3 determine the equation of a circle given: • its center and radius • its radius and the point of tangency with a given line • 3.4 solve routine and non- routine problems involving circles

  27. …Summary… Importance: To function effectively in the three- dimensional world, one should have a knowledge of the geometric concepts of points, lines and planes to ensure a better understanding of their relationships, properties and basic applications.

  28. …Summary… Objectives and Expectations: • The students should be able to illustrate, name and identify different geometrical figures based on a given specific measurements. • To be able to read or determine the measurements of figures by the use of mathematical tools like ruler, compass and protractor.

  29. One important fact in Life: “Mathematics may not teach us to inhale oxygen and exhale carbon dioxide. It may not teach us the right grammar when we are speaking in front. But beyond those brain cracking formulas and computations in Math, it is the only subject which always reminds us that for every difficult and confusing problems, there is always a solution…”

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