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#### Theorems and Formulas: Summary

• ANGLE RELATIONSHIPS
• TRIANGLES
• PROPORTIONALITY
• CIRCLES
• POLYGONS
• S.A. & VOLUME
• S.A. & VOL. COMPOSITE SOLIDS
• TRANSFORMATIONS

UNIT I

ANGLE RELATIONSHIPS

• Introduction to geometry: Through the work of this lesson students have the opportunity to learn about points, lines, rays, angles, and planes in the coordinate plane and as normally it is covered in plane geometry. For the coordinate geometry the distance formula, and midpoint formulas are introduced for one and two dimensions. The Pythagorean Theorem is briefly introduced since it is used early in some lessons. For it there is a more in depth coverage in more advanced sections of the geometry section in this website.
• Finding complementary, supplementary, and congruent angles: Students need to understand that in geometry you start with a point, then lines, then rays, and finally you get to angles. These in turn have some important relationships when they are in pairs. These are complementary, supplementary, and vertical angle relationships. This lesson introduces those concepts and definitions, including the case of supplementary angles forming a linear pair.
• In geometry as in any other discipline is of paramount importance that students learn to “read” word problems, and how to distinguish the relevant from the non-relevant information in the solution of a problem. This lessons highlights that process by underlining in different colors the information that corresponds to the right and left side of the equations that needs to be setup for the solution of the problems.
• Concrete to abstract activity: Modeling complementary, supplementary, linear pairs and vertical angles using Geo-Legs: Mathematics in general implies to deal with abstraction. For some students the concepts behind angle pairs: Complementary, supplementary and vertical. Turns to be somewhat “too abstract.” They are not to be blamed for that. Opportunities to develop this are different from individual to individual. The advantage of the lesson for students is to get a visual approach to deal with these concepts including addressing the issue of not knowing how to read angular scales. Geolegs are used to present the concepts. These a available online and chances are students themselves have them already at their homes.
• Introduction: Determining the difference with parallel lines and perpendicular lines in the coordinate plane includes to deal with the concept of the inclination of a line or slope. This in turn may be defined in terms of the coordinates of two points, or in terms of the change in “x” and the change in “y”. This lesson shows students the two ways of approaching the slope.

• Finding angles involving parallel lines. Once students were exposed to the concept of parallel lines, and the concepts behind angle pair relationships. They are ready to apply it to parallel lines cut by a transversal and the angle relationships that take place in both intersections and how they are related in both of them. Further in more advanced sections this knowledge will enable them to solve problems that involve triangles, quadrilaterals, circles, and, so on. In this lesson, those skills are fully developed.
• From conditionals: If...Then; to logical reasoning: deductive & inductive A big big in geometry is to develop the ability to use inductive and deductive reasoning in the solution of problems in geometry, and to extend it to the real life of the students. They need to know what is a logical “if …then” statement, its converse, its inverse, and its contrapositive. They need to learn to identify and use the Law of Syllogism and The Law of Detachment. These are vastly used in the geometric proofs: Both formal and informal, and are part of our natural thinking tools to interpret the world around us and our relationship with it. Therefore the working the lesson, you will develop this way of thought.

• Proofs: Involving segments and angle relationships. These proofs apply segment addition postulate, angle addition postulate and angle relationships like complementary, supplementary and vertical angles. Includingangle relationships in parallel lines. You will be able to setup the proof and see step by step how it is developed to reach the final conclusion.
• Writing across the curriculum: PARAGRAPH PROOFS. Proofs may be developed using a “T” table format, a flow chart, or in a paragraph. We do proofs everyday. When you want to put in order your room, you look around and check what is out of order, and then you determine the right place for each item and then carry out the necessary actions to put it in order. If you put these steps on a “T” table, and then use this to put it in a paragraph, then you would be doing exactly what is done here in the lesson, but with geometric proofs involving segments, angles, and angle relationships.

