Unit 2: Triangles and Polygons Background for Standard G.CO.9: Prove theorems about lines and angles. Objective: By the end of class, I should Using the diagram below, answer the following questions. Line w and line x are lines, which means that they Line y intersects both line w and line x. It is called a transversal. What angles do you know to be congruent because they are vertical angles? What angles do you know to be supplementary because they form a linear pair? This picture contains two intersections. Which angle on the intersection of line x and line y corresponds to 1 on the intersection of line w and line y? Which angle corresponds to 6 on the other intersection? Which angle corresponds to 7 on the other intersection? Corresponding Angle Postulate: If two parallel lines are intersected by a transversal, then corresponding angles are. A conjecture is a hypothesis that something is true. The hypothesis can later be proved or disproved. A postulate is a statement that is accepted without proof. A theorem is a statement that can be proven. If you can prove that a conjecture is true, it becomes a theorem.
Example 1: List which pairs of angles fall under each category. Then, keeping the Corresponding Angle Postulate in mind, write a conjecture about each pair of angles formed by parallel lines cut by a transversal. Explain how you made each conjecture. A. Alternate Interior Angles B. Alternate Exterior Angles C. Same-Side Interior Angles D. Same-Side Exterior Angles Example 2: Are the Converses true? Do the conjectures we made on this page work the other way? For example, if the corresponding angles are congruent, do the lines have to be parallel?
Alternate Interior Angle Theorem: If two parallel lines are intersected by a transversal, then alternate interior angles are. Alternate Exterior Angle Theorem: If two parallel lines are intersected by a transversal, then alternate exterior angles are. Same-Side Interior Angle Theorem: If two parallel lines are intersected by a transversal, then interior angles on the same side of the transversal are. Same-Side Exterior Angle Theorem: If two parallel lines are intersected by a transversal, then exterior angles on the same side of the transversal are. Example 3: Using the figure below, list all the pairs of the types of angle relationships. Corresponding Angles Alternate Interior Angles Same-Side Interior Angles Alternate Exterior Angles Same-Side Exterior Angles Vertical Angles Linear Pairs Example 4: Using the diagram from Example 3, if m 2 = 110, find the measures of all the other angles. Example 5: Using the diagram from Example 3, if m 7 = 50, find the measures of all the other angles.
Example 6: Given l 1 l 2 and l 3 l 4. Using the diagram, provide the appropriate theorem or postulate that supports each statement. The symbol used for congruence is For example, 3 13 is read: Angle 3 is congruent to angle 13. It means that angle 3 and angle 13 are the same size and have the same measure. Example 7: Solve for x in each figure. A. B. Example 8: Use the figure to determine the measure of each indicated angle. A. What is the value of x? B. m EGA C. m CHF D. m FHD E. m EGB
Background for Standard G.CO.10: Prove theorems about triangles. Objective: By the end of class, I should Triangle Sum Theorem: Draw any triangle on a piece of paper. Tear off the triangle s three angles. Arrange the angles so that they are adjacent angles. What do you notice about the sum of these three angles? The sum of the measures of the interior angles of any triangle is. Example 1: Use the triangle sum theorem to solve for x in each diagram. A. B. Example 2: Describe the following classifications of triangles: By Their Sides Scalene Acute By Their Angles Isosceles Right Equilateral Obtuse Example 2: Label the sides and angles of an isosceles triangle.
Isosceles Triangle Base Angle Theorem: If two sides of a triangle are congruent, then the angles opposite these sides are congruent. Isosceles Triangle Base Angle Converse Theorem: If two angles of a triangle are congruent, then the sides opposite these angles are congruent. Isosceles Triangle Base Theorem: The altitude to the base of an isosceles triangle bisects the base. Isosceles Triangle Vertex Angle Theorem: The altitude to the base of an isosceles triangle bisects the vertex angle. Isosceles Triangle Perpendicular Bisector Theorem: The altitude from the vertex angle of an isosceles triangle is the perpendicular bisector of the base. Example 3: Using Isosceles Triangle Theorems A. Calculate AP if the perimeter B. Calculate m T. of AYP is 43 cm. C. Solve for x. D. Solve for x. E. If m 2 = x + 94, solve for x. F. Solve for x and y.
Background for Standard G.CO.9: Prove theorems about triangles. Objective: By the end of class, I should Example 1: Trapezoid ABCD on the coordinate plane below has the following vertices: A(3,8), B(3,4), C(11,4), and D(11,10). Shade the interior of this shape with your pencil. H E G F A B D C ABCD was translated 14 units to the left and 12 units down to form trapezoid A B C D. List the coordinates of the new vertices. A : C : B : D : Is ABCD A B C D? Explain your reasoning. A D B C J I K L ABCD was rotated 90 counter-clockwise about the origin to form trapezoid EFGH. List the coordinates of the new vertices. E: G: F: H: Is ABCD EFGH Explain your reasoning. ABCD was reflected across the x-axis to form trapezoid IJKL. List the coordinates of the new vertices. I: K: J: L: Is ABCD IJKL? 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: JRB MNS A. Identify the congruent angles. B. Identify the congruent sides. Example 3: Given the following information, write a triangle congruency statement. A. B K B. OV SR W M VT RX P C OT SX
Example 4: 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? What pattern is this? Is it a triangle congruency theorem? 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? What pattern is this? Is it a triangle congruency theorem? C. Given one fixed angle and one fixed side length, how many different triangles can you create? What pattern is this? Is it a triangle congruency theorem? 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? What pattern is this? Is it a triangle congruency theorem? E. Given two fixed side lengths, how many different triangles can you create? What pattern is this? Is it a triangle congruency theorem? 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? What pattern is this? Is it a triangle congruency theorem? G. Given three fixed angles, how many different triangles can you create? What pattern is this? Is it a triangle congruency theorem? 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? What pattern is this? Is it a triangle congruency theorem?
