Vocabulary. Term Page Definition Clarifying Example altitude of a triangle. centroid of a triangle. circumcenter of a triangle. circumscribed circle

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1 CHAPTER Vocabulary The table contains important vocabulary terms from Chapter. As you work through the chapter, fill in the page number, definition, and a clarifying eample. Term Page Definition Clarifying Eample altitude of a triangle centroid of a triangle circumcenter of a triangle circumscribed circle concurrent equidistant incenter of a triangle 100 Geometry

2 CHAPTER Vocabulary The table contains important vocabulary terms from Chapter. As you work through the chapter, fill in the page number, definition, and a clarifying eample. Term Page Definition Clarifying Eample altitude of a triangle centroid of a triangle circumcenter of a triangle circumscribed circle A perpendicular segment from a verte to the line containing the opposite side. The point of concurrency of the three medians of a triangle. Also known as the center of gravity. The point of concurrency of the three perpendicular bisectors of a triangle. Every verte of the polygon lies on the circle. circumcenter of a triangle centroid of a triangle concurrent 307 Three or more lines that intersect at one point. concurrent lines equidistant 300 The same distance from two or more objects. If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment. incenter of a triangle 389 The point of concurrency of the three angle bisectors of a triangle. incenter of a triangle 100 Geometry

3 CHAPTER VOCABULARY CONTINUED Term Page Definition Clarifying Eample inscribed circle locus median of a triangle midsegment of a triangle orthocenter of a triangle point of concurrency Pythagorean triple 101 Geometry

4 CHAPTER VOCABULARY CONTINUED Term Page Definition Clarifying Eample inscribed circle 309 A circle in which each side of the polygon is tangent to the circle. locus median of a triangle A set of points that satisfies a given condition. A segment whose endpoints are a verte of the triangle and the midpoint of the opposite side. The perpendicular bisector is a segment can be defined as the locus of points that are equidistant from the endpoints of the segment. median midsegment of a triangle 322 A segment that joins the midpoints of two sides of the triangle. midsegment orthocenter of a triangle 316 The point of concurrency of the three altitudes of a triangle. orthocenter of a triangle point of concurrency 307 A point where three or more lines coincide. point of concurrency Pythagorean triple 349 A set of three nonzero whole numbers a, b, and c such that a 2 b 2 c 2. a 3, b 4, c Geometry

5 CHAPTER Chapter Review -1 Perpendicular and Angle Bisectors Find each measure. 1. BC 2. RS 3. XW B U A D 36 R n 3 C S 43 T W X 3n + 9 Y Z 4. Write an equation in point-slope form for the perpendicular bisector of the segment with endpoints K(10, 3) and L( 2, ). -2 Bisectors of Triangles. NP, OP, MP are the perpendicular bisectors of JKL. Find PK and JM. J N P M K O L 6. DH and DG are angle bisectors of EGH. Find m EHD and the distance from D to GH. E 4 19 F 42 D H G 7. Find the circumcenter of JKL with vertices J(0, 6), K(8, 0) and L(0, 0). y Geometry

6 CHAPTER Chapter Review -1 Perpendicular and Angle Bisectors Find each measure. 1. BC 2. RS 3. XW B U A D 36 R n 3 C S 43 T W X 3n + 9 Y Z Write an equation in point-slope form for the perpendicular bisector of the segment with endpoints K(10, 3) and L( 2, ). y 1 3 ( 4) 2-2 Bisectors of Triangles. NP, OP, MP are the perpendicular bisectors of JKL. Find PK and JM. J N P M K O L PK 9.9 JM DH and DG are angle bisectors of EGH. Find m EHD and the distance from D to GH. E 4 19 F 42 D H G m EHD 44 DH Find the circumcenter of JKL with vertices J(0, 6), K(8, 0) and L(0, 0). (4, 3) y Geometry

