Properties of a Triangle Student Activity Sheet 1; use with Overview

Transcription

1 Student: Class: Date: Properties of a Triangle Student Activity Sheet 1; use with Overview 1. REEVVI IEEW Suppose points A, B, and C are collinear, where B is between A and C. If AB = 2x + 3, BC = 6x 5, and AC = 22, solve for x. 2. Why is the triangle the shape of choice for structures such as bridges and the Eiffel Tower? 3. What does it mean for a triangle to be rigid? 4. Is the figure below a rigid figure? If not, what can you do to make it rigid? Activity sheet 1, Page 1 of 2

2 Student: Class: Date: Properties of a Triangle Student Activity Sheet 1; use with Overview 5. Do you remember what makes each of the following triangles special? a. Isosceles triangle b. Equilateral triangle c. Scalene triangle Activity sheet 1, Page 2 of 2

3 Student: Class: Date: Student Activity Sheet 2; use with Exploring A triangle, or not? 1. Write a good definition of a triangle. 2. It is possible to create a triangle with side lengths 6 units, 6 units, and 2 units. Decide whether the other combinations of side lengths in the table create triangles or not. Mark your answers in the table. 3. Explain why some of the side length combinations in the table above do not form triangles. Activity sheet 2, Page 1 of 2

4 Student: Class: Date: Student Activity Sheet 2; use with Exploring A triangle, or not? 4. Write a conjecture about the relationship among the lengths of the sides of a triangle. Name this conjecture the Triangle Inequality Conjecture. 5. REEI INFFORRCCEE The lengths of two sides of a triangle are 7 cm and 10 cm. What are the upper and lower bounds on the third side of the triangle? Activity sheet 2, Page 2 of 2

5 Student: Class: Date: Student Activity Sheet 3; use with Exploring Triangle angle theorems 1. In ABC, what is the sum of the measures of! A,! B, and! C? Explain why this is true. 2. In ABC, what is true about the measures of angles A and C if! B is a right angle? 3. How does EF relate to AC? 4. How do points E and F relate to AB and CB? 5. What is a midsegment of a triangle? 6. What does the Triangle Sum Theorem say? Activity sheet 3, Page 1 of 4

6 Student: Class: Date: Student Activity Sheet 3; use with Exploring Triangle angle theorems 7. Fill in the following flowchart proof of the Triangle Sum Theorem. 8. REEI INFFORRCCEE In CAT, m! C = 15x, m! A = 5x + 40, and m! T = 10x Find the measures of each interior angle of CAT. Activity sheet 3, Page 2 of 4

7 Student: Class: Date: Student Activity Sheet 3; use with Exploring Triangle angle theorems 9.! EFG is formed by an extended ray and the adjacent side of ΔDEF. a. What type of angle is! EFG? b. What are! D and! E called with respect to the exterior angle! EFG? c. What is! EFD called with respect to! EFG? 10. Write a conjecture relating the measures of an exterior angle of a triangle and its remote interior angles. Call this conjecture the Exterior Angle Conjecture. Activity sheet 3, Page 3 of 4

8 Student: Class: Date: Student Activity Sheet 3; use with Exploring Triangle angle theorems 11. Fill in the blanks to complete the paragraph proof of the Exterior Angle Conjecture. straight line Triangle Sum Theorem right angle substitution 180 m! EFG = m! E + m! D Angle Addition Postulate m! EDF 12. REEI INFFORRCCEE In the diagram below, m! EFG = 30x, m! E = x , and m! D = 10x Find the measure of! EFG. Activity sheet 3, Page 4 of 4

9 Student: Class: Date: Student Activity Sheet 4; use with Exploring Isosceles triangle conjectures 1. In the Patty Paper activity, what kind of triangle did you create? How do you know? 2. What is the name of the angle of an isosceles triangle formed by rays containing the two congruent sides of the triangle? 3. Use your Patty Paper exploration to help you decide if each statement is true or false for any triangle ABC. 4. List three conjectures about isosceles triangles. Activity sheet 4, Page 1 of 2

10 Student: Class: Date: Student Activity Sheet 4; use with Exploring Isosceles triangle conjectures 5. REEI INFFORRCCEE In ABC, suppose AB = 15 cm, BC = 15 cm, AD = 2x 8 cm, and DC = 4x 20 cm. Solve for x. 6. REEI INFFORRCCEE In ABC above, suppose m ABD = (x 2 5) and m CBD = 4x. Solve for x. Activity sheet 4, Page 2 of 2