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? 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 of the following right triangle theorems? 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 (*).
Example 5: Suppose AB AD and BC DC 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 AB DF, AC, DE and BE FC in the diagram shown. Are there congruent triangles in the diagram? If so, write a triangle congruence statement and name the theorem used. Example 7: Suppose M is the midpoint of AD and M is the midpoint of BC in the diagram shown. Are there congruent triangles in the diagram? If so, write a triangle congruence statement and name the theorem used. Example 8: Suppose AB MC, and BM CD in the diagram shown. Are there congruent triangles in the diagram? If so, write a triangle congruence statement and name the theorem used. Example 10: Suppose AC BC, and DC bisects C in the diagram shown. Are there congruent triangles in the diagram? If so, write a triangle congruence statement and name the theorem used.
Background for Standard G.CO.10: Prove theorems about triangles. Objective: By the end of class, I should **Remember that 3 pieces of information are needed to show that two triangles are congruent. You may need to use the picture to find enough pieces of information. Example 1: Is ABC DCB? Use a triangle congruence theorem to support your answer. Example 2: Is ABC ADC? Use a triangle congruence theorem to support your answer. Example 3: Is ABC DCB? Use a triangle congruence theorem to support your answer. Example 4: Is GAB SBA? Use a triangle congruence theorem to support your answer. Example 5: If JA MY and JY, AY is JYM AYM?
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 their must be congruent. This is what s known as C.P.C.T.C. 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 AG BS? Explain your reasoning. Example 7: In the diagram below, is DBC ACB? Explain your reasoning. Example 8: In the diagram below, BC, AE BD BE, and ABE and CDB are right angles. Is BEA BCD? Explain your reasoning.
Background for Standard G.CO.11: Prove theorems about parallelograms. 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 Example 1: Organize the following quadrilaterals into two categories using the characteristic of your choice. Write an A or a B inside each shape to categorize it each shape must be put into one of the groups. Hint: choose a characteristic, then put all of the quadrilaterals that have that characteristic into Group A and all of the quadrilaterals that don t have that characteristic into Group B. Group A: Group B: 1 2 3 4 5 10 7 6 9 8 13 11 12 14
The shapes on the previous page can be categorized into 6 different types of quadrilaterals: parallelogram, kite, rhombus, square, rectangle, and trapezoid. 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. Mark some of the properties of these quadrilaterals on their picture (for example parallel sides, congruent sides, congruent angles). Example 2: 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 3: Create a flowchart to classify any quadrilateral based on the filled-in questions. This chart will help you categorize any quadrilateral. 0 How many pairs of 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 Let s Focus on Parallelograms Example 4: Use the properties of parallelograms to solve for the variable in each picture. A. Opposite sides are congruent. B. Opposite angles are congruent. C. Consecutive angles are supplementary. D. Diagonals bisect each other
Background for Standard G.CO.11: Prove theorems about parallelograms. Background for Standard G.C.3: Construct the inscribed and circumscribed circles of a triangles, and prove properties of angles for a quadrilateral inscribed in a circle. 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) Sum of Interior Angles: Quadrilateral (4 sides) Number of Triangles: Sum of Interior Angles: Pentagon (5 sides) Number of Triangles: Sum of Interior Angles: Hexagon (6 sides) Number of Triangles: Sum of Interior Angles: Heptagon (7 sides) Number of Triangles: Sum of Interior Angles: Octagon (8 sides) Number of Triangles: Sum of Interior Angles: Nonagon (9 sides) Number of Triangles: Sum of Interior Angles: Decagon (10 sides) Number of Triangles: Sum of Interior Angles: 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 the interior angle measures of a polygon with 32 sides? Example 2: What is the sum of all of all the interior angle measures of a 100-sided polygon?
Example 3: If the sum of all the interior angle measures of a polygon is 9540, how many sides does the polygon have? Example 4: Use what you know about the sum of the interior angles of a polygon to solve for x in each picture. A. B. 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 triangle below, estimate the measure of each interior angle. Make sure that these numbers add up to 180 because the angles in every triangle 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 quadrilateral below, estimate the measure of each interior angle. Make sure that these numbers add up to 360 because the angles in every quadrilateral 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. Pentagon: Calculate the sum of the exterior angle measures of a pentagon 1. In the pentagon below, estimate the measure of each interior angle. Make sure that these numbers add up to 540 because the angles in every pentagon 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 five exterior angles. Are you seeing a pattern? What is the sum of all of the exterior angle measures of a polygon?
Example 5: 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. 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 (n). Measure of each interior angle of a regular polygon: 180(n 2) n Measure of each exterior angle of a regular polygon: 360 n Example 6: Use the formula to calculate each interior angle measure of a regular 100-sided polygon. Example 7: 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 8: Calculate the measure of each exterior angle of an equilateral triangle (a regular triangle). Explain your reasoning. Example 9: Calculate the measure of each exterior angle of a 25-sided polygon. Explain your reasoning. Example 10: 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.