UNIT II

CONGRUENT TRIANGLES

• Proving triangles are congruent: To determine if two triangles are congruent or not, you compare the 3 corresponding sides and 3 corresponding angles. There are some shortcuts to simplify this process. Anyhow, once you know that two triangles are congruent, you might want to find the numeric value for the length of the sides, or the angular measure of the angles. The lesson starts when you are given triangle pairs where sides, or angles are in terms of algebraic expressions and then students need to setup equations using corresponding congruent angles or sides. Sometimes they need to resort to the Interior Angle Sum Theorem to accomplish this.

• Proving triangles are congruent (SSS, SAS, ASA, AAS, HL, HA, LA): If you are asked to compare two cars, and determine if they are identical. You would go to compare doors, windows, seats, etc… What about if they tell you that once you determine that the seats and the dashboard are identical then you may jump to the conclusion that the two cars are identical. That would save time. You do the same when determining if two triangles are congruent. You may compare all 6 corresponding part pairs, or you may compare just 3 corresponding part pairs using shortcuts: Side-Side-Side, Side-Angle-Side, Angle-Side-Angle, or Angle-Angle-Side. These in turn are widely used in proofs with triangles, triangles and quadrilaterals, circles, and so on. This lessons exposes you to identify the missing corresponding part to be able to state that two triangles are congruent for any of the given shortcuts.

• Finding triangle side and angle inequality relationships: If you look at the roof of most american homes, you will see triangles made with interconnected beams. Carpenters need to know the length of those beams to actually get a triangle and not ending up with three pieces of wood that can’t be connected to complete a triangle because they are too short. This lesson deals with the concepts that help you to determine when three segments may be interconnected at the endpoints to complete a triangle. Now, you might be interested in ordering the sides, or the angles of a triangle in ascending or descending order. This is also taught when you know the sides to order the unknown angles, or when you know the angles to order the unknown sides. It is also relevant to compare triangles when they have two congruent pairs of corresponding sides. This is addressed as well in the lesson: Side-Side-Side and Side-Angle-Side Inequality Theorems in triangles.

• Proofs: Involving congruence theorems in triangles. Once, you have mastered the concepts necessary to determine in an effective way if two triangles are congruent or not. You are capable of completing proofs that ask you to determine if two triangles are congruent given a set of known parts of two triangles that may be given separated or embedded in a figure. But this is extended to cases where you use these concepts to complete proofs for which the end statement is not to prove if two triangles are congruent but to use this as an intermediate step in proving some other type of statement. These proofs apply SSS, ASA, SAS, etc.

 UNIT IV SIMILARITY AND RIGHT TRIANGLES Proving triangles are similar PART1: This lesson deals with helping you to use similarity theorems to determine side lengths. You will apply Side-Side-Side, Angle-Angle, and Side-Angle-Side similarity theorems to the solution of problems that try to find segments in pairs of similar triangles. Students will extend these concepts to a parallel to the third side in a triangle (including when this goes from the midpoints of the other two sides) and to parallels cut by transversals. Students will apply it to polygons in general. Proving triangles are similar PART2: You have special segments in triangles: Altitudes, angle bisectors, medians, and perpendicular bisectors. When in any triangle you draw any of these; you generate a set of similar triangles embedded in the figure. Knowing the similarity theorems in turn enables you to find segment lengths in the perimeter of the figure or in the interior. These lesson exemplifies how to do it in the figure itself or how to draw the generated triangles and use them to carry out the solution. Finding corresponding parts of similar triangles: You are given a right triangle. Then you are asked to draw the altitude from the right angle to the opposite side. This will generate three similar triangles. The original one plus the two smaller ones in the interior of the original triangle. This lesson approaches the solutions separately drawing the three triangles and highlighting in them the parts to be used to setup proportions to find their lengths, and or using the Pythagorean Theorem. Proofs: Involving triangles and similarity. In previous sections you have learned how to determine if two triangles are congruent or similar, and how to use theorems related to segment or angle addition postulate, and angle relationships. This body of knowledge and skills will enable you to setup and carry out full proofs involving triangles and other polygons in standalone triangles, or figure embedded ones. 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°. You will learn to find the corresponding relationship of the three sides in a triangle when you have any of the two angle sets listed above. This at the same time will help you to setup equations to find segment lengths in them. Using trigonometric ratios and inverse trigonometric ratios: One of the most useful topics in geometry for both: Career oriented students or college bound ones, is trigonometry applied to right triangles. Sine, cosine, and tangent are widely used to get indirect measurements of angles and segments in situations in which it wouldn’t be practical, or even impossible to attempt it otherwise. Navigation is one of the ones that demands more use of trigonometry knowledge, and in particular the application of Angles of Elevation and Depression. In the past when calculators were nonexistent, then only way to work with trigonometric problems was using tables or graphs. This is no longer in use, but the lesson does a brief review of this. This lesson dwells into all these concepts. Using Law of Sines and Law of Cosines: Not all triangles are right triangles. You may have acute or obtuse triangles. In these situations the Sine, Cosine, and Tangent uses learned in solution of right triangles does not help. Nevertheless, you have the Law of Sines and the Law of Cosines. Both deal with finding segments and angles in triangles that are non-right triangles. The lesson offers the proof for them.

