Example 1 Example 2 −− Answers −−. Find

1 AnswersCopyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. PDF 2ndChapter 5 A1 Glencoe GeometryCopyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, DATE PERIOD Chapter ResourcesChapter 5 3 Glencoe Geometry5 Before you begin Chapter 5 Read each statement. Decide whether you Agree (A) or Disagree (D) with the statement. Write A or D in the first column OR if you are not sure whether you agree or disagree, write NS (Not Sure). After you complete Chapter 5 Reread each statement and complete the last column by entering an A or a D. Did any of your opinions about the statements change from the first column?

2 For those statements that you mark with a D, use a piece of paper to write an Example of why you GuideRelationships in TrianglesSTEP 1 A, D, or NSStatementSTEP 2A or D 1. Any point that is on the perpendicular bisector of a segment is equidistant from the endpoints of that 2. The circumcenter of a triangle is equidistant from the midpoints of each side of the 3. The altitudes of a triangle meet at the 4. Three altitudes can be drawn for any one 5. A median of a triangle is any segment that contains the midpoint of a side of the 6. The measure of an exterior angle of a triangle is always greater than the measures of either of its corresponding remote interior 7. The longest side in a triangle is opposite the smallest angle in that 8. To write an indirect proof that two lines are perpendicular, begin by assuming the two lines are not 9.

3 The length of the longest side of a triangle is always greater than the sum of the lengths of the other two In two triangles, if two pairs of sides are congruent, then the measure of the included angles determines which triangle has the longer third 1 Step 36/6/08 12:51:46 PMLesson 5-1 Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, DATE PERIOD Chapter 5 5 Glencoe GeometryPerpendicular Bisector A perpendicular bisector is a line, segment, or ray that is perpendicular to the given segment and passes through its midpoint. Some theorems deal with perpendicular Bisector TheoremIf a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the of Perpendicular Bisector TheoremIf a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the TheoremThe perpendicular bisectors of the sides of a triangle intersect at a point called the circumcenter that is equidistant from the vertices of the Guide and InterventionBisectors of Triangles Find the measure of (', FK is the perpendicular bisector of GM.)

4 FG = = FM BD is the perpendicular bisector of AC . Find + 85x- 6 BCDA AD = DC 3x + 8 = 5x - 6 14 = 2x 7 = xExample 1 Example 2 ExercisesFind each measure. 1. XW 2. ;: '"&# P is the circumcenter of EMK. List any segment(s) congruent to each segment below. 3. MY 4. KP 5. MN 6. ER &:31.,/ YE NK RK MP , EP 54/11/08 8:13:29 AMAnswers (Anticipation Guide and Lesson 5-1) A16/19/08 5:34:11 PM6/19/08 5:34:11 PMCopyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

5 PDF 2ndChapter 5 A2 Glencoe GeometryCopyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, DATE PERIOD Chapter 5 6 Glencoe GeometryAngle Bisectors Another special segment, ray, or line is an angle bisector, which divides an angle into two congruent Bisector TheoremIf a point is on the bisector of an angle, then it is equidistant from the sides of the of Angle Bisector TheoremIf a point in the interior of an angle if equidistant from the sides of the angle, then it is on the bisector of the TheoremThe angle bisectors of a triangle intersect at a point called the incenter that is equidistant from the sides of the Guide and Intervention (continued)Bisectors of Triangles MR is the angle bisector of NMP.

6 Find x if m 1 = 5x + 8 and m 2 = 8x - MR is the angle bisector of NMP, so m 1 = m + 8 = 8x - 16 24 = 3x 8 = xExercisesFind each measure. 1. ABE 2. YBA43 $#"& '#:"47 88 3. MK 4. EWL 7.,53x-82x+1 8&-+(3x+21) (7x+5) Point U is the incenter of GHY. Find each measure below. 5. MU 6. UGM 7. PHU 8. HUExample(#V:1).28 21 125 43 47 19 64/11/08 8:13:37 AMLesson 5-1 Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, DATE PERIOD Chapter 5 7 Glencoe Geometry5-1 Skills PracticeBisectors of TrianglesFind each measure.

7 1. FG 2. KL'(%&5x- 1713133x+ 1 , 3. TU 4. LYF56342x+ 245x- 30 +:'-58 5. IU 6. MYW35*619 19 2x+ 57x 8:.1(2x+ 5) (4x- 1) Point P is the circumcenter of ABC. List any segment(s) congruent to each segment below. 7. BR 8. CS 9. BP Point A is the incenter of PQR. Find each measure ARU 11. AU 12. QPK 5$#341"56"123,40 20 (4x- 9) (3x+ 2) 28 60 58 7 11 AR AP , CP 402035 AS 76/6/08 12:52:24 PMAnswers (Lesson 5-1) A26/19/08 5:34:17 PM6/19/08 5:34:17 PMCopyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.)

8 PDF 2ndChapter 5 A4 Glencoe GeometryCopyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, DATE PERIOD Chapter 5 10 Glencoe GeometryInscribed and Circumscribed CirclesThe three angle bisectors of a triangle intersect in a single point called the incenter. This point is the center of a circle that just touches the three sides of the triangle. Except for the three points where the circle touches the sides, the circle is inside the triangle. The circle is said to be inscribed in the triangle. 1. With a compass and a straightedge, construct the inscribed circle for PQR by following the steps below.

9 Step 1 Construct the bisectors of R and Q. Label the point where the bisectors meet, A. Step 2 Construct a perpendicular segment from A to RQ . Use the letter B to label the point where the perpendicular segment intersects RQ . Step 3 Use a compass to draw the circle with center at A and radius AB .Construct the inscribed circle in each triangle. 2. 3. The three perpendicular bisectors of the sides of a triangle also meet in a single point. This point is the center of the circumscribed circle, which passes through each vertex of the triangle. Except for the three points where the circle touches the triangle, the circle is outside the triangle. 4. Follow the steps below to construct the circumscribed circle for FGH. Step 1 Construct the perpendicular bisectors of FG and FH.

10 Use the letter A to label the point where the perpendicular bisectors meet. Step 2 Draw the circle that has center A and radius AF .Construct the circumscribed circle for each triangle. 5. 6. 104/11/08 8:14:07 AMLesson 5-2 Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, DATE PERIOD Chapter 5 11 Glencoe GeometryMedians A median is a line segment that connects a vertex of a triangle to the midpoint of the opposite side. The three medians of a triangle intersect at the centroid of the triangle. The centroid is located two thirds of the distance from a vertex to the midpoint of the side opposite the vertex on a median.