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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