 UNIT V CIRCLES Finding central angles and their arcs:The spokes of bike tyre are examples of radii, the partial arc seen at the top of the arches in a colonial building are minor arcs, the pie you eat to celebrate your birthday is a circle’s sector. 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: If we make two chords to intersect at a right angle; so that one of them is the largest chord possible in the circle (Diameter) then the other chord will cut both the chord and the intersected arc by this chord in congruent parts. This lesson will apply these concepts to solution of problems that will make use of the Pythagorean theorem to complete the solutions. At the end you will see how this is applied to two column proofs where concepts like triangle congruence will be very handy. Finding inscribed angles and their arcs: Usually we cut pizza slices by placing an imaginary point at the center. This brings about central angles. If rather that placing that imaginary point in the center we place it at the circumference then we get an inscribed angle. Now the segments that make up the inscribed angle are chords. We use these concepts to find angle measure and segment length in figures embedded into a circle. It is a multistep problem that uses all the information that is generated along the solution. Finding arcs or angles formed by intersecting lines or segments in, on, or out the circle: If you throw an arrow to a rounded watermelon and you perforate it in a way that the arrow protrudes at both sides of the watermelon. Then you have placed a secant to the sphere in the minor circle that contains the segment of the arrow that is at the interior of the watermelon. If now you take a knife and insert it in the watermelon so that it touches the arrow in one point. They you may generate two secants intersecting outside or on the circle. Depending where the knife is touching the arrow. That is why 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: You may take two Chinese sticks and put them on top of a circle. They might intersect at the interior, on the circle itself, or outside the circle. Now they might generate two secants, a secant and a tangent, or two tangents. This unit finalizes presenting to students how to find special segments in circles, when tangents and secants intersect inside and outside the circle. Proofs: Involving segments, angles and arcs in circles. Throughout the lessons given before you have learned how to work with angle relationships, segment and angle addition postulate, triangle congruence, inequality, and similarity, polygons, and of course the theorems and postulates that are generated by segments and lines intersecting inside, on, or outside the circle. Now, you will apply all this knowledge to complete full two column proofs that use all these elements of the geometry course. Proofs: Involving chords, secants, tangents, angles and arcs in circles. Same like the lesson before, you will be given the opportunity to work with all previous knowledge related to angle relationships, congruence, inequality, and similarity in triangles, segments and angles in quadrilaterals, and polygons in general. All this in conjunction with the circle theorems and postulates that involve finding segments, angles, and arcs formed by intersecting chords, tangents, and secants inside, on, or outside the circle.

