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Theorems and Formulas: Summary

UNIT I

ANGLE RELATIONSHIPS 

  • Introduction: This lesson covers points, lines, line segments, rays, angles and planes. It introduces the coordinate plane, the distance and midpoint formulas, and Pythagorean Theorem.
  • Word Problems: This lesson covers word problems for complementary and supplementary angles.
  • Introduction: This lesson deals with introducing slope, parallel and perpendicular lines and segments.

  • Proofs: Involving segments and angle relationships. These proofs apply segment addition postulate, angle addition postulate and angle relationships like complementary, supplementary and vertical angles. Including  angle relationships in parallel lines.

 

 

 

 

 

UNIT II

CONGRUENT TRIANGLES 

  • Proving triangles are congruent: Learn how to classify triangles as polygons and use reflexive, symmetrica and transitive properties in triangles. Solve problems with exterior angle sum theorem and problems that involve congruence of triangles by corresponding parts of congruent triangles are congruent. Learn to use the angle sum theorem to find angles in congruent triangles.

  • Finding triangle side and angle inequality relationships: Learn how to determine if three given sides are the sides of a triangle, and to order the angles and sides of a triangle when the sides or the angles are given. Learn the concept behind side-angle-side and side-side-side inequality theorems.  

 

 

 

 

 

 

UNIT III

QUADRILATERALS

  • Quadrilaterals: You will learn properties for quadrilaterals shown in the figures. Parallelograms, squares, rectangles, rhombi, and trapezoids.

  • Finding angles and segments on parallelograms and rectangles: This lesson will show you how to find segments and angles in parallelograms and rectangles. You will start covering the properties for parallelograms, and then a set of problems that generate linear equations, and systems of linear equations will follow.
  • Finding angles and segments on trapezoids:  The first set of problems will need that you find segments in isosceles trapezoids and the second set will require that you find angles. Properties of trapezoids together with properties for angles formed by a transversal cutting two parallel lines will be applied.

 

 

 

 

 

 

 

UNIT IV

SIMILARITY AND RIGHT TRIANGLES

  • Finding corresponding parts of similar triangles: In this lesson students will apply triangle similarity to find segments in a right triangle drawn with an altitude from his right angle to the hypotenuse. Students will draw the three similar triangles that this figure forms and will use proportionality and similarity to solve the problems.

  • Using 30-60-90 and 45-45-90 ratios: This lesson is dedicated to solve special right triangles: 30°-60°-90° and 45°-45°-90°. Students will learn how to prove these theorems and will solve problems that involve finding the sides of these kind of triangles.        

 

 

 

 

 

 

UNIT V

CIRCLES

  • Finding central angles and their arcs:    This lesson starts by introducing the basic concepts of chord, central angle, major and minor arcs, and circumference. Students then find central angles and arc's length. Students learn how to make a pie graph applying these concepts. The lesson is extended to cover angle addition postulate problems.

  • Finding congruent chords: Students will apply the Pythagorean theorem to problems that involve congruent arcs formed by a chord intersected by a diameter in a right triangle. The lesson also covers finding angles in problems with equidistant chords from the center of the circle.

  • Finding inscribed angles and their arcs:  The lesson starts by presenting inscribed angles and their intercepted arc; then it solves problems involving the concept. Students learn about inscribed and circumscribed regular polygons in  a circle. This is done in problems that generate linear equations as in two column proofs.

  • Finding arcs or angles: Students learn to solve problems about secants and tangents and the angles and arcs that may be formed when they intersect inside and outside the circle.

  • Finding segments of tangents/secants: This unit finalizes presenting to students how to find special segments in circles, when tangents and secants intersect inside and outside the circle.  

 

 

 

 

 

 

 

 

UNIT VI 

POLYGONS

  • Finding angle measure in polygons:  Learn about interior and exterior angles y polygons. You will find the number of sides of a polygon given the interior or exterior angle, and be able to calculate the interior and exterior angle sum of any polygon. 

  • Finding areas of parallelograms and rectangles: Students will review the properties of parallelograms, squares and rectangles, and then; they will learn how to find the area when the dimensions for the sides are given, or how to find one side if the dimension of the other side is given together with the area. Students will advance from simple problems that generate linear equations to problems that require to solve a system of linear equations or to use trigonometry to find the areas.

  • Finding area of special triangles. In the lesson before students learnt how to use the formula for a triangle. In this lesson they will learn how to do it in special cases where the base or the height or both are unknowns. Students will use special right triangles and trigonometry to find the missing dimensions and to be able to apply the area formula for triangles. Students will extend this problems to calculate the area of regular polygons broken into isosceles or equilateral triangles. 

  • Finding areas of regular polygons: The lesson before this, taught how to find the area of triangles and thus the area of a regular polygon broken into isosceles or equilateral triangles. In this lesson they will develop the general formula to find the area of regular polygons, based in the apothem and the perimeter of the regular polygon. A review will be given in how to find perimeter and area of circles. The lesson will finalize presenting two project based problems where students will calculate the square feet of paint necessary to paint the front of a home with windows and door that involve areas of parallelograms, circles and regular polygons.

 

 

 

 

 

 

UNIT VII 

SOLIDS

  • Finding surface area & volume of cylinders: This lesson  teaches the classification of the most common solids and concentrates in the right cylinder. A formula is developed to find the surface area and with the formula for the volume is applied in problems that go from finding both when the radius and the height are known to problems where it is necessary to apply the Pythagorean theorem to find them. It is extended to problems that involve finding the radius or the height when one and the volume is known. It finalizes using similarity in solids.
  • Surface Area and Volume with Base 10 Blocks: This activity presents the use of base 10 blocks to find surface area and volume of solids made of prisms. It may be taught with these manipulatives. Students learn how to project the views and how to get the minimum amount of these views to represent the solid figure. They learn how to do the isometric view and how to get the figure from the views. It is a long file to download but it is worthwhile the wait.


  • Finding surface area and volume of prisms:This lesson  involves finding the surface area formula for prisms and presenting the one for the volume, and using them in problems. The activity is extended to calculate, geometrically, the product of three binomials. It presents similarity in solids at the end of the lesson.

  • Finding surface area and volume of cones: Students develop the surface area formula and learn the volume formula. They apply both to the solution of problems where the radius and the height are given or where they need to be found, before using these formulas.
  • Finding surface area and volume of pyramids: Pyramid surface area and volume are found in this lesson. Students need to find the perimeter of the bases and then they apply the developed formula for surface area and the one for the volume.  
  • Finding surface area and volume of spheres:Students will be able to find surface area and volume of spheres when the given information is the radius, the diameter, the circumference, or the surface area and volume. They will have to find the radius, diameter and circumference when the surface area and volume are given.

 

 

 

 

 

 

 

 

 

UNIT VIII 

COMPOSITE SOLIDS