Curriculum Guide
Geometry
- Identifies quadrilaterals that are parallelograms.
- Determines the conditions that make a quadrilateral a parallelogram.
- Uses properties to find measures of angles, sides and other quantities involving parallelograms.
- Proves theorems on the different kinds of parallelogram (rectangle, rhombus, square).
- Proves theorems on the different kinds of parallelogram (rectangle, rhombus, square).
- Proves the Midline Theorem.
- Proves theorems on trapezoids and kites.
- Solves problems involving parallelograms, trapezoids and kites.
- Describes a proportion.
- Applies the fundamental theorems of proportionality to solve problems involving proportions.
- Illustrates similarity of figures.
- Proves the conditions for similarity of triangles.
- Applies the theorems to show that given triangles are similar.
- Proves the Pythagorean Theorem.
Lesson 1: Quadrilaterals
This module shall focus on quadrilaterals that are parallelograms, properties of a parallelogram, theorems on the different kinds of parallelogram, the Midline theorem, theorems on trapezoids and kites, and problems involving parallelograms, trapezoids, and kites.
Properties of Parallelogram
- In a parallelogram, any two opposite sides are congruent.
- In a parallelogram, any two opposite angles are congruent.
- In a parallelogram, any two consecutive angles are supplementary.
- The diagonals of a parallelogram bisect each other.
- A diagonal of a parallelogram forms two congruent triangles.
A = bh
Theorems of Rectangle
- If a parallelogram has a right angle, then it has four right angles and the parallelogram is a rectangle.
- The diagonals of a rectangle are congruent.
A = bh
Theorems on Rhombus
- The diagonals of a rhombus are perpendicular.
- Each diagonal of a rhombus bisects opposite angles.
The Midline Theorem
- The segment that joins the midpoints of two sides of a triangle is parallel to the third side and half as long.
Solving a Problem Using the Middle Theorem
Given: point A, point B
Find: middle X – axis, middle Y- axis, midpoint
Solve using this formula:
x = x + x ÷ 2 = X – axis
y = y + y ÷ 2 = Y – axis
(X – axis, Y – axis) = midpoint
The Midsegment Theorem of Trapezoid
- The segment joining the midpoints of the legs of a trapezoid is called median.
- The median of a trapezoid is parallel to each base and its length is one half the sum of the lengths of the bases.
Theorems on Isosceles Trapezoid
- The base angles of an isosceles trapezoid are congruent.
- Opposite angles of an isosceles trapezoid are supplementary.
- The diagonals of an isosceles trapezoid are congruent.
Solving Problems Involving Theorems on Trapezoids
Solving the area of a trapezoid:
Given:
Short base, Long base, Height
Problem:
What is the area of the trapezoid?
Formula:
A = (a+b) / 2 x h
Theorems on Kite
- In a kite, the perpendicular bisector of at least one diagonal is the other diagonal.
- The area of a kite is half the product of the lengths of its diagonals.
Solving Problems Involving Kites
Given:
Width, Height
Problem:
What is the area of the Kite?
Formula:
A = w * h/2
Solving Problems Involving Parallelograms, Trapezoids, and Kites
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