7 CHAPTER REVIEW CONTINUED -3 Medians and Altitudes of Triangles 8. Nathan cuts a triangle with vertices at coordinates (0, 4), (6, 0), and (3, 2) from a piece of grid paper. At what coordinates should he place the tip of a pen to balance the triangle? 9. Find the orthocenter of WYX with vertices W(1, 2), X(7, 2), and Y(3, ). y The Triangle Midsegment Theorem 10. Find BA, JC, and m FBA 11. What is the distance MP across in KCJ. the lake? 8 F 67 K B C A 40 J N 8m 47m G H 8m 1m 47m M P 12. Write an indirect proof that an equiangular triangle can not have an obtuse angle. 118 Geometry

8 CHAPTER REVIEW CONTINUED -3 Medians and Altitudes of Triangles 8. Nathan cuts a triangle with vertices at coordinates (0, 4), (6, 0), and (3, 2) from a piece of grid paper. At what coordinates should he place the tip of a pen to balance the triangle? 3, Find the orthocenter of WYX with vertices W(1, 2), X(7, 2), and Y(3, ). y , The Triangle Midsegment Theorem 10. Find BA, JC, and m FBA 11. What is the distance MP across in KCJ. the lake? 8 F 67 K B C A 40 J BA 29 JC 134 m FBA 40 N 8m 47m G H 8m 1m 47m M P 102 meters 12. Write an indirect proof that an equiangular triangle can not have an obtuse angle. Answers may vary. 1. Given: An equiangular triangle Prove: The triangle does not have an obtuse angle 2. Assume the equiangular triangle has an obtuse angle 3. An equilateral triangle, by definition, is a triangle whose angles are all congruent. Since the sum of the angles in a triangle measure 180, then each angle in an equiangular triangle must measure 60. An obtuse angle, by definition, is an angle whose measure is greater than 90. Since 60 is less then 90, none of the angles are obtuse. 4. Original conjecture is true. An equiangular triangle cannot have an obtuse angle. 118 Geometry

9 CHAPTER REVIEW CONTINUED - Indirect Proof and Inequalities in One Triangle 13. Write the angles of QRS 14. Write the sides of ABC in order in order from smallest to largest. from shortest to longest R 8.9 C Q 62.4 S B 39 A Tell whether a triangle can have sides with given lengths. Eplain , 9.4, r, r 4, r 2 when r 17. The distance from Tyler s house to the library is 3 miles. The distance from his home to the park is 12 miles. If the three locations form a triangle, what is the range of distance from the library to the park? -6 Inequalities in Two Triangles 18. Compare KL and 19. Compare m BAD and 20. Find the range of NP. m CAD. values of. K N 118 J 126 O L P B A D C S 34 V T (4 6) U 119 Geometry

10 CHAPTER REVIEW CONTINUED - Indirect Proof and Inequalities in One Triangle 13. Write the angles of QRS 14. Write the sides of ABC in order in order from smallest to largest. from shortest to longest R 8.9 C Q 62.4 S B 39 A m RSQ m RQS m QRS AC, AB, BC Tell whether a triangle can have sides with given lengths. Eplain , 9.4, r, r 4, r 2 when r Yes, the sum of each pair of lengths is greater than the third length. No, the sum of the pair of lengths is not greater than the third length in all cases. 17. The distance from Tyler s house to the library is 3 miles. The distance from his home to the park is 12 miles. If the three locations form a triangle, what is the range of distance from the library to the park? 9 d 1 miles -6 Inequalities in Two Triangles 18. Compare KL and 19. Compare m BAD and 20. Find the range of NP. m CAD. values of. K N 118 J 126 O L P B A D C S 34 V T (4 6) U NP KL m BAD m CAD Geometry

