Geometry (3rd Year High School)

CORE COMPETENCIES MATH IIIPlane Geometry

FIRST QUARTER

A. Geometry of Shape and Size

1. Demonstrate knowledge and skills related to undefined terms, angles, polygons and circle1.1 describe the ideas of Point

Line

plane

1.2 name the subsets of a line

segment

ray

1.3 name the parts of angle

1.4 determine the measure of an angle using a protractor

1.5 illustrate different kinds of angles

acute

right

obtuse

1.6 illustrate different kinds of polygons according to the number sides

1.6.1 identify the parts of a regular polygon ( vertex angle, central angle, exterior angle )

1.7 differentiate convex and non-convex polygons1.8 illustrate a triangle

its basic parts

its secondary parts1.9 classify triangles according to

angles

sides

sides and angles1.10 illustrate a quadrilateral and its parts

1.11 illustrate the different kinds of quadrilaterals

1.12 determine:

sum of the measures of the angles of a triangle

sum of the measures of the exterior angles of quadrilateral

sum of the measures of the interior angles of polygon

1.13 name the terms related to the circle ( radius, diameter and chord )

2. Manifest knowledge and skills in identifying and measuring plane and solid figures and applying these in solving real life problems.

2.1 apply the formulas for the measurements of plane and solid figures] perimeter of a triangle

circumference of a circle

area of triangle, square, parallelogram, trapezoid, and circle

surface area of a cube, rectangular prism, square pyramid, cylinder, cone, and sphere

volume of rectangular prism, triangular prism, pyramid, cylinder, cone, and sphere2.2 solve problems involving plane and solid figuresB. Geometric Relations

1. Demonstrate knowledge and skills involving relations of segments and angles, sides and angles of a triangle, and angles formed by parallel lines cut by a transversal, and solve problems on the relationships between segments and between angles.1.1 illustrate and define betweeness and collinearity of points

1.2 Illustrate the following:

congruent segments

midpoint of a segment

bisector of an angle

1.3 illustrate the different kinds of angle pairs: supplementary

complementary

congruent

adjacent

linear pair

vertical angles

1.4 illustrate perpendicularity1.5 illustrate the perpendicular bisector of a segment1.6 apply relationships among the sides and angles of a triangle

exterior and corresponding remote interior angles of a triangle

triangle inequality

1.7 illustrate parallel lines1.8 Illustrate transversal

1.9 Identify the angles formed by parallel lines cut by a transversal

1.10 determines the relationships 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 transversal1.11 solve problems using the definition and properties involving relationships between segments and between anglesSECOND QUARTERC. Triangle Congruence

1. Manifest ability to illustrate and apply the conditions for triangle congruence in solving real life problems.1.1 apply the properties of congruence

Reflexive Property

Symmetric Property

Transitive Property

1.2 Use inductive skills to prove congruence between triangles

1.3 Apply deductive skills to show congruence between triangles

SSS Congruence

SAS Congruence

ASA Congruence

SAA Congruence

1.4 prove congruence 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 congruent1.5 prove inequality properties in an isosceles triangle1.6 use condition for triangle congruence to prove:

congruent segments

congruent angles

1.7 solve routine and non-routine problems

2. Apply inductive and deductive skills to derive other conditions for congruence between two right triangles

LL Congruence

LA Congruence

HyL Congruence

HyA Congruence

THIRD QUARTER

D. Properties of Quadrilaterals

1. Manifest ability to solve practical problems involving types of quadrilaterals and their properties and the conditions that guarantee that a quadrilateral is a parallelogram1.1 apply inductive and deductive skills to derive certain properties of the trapezoid

median of a trapezoid

base angles of an isosceles trapezoid

diagonals of an isosceles trapezoid

1.2 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

1.3 apply inductive and deductive skills to derive the properties of the diagonals of special quadrilaterals

rectangle

square

rhombus 1.4 verify sets of sufficient conditions which guarantee that a quadrilateral is a parallelogram

1.5 apply the conditions to prove that a quadrilateral is a parallelogram

1.6 solve routine and non-routine problems2. Enrichment

2.1 apply inductive and deductive skills to discover certain properties of the KiteE. Similarity

1. Demonstrate knowledge and skills in verifying and applying ratio and proportion, proportionality theorems, similarity between triangles and similarities in a right triangle.

1.1 Apply fundamental law of proportion

product of the means is equal to the product of the extremes1.2 apply the definition of proportionality segments to find unknown lengths

1.3 Illustrate and verify the Basic Proportionality Theorem and its Converse1.4 Apply the definition of similar triangles in:

determining if two triangles are similar

finding the length of a side or measure of an angle of a triangle

1.5 verify the Similarity Theorems:

AA similarity

SSS similarity

1.6 apply the properties of similar triangles and the proportionality theorems to calculate lengths of certain line segments

1.7 apply the AA Similarity Theorem to determine similarities in a right triangle

In a right triangle the altitude to the hypotenuse divided it into two right triangles which are similar to each other and to the given right triangle1.8 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

1.9 Derive the relationships between the sides of :

Isosceles triangle

30-60-90 triangle

Using the Pythagorean theorem2. Enrichment

2.1 verify consequences of the basic Proportionality Theorem parallel lines cut by two or more transversals make proportional segments

the bisector of an angle of a triangle separates the opposite side into segments whose lengths are proportional to the lengths of the other 2 sides

2.2 apply the ratio between the perimeters and the areas of similar triangles

FOURTH QUARTERF. Circles

1. Demonstrate knowledge and skills related to circles, arcs and angles, tangent circles, and angles formed by tangents and secant lines.1.1 Identify minor and major arcs1.2 determine the degree measure of an arc of a circle

1.3 Identify a central angle

1.4 Determine the measure of a central angle

1.5 Identify an inscribed angle

1.6 Determine the measure of an inscribed angle

1.7 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

2. Enrichment

2.1 Illustrate externally tangent circles

2.2 illustrate internal tangent circles

2.3 illustrate common internal tangents

2.4 identify and illustrate common external tangents

2.5 apply geometric constructions:

duplicating a segment duplicating an angle

construction of perpendicular bisector and the midpoint of a segment e.g. deriving the perpendicular Bisector Theorem

construction of the perpendicular to a line e.g. from a point on a line, from a pint nor on the line

construction of the bisector of an angle

construction of parallel lines

2.6 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. Demonstrate knowledge and skills related to plane coordinate1.1 derive the equation of a line given two pints of the line1.2 determine algebraically the point of intersection of two lines

1.3 apply the definition of

parallel line

perpendicular lines

1.4 derive the distance formula using the Pythagorean Theorem

1.5 derive midpoint formula

1.6 apply the Distance and Midpoint Formulas to find the:

lengths of segments

unknown vertices or points

1.7 verify properties of:

triangles

quadrilaterals

using coordinate proof

1.8 derive the standard form of the equation of a circle from the general form

1.9 given the equation of a circle, find its center and radius

1.10 determine the equation of a circle given:

its center and radius

its radius and the point of tangency with a given line

1.11 solve routine and non-routine problems involving circles