11 Teacher Version Student Activity Sheet 1; use with Overview 1. REEVVI IEEW Suppose points A, B, and C are collinear, where B is between A and C. If AB = 2x + 3, BC = 6x 5, and AC = 22, solve for x. By the Segment Addition Postulate, AB + BC = AC. 2x x 5 = 22 8x 2 = 22 8x = 24 x = 3 Note to teacher: Students will use the Segment Addition Postulate to explain the Triangle Inequality Conjecture in the first Exploring. This problem reviews the Segment Addition Postulate, some vocabulary ("collinear" and "between"), and solving equations. 2. Why is the triangle the shape of choice for structures such as bridges and the Eiffel Tower? [OV, screens 2 and 3] Triangles are the only polygons that are rigid. 3. What does it mean for a triangle to be rigid? [OV, screen 3] A figure is rigid if it cannot be distorted under stress. 4. Is the figure below a rigid figure? If not, what can you do to make it rigid? [OV, screens 3 and 4] This quadrilateral is not rigid. In order to make it rigid, you need to add a brace so that the quadrilateral is made up of two rigid triangles. Activity sheet 1, Page 1 of 2

12 Teacher Version Student Activity Sheet 1; use with Overview 5. Do you remember what makes each of the following triangles special? [OV, screen 5] a. Isosceles triangle An isosceles triangle is a triangle with at least 2 congruent sides. b. Equilateral triangle An equilateral triangle is a triangle with 3 congruent sides. c. Scalene triangle A scalene triangle is a triangle with no sides congruent to each other. Activity sheet 1, Page 2 of 2

13 Teacher Version Student Activity Sheet 2; use with Exploring A triangle, or not? 1. Write a good definition of a triangle. [EX1, screen 1] A triangle is a closed figure (or a polygon) with three sides. 2. It is possible to create a triangle with side lengths 6 units, 6 units, and 2 units. Decide whether the other combinations of side lengths in the table create triangles or not. Mark your answers in the table. [EX1, screen 2] 3. Explain why some of the side length combinations in the table above do not form triangles. [EX 1, screen 4] The Segment Addition Postulate says that if the sum of two segments is equal to a third segment with a common endpoint, then the points are collinear. If two sides of a figure have a sum equal to the third side, then the three vertices of the figure would be collinear and no triangle would be formed. If the sum of two sides of a figure is less than the third side, then there is no way the two shorter sides can meet to form a triangle. Activity sheet 2, Page 1 of 2

14 Teacher Version Student Activity Sheet 2; use with Exploring A triangle, or not? 4. Write a conjecture about the relationship among the lengths of the sides of a triangle. Name this conjecture the Triangle Inequality Conjecture. [EX1, screen 5] Triangle Inequality Conjecture: The sum of the lengths of any two sides of a triangle is greater than the length of the third side. 5. REEI INFFORRCCEE The lengths of two sides of a triangle are 7 cm and 10 cm. What are the upper and lower bounds on the third side of the triangle? Let the third side of the triangle be s. One possibility is that the 10 cm side is the longest side of the triangle. In that case, s + 7 > 10. So, s > 3 cm. Another possibility is that the third side, s, is the longest side. In that case, > s. So, s < 17 cm. Therefore, the third side, s, must have a length between 3 cm and 17 cm. In other words, 3 cm < s < 17 cm. Activity sheet 2, Page 2 of 2

15 Teacher Version Student Activity Sheet 3; use with Exploring Triangle angle theorems 1. In ABC, what is the sum of the measures of! A,! B, and! C? Explain why this is true. [EX2, screens 1 and 2] The sum of the measures of the interior angles of a triangle is 180. Student explanations may vary. Some examples are: Through paper folding, angles A, B, and C form a straight angle. By tearing off angles A, B, and C, you can reposition them so that they are adjacent angles. When you do this, the three angles form a straight angle. 2. In ABC, what is true about the measures of angles A and C if! B is a right angle? [EX2, screens 1 and 2] If! B is a right angle, then angles A and C are complementary. The sum of their measures is How does EF relate to AC? [EX2, screens 1 and 2] EF = 1 2 AC 4. How do points E and F relate to AB and CB? [EX2, screens 1 and 2] E and F are midpoints of AB and CB, respectively. 5. What is a midsegment of a triangle? [EX2, screen 2] A midsegment of a triangle is a line segment that connects the midpoints of two sides of the triangle. 6. What does the Triangle Sum Theorem say? [EX2, screen 3] The sum of the measures of the interior angles of a triangle is 180. Activity sheet 3, Page 1 of 4