11 CHAPTER REVIEW CONTINUED -7 The Pythagorean Theorem 21. Find the value of. Give the 22. Find the missing side length. Tell if answer in simplest radical form. the side lengths form a Pythagorean. Eplain Tell if the measures 9, 40, and 41 can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse or right. 24. A carpenter wants to cut a rectangular piece of wood. He checks to see if the diagonal measurements are the same. What should they measure? -8 Applying Special Right Triangles 24 cm 10 cm 2. A flag is an equilateral triangle with the side length of 24 inches. What is the height h of the flag? Round your answer to the nearest inch. 24 in. 60 h GO TEAM! Find the values of the variables. Give your answer in simplest radical form y 60 y 120 Geometry

12 CHAPTER REVIEW CONTINUED -7 The Pythagorean Theorem 21. Find the value of. Give the 22. Find the missing side length. Tell if answer in simplest radical form. the side lengths form a Pythagorean. Eplain Tell if the measures 9, 40, and 41 can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse or right. 17; Yes, the side lengths are whole numbers that satisfy the equation a2 b2 c2. Yes; right triangle 24. A carpenter wants to cut a rectangular piece of wood. He checks to see if the diagonal measurements are the same. What should they measure? 26 cm -8 Applying Special Right Triangles 24 cm 10 cm 2. A flag is an equilateral triangle with the side length of 24 inches. What is the height h of the flag? Round your answer to the nearest inch. 21 inches 24 in. 60 h GO TEAM! Find the values of the variables. Give your answer in simplest radical form y 60 y , y 28 6, y Geometry

13 CHAPTER Postulates and Theorems Theorem -1-1 Converse -1-2 Theorem -1-3 Converse -1-4 Theorem -2-1 Theorem -2-2 (Perpendicular Bisector Theorem) If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. (Converse of the Perpendicular Bisector Theorem) If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment. (Angle Bisector Theorem) If a point is on the bisector of an angle, then it is equidistant from the sides of the angle. (Converse of the Angle Bisector Theorem) If a point in the interior of an angle is equidistant from the sides of the angle, then it is on the bisector of the angle. (Circumcenter Theorem) The circumcenter of a triangle is equidistant from the vertices of the triangle. (Incenter Theorem) The incenter of a triangle is equidistant from the sides of the triangle. Theorem -3-1 (Centroid Theorem) The centroid of a triangle is located 2 of 3 the distance from each verte to the midpoint of the opposite side. Theorem -4-1 Theorem --1 Theorem --2 Theorem --3 Theorem -6-1 Converse -6-2 (Triangle Midsegment Theorem) A midsegment of a triangle is parallel to a side of the triangle, and its length is half the length of that side. If two sides of a triangle are not congruent, then the larger angle is opposite the longer side. If two angles of a triangle are not congruent, then the longer side is opposite the larger angle. (Triangle Inequality Theorem) The sum of any two side lengths of a triangle is greater than the third side length. (Hinge Theorem) If two sides of one triangle are congruent to two sides of another triangle and the included angles are not congruent, then the longer third side is across from the larger included angle. (Converse of the Hinge Theorem) If two sides of one triangle are congruent to two sides of another triangle and the third sides are not congruent, then the larger included angle is across from the longer third side. 121 Geometry

14 CHAPTER Postulates and Theorems Theorem -1-1 Converse -1-2 Theorem -1-3 Converse -1-4 Theorem -2-1 Theorem -2-2 (Perpendicular Bisector Theorem) If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. (Converse of the Perpendicular Bisector Theorem) If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment. (Angle Bisector Theorem) If a point is on the bisector of an angle, then it is equidistant from the sides of the angle. (Converse of the Angle Bisector Theorem) If a point in the interior of an angle is equidistant from the sides of the angle, then it is on the bisector of the angle. (Circumcenter Theorem) The circumcenter of a triangle is equidistant from the vertices of the triangle. (Incenter Theorem) The incenter of a triangle is equidistant from the sides of the triangle. Theorem -3-1 (Centroid Theorem) The centroid of a triangle is located 2 of 3 the distance from each verte to the midpoint of the opposite side. Theorem -4-1 Theorem --1 Theorem --2 Theorem --3 Theorem -6-1 Converse -6-2 (Triangle Midsegment Theorem) A midsegment of a triangle is parallel to a side of the triangle, and its length is half the length of that side. If two sides of a triangle are not congruent, then the larger angle is opposite the longer side. If two angles of a triangle are not congruent, then the longer side is opposite the larger angle. (Triangle Inequality Theorem) The sum of any two side lengths of a triangle is greater than the third side length. (Hinge Theorem) If two sides of one triangle are congruent to two sides of another triangle and the included angles are not congruent, then the longer third side is across from the larger included angle. (Converse of the Hinge Theorem) If two sides of one triangle are congruent to two sides of another triangle and the third sides are not congruent, then the larger included angle is across from the longer third side. 121 Geometry

