GEOMETRY PSSLC

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    GEOMETRY (3RD YEAR HIGH SCHOOL)

    A. Geometry of Shape and Size

    1. Undefined Terms1.1. Describe the ideas of point, line, and plane1.2. Define, identify, and name the subsets of a line

    Segment

    Ray2. Angles

    2.1. Illustrate, name, identify and define an angle

    2.2. Name and identify the parts of an angle2.3. Read or determine the measure of an angle using a protractor2.4. Illustrate, name, identify and define different kinds of angles

    Acute

    Right

    Obtuse3. 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 (e.g., vertices, sides, angles,median, angle bisector, altitude)

    3.3. Illustrate, name and identify different kinds of triangles and their parts (e.g., legs, base, hypotenuse)

    classify triangles according to their angles and according to their sides3.4. Illustrate, name and define a quadrilateral and its parts3.5. 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)1804. Circle

    4.1. Define a circle

    4.2. Illustrate, name, identify, and define the terms related to the circle (radius, diameter and chord)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

    Volume of a rectangular prism, triangular prism, pyramid, cylinder, cone, and a sphere5.3. Solve problems involving plane and solid figures

    B. Geometric Relations1. Relations involving Segments and Angles

    1.1. Illustrate and define betweeness and collinearity of points1.2. Illustrate, identify and define congruent segments1.3. Illustrate, identify and define the midpoint of a segment1.4. 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 angles1.6. Illustrate, identify and define perpendicularity1.7. Illustrate and identify the perpendicular bisector of a segment

    2. Angles and Sides of a Triangle2.1. Derive/apply relationships among the sides and angles of a triangle

    Exterior and corresponding remote interior angles of a triangle

    Triangle inequality3. Angles formed by Parallel Lines cut by a Transversal

    3.1. Illustrate and define Parallel Lines3.2. Illustrate and define a Transversal3.3. 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

    Alternate interior angles

    Alternate exterior angles

    Corresponding angles

    Angles on the same side of the transversal4. Problem Solving involving the Relationships between Segments and between Angles

    4.1. Solve problems using the definitions and properties involving relationships between segments andbetween angles

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    C. Triangle Congruence1. Conditions for Triangle Congruence

    1.1. Define and illustrate congruent triangles1.2. State and apply the Properties of Congruence

    Reflexive Property

    Symmetric Property

    Transitive Property

    1.3. Use inductive skills to establish the conditions or correspondence sufficient to guarantee congruence

    between triangles1.4. Apply deductive skills to show congruence between triangles

    SSS Congruence SAS Congruence

    ASA Congruence

    SAA Congruence2. Applying the Conditions for Triangle Congruence

    2.1. Prove congruence and inequality properties in an isosceles triangle using the congruence conditions in 1.3

    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 congruentsegments and congruent angles between two triangles

    2.3. Solve routine and non-routine problems

    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 Quadrilaterals1. Different type 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

    Median of a trapezoid

    Base angles and diagonals of an isosceles trapezoid1.3. 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

    2. Conditions that guarantee that a Quadrilateral is a Parallelogram

    2.1. Verify sets of sufficient conditions which guarantee that a quadrilateral is a parallelogram2.2. 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 KiteE. Similarity

    1. Ratio and Proportion1.1. State and apply the definition of a ratio1.2. Define a proportion and identify its parts1.3. State and apply the fundamental law of proportion

    Product of the means is equal to the product of the extremes1.4. Define and identify proportional segments1.5. Apply the definition of proportional segments to find unknown lengths

    2. Proportionality Theorems2.1. State and verify the Basic Proportionality Theorem and its Converse

    3. Similarity between Triangles3.1. Define similar figures3.2. Define similar polygons3.3. Define similar triangles3.4. Apply the definition of similar triangles

    Determining if two triangles are similar

    Finding the length of a side or measure of an angle of a triangle3.5. State and verify the Similarity Theorems

    3.6. Apply the properties of similar triangles and the proportionality theorems to calculate lengths of certainline segments, and to arrive at other properties

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    4. Similarities in a Right Triangle4.1. 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 toeach 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 30-60-90triangle using the Pythagorean Theorem

    Enrichment

    State and verify consequences of the Basic Proportionality Theorem

    Parallel lines cut by two or more transversals make proportional segments

    Bisector of an angle of a triangle separates the opposite side into segments whose lengths areproportional to the lengths of the other 2 sides

    State, verify, and apply the ratio between the perimeters and areas of similar triangleApply 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 ofthe legs

    5. Word Problems involving Similarity5.1. Apply knowledge and skills related to similar triangles to word problems

    F. Circles1. The circle

    1.1. Recall the definition of a circle and the terms related to it

    Radius

    Diameter

    Chord Secant

    Tangent

    Interior and exterior2. Arcs and Angles

    2.1. Define and identify a central angle2.2. Define and identify a minor and major arc of a circle2.3. Determine the degree measure of an arc of a circle2.4. Define and identify an inscribed angle2.5. Determine the measure of an inscribed angle

    3. Tangent Lines and Tangent Circles3.1. 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

    congruent4. 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 circlesIllustrate and identify a common internal tangent or a common external tangentGeometric 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 lineFrom a point on the lineFrom a point not on the line

    Construct the bisector of an angle

    Construct parallel lines

    Perform construction exercises using the constructions in 4.1 to 4.6

    Use construction to derive some other geometric properties (e.g., shortest distance from an externalpoint to a line, points on the angle bisector are equidistant from the sides of the angle)

    G. Plane Coordinate Geometry

    1. Review of the Cartesian Coordinate System, Linear Equations and Systems of Linear Equations in 2 Variables

    1.1. Name the parts of a Cartesian Plane1.2. Represent ordered pairs on the Cartesian Plane and denote points on the Cartesian Plane1.3. Define the slope of a line and compute for the slope given the graph of a line1.4. Define a Linear Equation1.5. Define the y-intercept1.6. Derive the equation of a line given two points of the line1.7. Determine algebraically the point of intersection of two lines

    1.8. State and apply the definitions of Parallel and Perpendicular Lines2. Coordinate Geometry

    2.1. Derive and state the Distance Formula using the Pythagorean Theorem2.2. Derive and state the Midpoint Formula

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    2.3. Apply the Distance and Midpoint Formulas to find or verify the lengths of segments and find unknownvertices or points

    2.4. Verify properties of triangles and quadrilaterals using coordinate proof3. 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 radius3.3. Determine the equation of a circle given:

    Its center and radius

    Its radius and the point of tangency with a given line3.4. Solve routine and non-routine problems involving circles