Unit 3: Triangles and Polygons
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1 Unit 3: Triangles and Polygons Background for Standard G.CO.9: Prove theorems about triangles. Objective: By the end of class, I should Example 1: Trapezoid on the coordinate plane below has the following vertices: 3,8, 3,4, 11,4, and 11,10. Shade the interior of this shape with your pencil. H E G F A B D C was translated 14 units to the left and 12 units down to form trapezoid. List the coordinates of the new vertices. : : : : Is? Explain your reasoning. A D B C J I K L was rotated 90 counter clockwise about the origin to form trapezoid. List the coordinates of the new vertices. Is Explain your reasoning. was reflected across the x axis to form trapezoid. List the coordinates of the new vertices. Is? Explain your reasoning. **In order for two shapes to be congruent (exactly the same), corresponding angles must be congruent and corresponding sides must be congruent. Example 2: Consider the congruence statement: A. Identify the congruent angles. B. Identify the congruent sides.
2 Example 3: How much information do we need before knowing that two triangles are congruent? A. Given three fixed side lengths, how many different triangles can you create? B. Given two fixed side lengths and a fixed angle (where the angle is in between the sides, how many different triangles can you create? C. Given one fixed angle and one fixed side length, how many different triangles can you create? D. Given two fixed angles and one fixed side length (where the side is between the two angles), how many different triangles can you create? E. Given two fixed side lengths, how many different triangles can you create? F. Given two fixed side lengths and one fixed angle (where the angle is not in between the sides), how many different triangles can you create? G. Given three fixed angles, how many different triangles can you create? H. Given two fixed angles and one fixed side (where the side is not in between the two angles), how many different triangles can you create?
3 Use the previous page to summarize your findings about triangle congruence theorems. Triangle Congruence Theorems NOT Triangle Congruence Theorems How many pieces of information are necessary to have congruent triangles? What two patterns with three pieces of given information are NOT theorems? If not congruency, what can they guarantee? Example 4: Suppose and in the diagram shown. Are there congruent triangles in the diagram? If so, write a triangle congruence statement and name the theorem used. Example 5: Suppose is the midpoint of and is the midpoint of in the diagram shown. Are there congruent triangles in the diagram? If so, write a triangle congruence statement and name the theorem used. Example 6: Suppose, and bisects in the diagram shown. Are there congruent triangles in the diagram? If so, write a triangle congruence statement and name the theorem used.
4 Use the diagrams below and the provided information to prove the two triangles are congruent by SSS, SAS, ASA, and AAS. Example 7: Example 8: Example 9: Given and Example 10:
5 Background for Standard G.CO.9: Prove theorems about triangles. Objective: By the end of class, I should List the 4 triangle congruence theorems you explored previously. These congruence theorems apply to all triangles. There are also theorems that only apply to right triangles. To prove that two right triangles are congruent, only two pieces of information (side lengths or angles) are necessary because you are always given one angle the right angle. Which general triangle theorem (listed in the boxes at the top of this page) correlates with each right triangle theorem? Leg Leg (LL) Congruence Theorem Hypotenuse Angle (HA) Congruence Theorem Leg Angle (LA) Congruence Theorem Hypotenuse Leg (HL) Congruence Theorem *Because most of the right triangle congruency theorems are repeats of the general triangle congruency theorems, the only one we really care about is. Add this special theorem to your list above with an asterisk (*). Explain why only two pairs of corresponding parts are needed to prove that two right triangles are congruent. Use the diagrams below and the provided information to prove the two right triangles are congruent. Example 1:
6 Example 2: Example 3: Example 4: Given:, Prove: : Review : What kind of triangle is GBD? What are the congruent base angles of triangle GBD? What does it mean for two triangles to be congruent? Corresponding must be congruent and corresponding must be congruent. Therefore, if two triangles are congruent, then. This is what s known as C.P.C.T.C.
7 To use CPCTC to explain your reasoning, follow these steps: Step 1: Identify two triangles in which segments or angles are corresponding parts. Step 2: Use a triangle congruency theorem to prove the triangles congruent. Step 3: State that the two parts are congruent using CPCTC as the reason. Example 6: In the diagram below, is? Explain your reasoning. Example 7: In the diagram below, is? Explain your reasoning. Example 8: In the diagram below,,, and are right angles. Is? Explain your reasoning.
8 Background for Standard G.CO.11: Prove theorems about triangles. Objective: By the end of class, I should The Isosceles Triangle Base Theorem states: The altitude to the base of an isosceles triangle bisects the base. Draw an altitude from the vertex angle to the base. Remember altitudes are perpendicular to bases. The altitude to the base also the vertex angle. The altitude is to the base. In an isosceles triangle, the altitudes to the congruent sides are. In an isosceles triangle, the angle bisectors to the congruent sides are. Example 1: Given:,, 12., 20. Find the width of the doghouse.
9 Logic s Conditional: If p, then q. Converse: If q, then p. Inverse: If not p, then not q. Contrapositive: If not q, then not p. Truth Value: True means always true! Example 2: Given the following conditional statement, complete the remaining logic statements and give the truth value for each. Conditional: If a triangle is equilateral, then it is isosceles. T/F? Converse: T/F? Inverse: T/F? Contrapositive: T/F? Example 3: Create a conditional statement that is not related to Geometry. Example: If my alarm goes off, then I get out of bed. Write out the converse, inverse and contrapositive. Then give the truth value for all four statements. When you have completed this, compare your example with a neighbor and be prepared to share your example with the class. Be creative! Conditional: T/F? Converse: T/F? Inverse: T/F? Contrapositive: T/F? A Biconditional (if and only if) is true only when the conditional and the converse are both true. Example: A trapezoid is isosceles if and only if its diagonals are congruent. Is the conditional true? If the converse true?