15 CHAPTER POSTULATES AND THEOREMS CONTINUED Theorem -7-1 Theorem -7-2 Theorem -8-1 Theorem -8-2 (Converse of the Pythagorean Theorem) If the sum of the squares of the lengths of two sides of a triangle is equal to the square of the length of the third side, then the triangle is a right triangle. (Pythagorean Inequalities Theorem) In ABC, c is the length of the longest side. ( Triangle Theorem) In a triangle, both legs are congruent, and the length of the hypotenuse is the length of a leg times 2. ( Triangle Theorem) In a triangle, the length of the hypotenuse is 2 times the length of the shorter leg, and the length of the longer leg is the length of the shorter leg times Geometry

16 CHAPTER POSTULATES AND THEOREMS CONTINUED Theorem -7-1 Theorem -7-2 Theorem -8-1 Theorem -8-2 (Converse of the Pythagorean Theorem) If the sum of the squares of the lengths of two sides of a triangle is equal to the square of the length of the third side, then the triangle is a right triangle. (Pythagorean Inequalities Theorem) In ABC, c is the length of the longest side. ( Triangle Theorem) In a triangle, both legs are congruent, and the length of the hypotenuse is the length of a leg times 2. ( Triangle Theorem) In a triangle, the length of the hypotenuse is 2 times the length of the shorter leg, and the length of the longer leg is the length of the shorter leg times Geometry

17 CHAPTER Big Ideas Answer these questions to summarize the important concepts from Chapter in your own words. 1. Eplain the difference in location between the circumcenter and the incenter of a triangle. 2. Why is the centroid of a triangle also called the center of gravity? 3. Describe how many midsegments each triangle has and what is formed by them. 4. Use the Pythagorean theorem to describe an obtuse and acute triangle such as ABC. For more review of Chapter : Complete the Chapter Study Guide and Review on pages of your tetbook. Complete the Ready to Go On quizzes on pages 329 and 36 of your tetbook. 123 Geometry

18 CHAPTER Big Ideas Answer these questions to summarize the important concepts from Chapter in your own words. 1. Eplain the difference in location between the circumcenter and the incenter of a triangle. Answers may vary. Possible answer: The circumcenter can be inside the triangle, outside the triangle or on the triangle. Unlike the circumcenter, the incenter is always inside the triangle. 2. Why is the centroid of a triangle also called the center of gravity? Answers may vary. Possible answer: The centroid of a triangle is also called the center of gravity because it is a point where a triangular region will balance. 3. Describe how many midsegments each triangle has and what is formed by them. Answers may vary. Possible answer: Every triangle has three midsegments, which form the midsegment triangle. 4. Use the Pythagorean theorem to describe an obtuse and acute triangle such as ABC. If c 2 a 2 b 2, then ABC is an obtuse triangle. If c 2 a 2 b 2, then ABC is an acute triangle. For more review of Chapter : Complete the Chapter Study Guide and Review on pages of your tetbook. Complete the Ready to Go On quizzes on pages 329 and 36 of your tetbook. 123 Geometry

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