 UNIT VI  POLYGONS Finding angle measure in polygons: If you take a circle and randomly place points on its circumference so that they are not one on top of the other or right next to the other, and then you connect them with the immediate adjacent points with segments. You get an inscribed polygon to a circle. Or you may say that you have a circle circumscribing a polygon. Each one of these segments is a chord that with the contiguous chord will form an interior angle, that in turn will have an exterior angle associated to it. You will learn about interior and exterior angles in 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: Most living quarters are a type of parallelogram in their floor shape. If you had to replace the carpet of your house this proces would become very handy, if you need to calculate how much paint to buy to paint your house the same is true. 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 in terms of the given side. 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 rhombi, trapezoids, and triangles: It doesn't matter if you are career or college bound. In any case you will encounter the need to apply formulas to find areas to apply directly or to surface area and volume. A project at home might need that you calculate the interior volume of the house, or in college in a calculus class you might need to apply one of this formulas to find the answer of a more complex problem. Once you have harnessed the properties for triangles, rhombi, and trapezoids then you will put them to use in this lesson to find areas of figures in these shapes. Some of the problems will generate second degree equations that will need that you use the Quadratic Formula. Finding area of special triangles. You should know that in real life applications, many times you are not able to get in a direct way all the measures you need to have to apply the formulas with direct substitution. 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. Therefore you will see how useful the sequence and animations in the lesson are. It 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: Do you know why the formula for the cylinder is given as is? 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: For some students finding volumes and surface area of composite figures carries some difficulty for the abstraction involved in the solution of the problem. Base 10 blocks were designed with the purpose to provida a concret to representational to abstract approach. In other words a hands on way of tackiling the solution. You will have that 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: You will find very useful to learn the process of 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: Had you encountered problems in the past to find and develop the surface area formula of a cone, and learn how to apply it together with the volume formula? It is applied 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. So you will need to find the perimeter of the bases and then apply the developed formula for surface area and the one for the volume. You will see how the formula for surface area is developed. Finding surface area and volume of spheres: If you play baseball, or soccer. You might be interested in finding the surface area of the baseball in case you had to replace the leather in the outer layer, or should you need to calculate the amount of air necessary fill in the volume of the soccer ball. You will discover that you 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 Finding surface area and volume of composite solids: Adding solids. Composite volumes are much closer to what in real life you might encounter when calculating volumes and surface area. A house is not "pure" prism or a cone, or even a shoe box has the lid and the box itself with different dimensions. In the following presentation students will learn how to find surface area and volume when it is necessary to add areas and volumes of simple plane figures and solids to find the final answer of the composite solids. Finding surface area and volume of composite solids: Subtracting solids. Objects have what is called "cavities" or hollow spaces inside them. In this lesson you will learn how to find surface area and volume when it is necessary  to subtract areas and volumes of simple plane figures and solids to find the final answer of the composite solids. Finding surface area and volume of composite solids: Adding and Subtracting solids. Students will learn than to find surface area and volume when it is necessary to add and subtract areas and volumes of simple plane figures and solids to find the final answer of the composite solids where the solid has cavities and protuberances to be taken into account when the volume or surface area are calculated. Finding Volumes of Solids of Revolution Have you seen the rotating blades of an Egg Beater Blender or the ones of a hand mixer. When they rotate very fast they seem to form a solid (cylinder or cone, etc.) Those are solids of revolution because they are generated by rotating an area. This lesson focuses on calculating the volume for them with the simple tools learned in plane geometry. Calculus has a more accurate approach but requires algebra 2 and other tools not expected to have been taught up to now. Finding Surface Area of Solids of Revolution. The previous lesson presented to you how to find the volume for a solid of revolution. Now using a very similar process we are going to obtain the surface area of the solid. These solids may be generated by rotating a segment line, arc, set of connected lines of different type, etc. Their application is very important with dealing in finding surface areas of solids with circular transversal section but not constant diameter along the length, so that they get a complex profile. Using what you learned about regressions in Algebra 1 (if you had the opportunity) you may get polynomial expressions for the lines that make the contour of these kind of profiles and then using them to calculate the surface area and volume. This is done in calculus using integrals, the idea is that here in plane geometry you get a "glimpse" of the process, which conceptually is the same that when you use integrals.