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FeatureLesson Geometry Lesson Main In GHI, R, S, and T are midpoints. 1. Name all the pairs of parallel segments. 2. If GH = 20 and HI = 18, find RT. 3. If RH = 7 and RS = 5, find ST. 4. If m G = 60 and m I = 70, find m GTR. 5. If m H = 50 and m I = 66, find m ITS. 6. If m G = m H = m I and RT = 15, find the perimeter of GHI. RT || HI, RS || GI, ST || HG 90 64 70 7 9 Lesson 5-1 Quiz – Midsegments of Triangles 5-1 Lesson 5-2 Bisectors in Triangles

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FeatureLesson Geometry Lesson Main Lesson 5-2 Bisectors in Triangles 5-2 When a point is the same distance from two or more objects, the point is said to be equidistant from the objects. Triangle congruence theorems can be used to prove theorems about equidistant points.

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FeatureLesson Geometry Lesson Main Lesson 5-2 Bisectors in Triangles 5-2

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FeatureLesson Geometry Lesson Main Lesson 5-2 Bisectors in Triangles 5-2

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FeatureLesson Geometry Lesson Main Lesson 5-2 Bisectors in Triangles 5-2 The shortest segment from a point to a line is perpendicular to the line. This fact is used to define the distance from a point to a line as the length of the perpendicular segment from the point to the line.

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FeatureLesson Geometry Lesson Main Lesson 5-2 Bisectors in Triangles 5-2

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FeatureLesson Geometry Lesson Main Use the map of Washington, D.C. Describe the set of points that are equidistant from the Lincoln Memorial and the Capitol. The Converse of the Perpendicular Bisector Theorem states If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment. Lesson 5-2 Bisectors in Triangles Quick Check Additional Examples 5-2 Real-World Connection

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FeatureLesson Geometry Lesson Main (continued) A point that is equidistant from the Lincoln Memorial and the Capitol must be on the perpendicular bisector of the segment whose endpoints are the Lincoln Memorial and the Capitol. Therefore, all points on the perpendicular bisector of the segment whose endpoints are the Lincoln Memorial and the Capitol are equidistant from the Lincoln Memorial and the Capitol. Lesson 5-2 Bisectors in Triangles Additional Examples 5-2 Quick Check

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FeatureLesson Geometry Lesson Main Find x, FB, and FD in the diagram above. FD = FBAngle Bisector Theorem 7x – 37 = 2x + 5Substitute. 7x = 2x + 42 Add 37 to each side. 5x = 42 Subtract 2x from each side. x = 8.4 Divide each side by 5. FB = 2(8.4) + 5 = 21.8 Substitute. FD = 7(8.4) – 37 = 21.8 Substitute. Lesson 5-2 Bisectors in Triangles Additional Examples 5-2 Using the Angle Bisector Theorem Quick Check

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FeatureLesson Geometry Lesson Main Use this figure for Exercises 1–3. 1. Find BD. 2. Complete the statement: C is equidistant from ?. 3. Can you conclude that CN = DN? Explain. Use this figure for Exercises 4–6. 4. Find the value of x. 5. Find CG. 6. Find the perimeter of quadrilateral ABCG. 6 Lesson 5-2 points A and B 16 48 8 No; if CN = DN, CNB DNB by SAS and CB = DB by CPCTC, which is false. Bisectors in Triangles Lesson Quiz 5-2

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FeatureLesson Geometry Lesson Main (For help, go to Lesson 1-7.) Lesson 5-2 Bisectors in Triangles 1. Draw a triangle, XYZ. Construct STV so that STV XYZ. 2. Draw acute P. Construct Q so that 3. Draw AB. Construct a line CD so that CD AB and CD bisects AB. 4. Draw acute angle E. Construct the bisector of E. TM bisects STU so that m STM = 5x + 4 and m MTU = 6x – 2. 5. Find the value of x. 6. Find m STU. Use a compass and a straightedge for the following. Check Skills Youll Need Q P. Check Skills Youll Need 5-2

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FeatureLesson Geometry Lesson Main Lesson 5-2 Bisectors in Triangles Solutions 1.2. 3. 4. 5. Since TM bisects STU, m STM = m MTU. So, 5x + 4 = 6x – 2. Subtract 5x from both sides: 4 = x – 2; add 2 to both sides: x = 6. 6. From Exercise 5, x = 6. m STU = m STM + m MTU = 5x + 4 + 6x – 2 = 11x + 2 = 11(6) + 2 = 68. 1-4. Answers may vary. Samples given: Check Skills Youll Need 5-2

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