16 Teacher Version Student Activity Sheet 3; use with Exploring Triangle angle theorems 7. Fill in the following flowchart proof of the Triangle Sum Theorem. [EX2, screen 4] 8. REEI INFFORRCCEE In CAT, m! C = 15x, m! A = 5x + 40, and m! T = 10x Find the measures of each interior angle of CAT. By the Triangle Sum Theorem, m! C + m! A + m! T = x + 5x x + 20 = x + 60 = x = 120 x = 4 m! C = 15x = 15(4) = 60 m! A = 5x + 40 = 5(4) + 40 = = 60 m! T = 10x + 20 = 10(4) + 20 = = 60 Activity sheet 3, Page 2 of 4

17 Teacher Version Student Activity Sheet 3; use with Exploring Triangle angle theorems 9.! EFG is formed by an extended ray and the adjacent side of ΔDEF. [EX2, screen 5] a. What type of angle is! EFG? an exterior angle b. What are! D and! E called with respect to the exterior angle! EFG?! D and! E are called remote interior angles because neither angle is adjacent to! EFG. c. What is! EFD called with respect to! EFG?! EFD is called an adjacent interior angle because it is the angle of the triangle that shares a ray with! EFG. 10. Write a conjecture relating the measures of an exterior angle of a triangle and its remote interior angles. Call this conjecture the Exterior Angle Conjecture. [EX2, screen 7] Exterior Angle Conjecture: The measure of the exterior angle of a triangle is equal to the sum of the measures of the remote interior angles. Activity sheet 3, Page 3 of 4

18 Teacher Version Student Activity Sheet 3; use with Exploring Triangle angle theorems 11. Fill in the blanks to complete the paragraph proof of the Exterior Angle Conjecture. [EX2, screen 8] straight line Triangle Sum Theorem right angle substitution 180 m! EFG = m! E + m! D Angle Addition Postulate m! EDF 12. REEI INFFORRCCEE In the diagram below, m! EFG = 30x, m! E = x , and m! D = 10x Find the measure of! EFG. m! E + m! D = m! EFG x x + 35 = 30x x x + 75 = 30x x 2 20x + 75 = 0 (x 5)(x 15) = 0 x = 5 or x = 15 If x = 5, then m! EFG = 30(5) = 150. If x = 15, then m! EFG = 30(15) = 450. This angle measurement is not possible in the context of the problem, so m! EFG = 150 is the only solution. Activity sheet 3, Page 4 of 4

19 Teacher Version Student Activity Sheet 4; use with Exploring Isosceles triangle conjectures 1. In the Patty Paper activity, what kind of triangle did you create? How do you know? [EX3, screen 1, 2] This triangle must be isosceles because one of the sides is a reflection of the other side. Since reflections are congruence mappings, they preserve size. Therefore, at least two of the sides of the triangles are congruent. 2. What is the name of the angle of an isosceles triangle formed by rays containing the two congruent sides of the triangle? [EX3, screen 2] the vertex angle 3. Use your Patty Paper exploration to help you decide if each statement is true or false for any triangle ABC. [EX3, screen 3] 4. List three conjectures about isosceles triangles. [EX3, screen 4] Isosceles Triangle Conjectures: The base angles of an isosceles triangle are congruent. The segment from the vertex to the midpoint of the base of an isosceles triangle is perpendicular to the base and bisects the vertex angle. The segment from the vertex to the midpoint of the base of an isosceles triangle divides the isosceles triangle into two congruent triangles. Activity sheet 4, Page 1 of 2

20 Teacher Version Student Activity Sheet 4; use with Exploring Isosceles triangle conjectures 5. REEI INFFORRCCEE In ABC, suppose AB = 15 cm, BC = 15 cm, AD = 2x 8 cm, and DC = 4x 20 cm. Solve for x. Since ABC is isosceles, AD = DC. 2x 8 = 4x 20 2x = 4x 12 ß12 = 2x x = 6 6. REEI INFFORRCCEE In ABC above, suppose m ABD = (x 2 5) and m CBD = 4x. Solve for x. Since ABC is isosceles, m ABD = m CBD. x 2 5 = 4x x 2 4x 5 = 0 (x 5)(x + 1) = 0 x = 5 or x = -1 If x = -1, then m ABD = -4. So, x -1. Therefore, x = 5. Activity sheet 4, Page 2 of 2