10 Background for Standard G.CO.11: Properties of Squares, Rectangles, Parallelograms, Rhombi, Kites and Trapezoids. Objective: By the end of class, I should Polygon: a two dimensional, plane shape that is made of straight lines and is closed (all the lines connect). The root of the word polygon is Greek: Poly means many and gon means angle. So many angle there you go! Quadrilateral: a four sided polygon We will discuss 7 different quadrilaterals: Squares, Rectangles, Parallelograms, Rhombi, Kites, Trapezoid, and Isosceles Trapezoid. (Refer to your shapes for details about each quadrilateral). Some of the shapes on the previous page are special cases of the other shapes. This flowchart details which categories are subset of which other categories.
11 Example 1: Name at least one quadrilateral that fit each description. A. Both pairs of opposite sides are congruent B. All angles are congruent C. Each diagonal bisects opposite angles D. Exactly one pair of opposite sides are parallel E. Consecutive angles are supplementary F. Exactly one pair of opposite angles are congruent G. Exactly one pair of opposite sides are parallel and each pair of base angles are congruent H. All sides are congruent Example 2: Create a flowchart to classify any quadrilateral based on the filled in questions. This chart will help you categorize any quadrilateral. How many pairs of 0 parallel sides are there? 1 2 Are there four right angles? Yes Are the four sides congruent? No Are the four sides congruent? Yes No Yes No
12 Background for Standard: Use geometric shapes, their measures, and their properties to describe objects. Objective: By the end of class, I should Interior Angles An interior angle faces the inside of the polygon and is formed by consecutive sides of the polygon. Let s start with the polygon with the fewest number of sides (a triangle) and work up to a polygon with 10 sides (called a decagon) and look at the sum of all of the interior angles. Triangle (3 sides) Pentagon (5 sides) Number of Triangles: Quadrilateral (4 sides) Number of Triangles: Hexagon (6 sides) Number of Triangles: Heptagon (7 sides) Number of Triangles: Nonagon (9 sides) Number of Triangles: Octagon (8 sides) Number of Triangles: Decagon (10 sides) Number of Triangles: What is the formula for the sum of the interior angles of an n sided polygon? Explain your reasoning. Example 1: What is the sum of all of all the interior angle measures of a 100 sided polygon? Example 2: If the sum of all the interior angle measures of a polygon is 9540, how many sides does the polygon have?
13 Exterior Angles The picture on the right shows what exterior angles look like on a triangle. Each exterior angles forms a linear pair with an interior angle. Let s see if we can come up with a formula for the sum of all of the exterior angle measures of a polygon, starting with a triangle. Triangles: Calculate the sum of the exterior angle measures of a triangle with the following steps. 1. In the provided triangle estimate the degree measure of each angle. You can estimate these numbers, but make sure they add up to 180 like all triangles do. 2. Use the angle measures you estimated to find the measure of each exterior angle. Remember that each interior angle is supplementary to its exterior angle. 3. Add up the three exterior angles. Quadrilaterals: Repeat the steps above to calculate the sum of the exterior angle measures 1. In the provided quadrilateral estimate degree measure of each angle. You can estimate these numbers, but make sure they add up to 360 like all quadrilaterals do. 2. Use the angle measures you estimated to find the measure of each exterior angle. Remember that each interior angle is supplementary to its exterior angle. 3. Add up the four exterior angles. Are you seeing a pattern? What is the sum of all of the exterior angle measures of a polygon? Example 3: Use what you know about the sum of the exterior angles of a polygon to solve for the variable(s) in each picture. A. B.
14 Regular Polygons In a regular polygon, all of the sides are congruent. This means that all of the angles are also congruent. If all of the angles of a regular polygon are congruent, then we can find the measure of one interior or exterior angle by dividing the formula by the number of angles. Measure of each interior angle of a regular polygon: Measure of each exterior angle of a regular polygon: 360 Example 4: Use the formula to calculate each interior angle measure of a regular 100 sided polygon. Example 5: If each interior angle measure of a regular polygon is equal to 150, determine the number of sides. Explain how you calculated your answer. Example 6: Calculate the measure of each exterior angle of an equilateral triangle (a regular triangle). Explain your reasoning. Example 7: Calculate the measure of each exterior angle of a 25 sided polygon. Explain your reasoning. Example 8: If the measure of each exterior angle of a regular polygon is 18, how many sides does the polygon have? Explain how you calculated your answer.
15 Review Example 1: With your partner, complete the table by placing a checkmark in the appropriate row and column to associate each figure with its properties. Characteristic No parallel sides Exactly one pair of parallel sides Two pairs of parallel sides One pair of sides are both congruent and parallel Two pairs of opposite sides are congruent Exactly one pair of opposite angles are congruent Two pairs of opposite angles are congruent Consecutive angles are supplementary Diagonals bisect each other All sides are congruent Diagonals are perpendicular to each other Diagonals bisect the vertex angles All angles are congruent Diagonals are congruent Quadrilateral Trapezoid Kite Parallelogram Rhombus Rectangle Square Example 2: As a class, create a Venn diagram that describes the relationships between all of the quadrilaterals listed: trapezoid, rhombi, parallelograms, quadrilaterals, kites, rectangles, squares. Example 3: True or False? A square is also a rectangle. A rectangle is also a square. The base angles of a trapezoid are congruent. A parallelogram is also a trapezoid. A square is a rectangle with all sides congruent. The diagonals of a trapezoid are congruent. A kite is also a parallelogram. The diagonals of a rhombus bisect each other.
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