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Transcript
Geometry Name: _______________________________ Date: _____________ Hour: _____ Target 5a (Day 1) Identify bisectors of angles and segments and use them to find segment measures. Perpendicular Bisectors Theorem 5.1 – Any point on the perpendicular ______________ of a segment is equidistant from the ______________ of the segment. Theorem 5.2 – Any point _________________ from the endpoints of a segment _______ on the perpendicular bisector of the segment. Concurrent lines – When ___ or more lines intersect at a common point. Point of Concurrency – The ________________ point where three or more lines meet. Circumcenter – The point of concurrency of the perpendicular ______________ of a triangle. Theorem 5.3 – (Circumcenter Theorem) The circumcenter of a triangle is __________________ from the ________________ of the triangle. Practice Problems SY = 6 yd. Find TS. Find CD if DX = 17 mm. Each figure shows a triangle with its three perpendicular bisectors intersecting at point D. LJ = 6 and DJ = 11. Find LD. Find x if GQ = 9x - 12 and QF = 7x - 8. Homework: 5a Day 1 Worksheet (from 5.1 in text, page 238) Target 5a - continued (Day 2) Identify bisectors of angles and segments and use them to find segment measures. Angle Bisectors Theorem 5.4 – Any point on the ____________ bisector is equidistant from the ______________ of the angle. Theorem 5.5 – Any point equidistant from the sides of an _________ lies on the angle _________________. Incenter – The point of concurrency for _________ bisectors. Theorem 5.6 – (Incenter Theorem) The _____________ of a triangle is equidistant from each ____________ of the triangle. Practice Problems Each figure shows a triangle with its three angle bisectors intersecting at point P. mCBD 68°. Find m1. Find PB if CP = 3. HP = 9 and PU = 4. Find HS. Find x if m1 = 9x + 23 and m2 =15x - 19. Homework: 5a Day 2 Worksheet (from 5.1 in text, page 238) Target 5b Identify medians and altitudes of triangles and use them to find segment measures. Medians Median – A segment whose ____________________ are one vertex of a triangle and the _________________ of the side opposite the vertex. Centroid – The point of concurrency for the _________________ of a triangle. Theorem 5.7 – (Centroid Theorem) The centroid of a triangle is located ______________ of the distance from a ______________ to the midpoint of the side _________________ the vertex on a median. Practice Problems a) is a median and ME = 4.7. Find MN. c) Point D is a centroid. Find x Find y b) is a median. Find x. d) Point T is a centroid. CT = 6, TJ = 20, and KB = 33. Find z Find TL Find AJ Find TB Altitudes Altitude – A segment in a triangle from a ______________ to the line containing the opposite side and _________________________________ to the line containing that side. Orthocenter – The point of concurrency for the ____________________ of a triangle. Practice Problems a) is an altitude. mXWY is 4x – 6. Find x. b) mTRS is 6x. Find x. Review – Targets 5a and 5b: Directions: Fill in the blanks with the terms for each special segment or point of concurrency that the arrows are pointing to. Homework: 5b Worksheet (from 5.1 in text, page 238) Target 5c Solve problems using triangle inequality theorems. Quiz 5a & 5b next time Theorem 5.9 – If one side of a triangle is ____________ than another side, then the angle ___________ the longer side has a _________ measure than the angle opposite the shorter side. Directions: Determine the relationship between the measures of the given angles. a) ADB and DBA. b) CDB and CBD. c) RSU and SUR. d) TSV and STV. Theorem 5.10 – If one _______________ of a triangle has a greater measure than another angle, then the ___________ opposite the greater angle is longer than the side opposite the lesser angle. Directions: Determine the relationship between the lengths of the given sides. a) and . b) and . c) d) and and . . Triangle Inequality Theorem (Theorem 5.11) – the _______ of the lengths of the two smallest sides of a triangle is __________________ than the length of the third side. Directions: Determine whether the given measures can be lengths of the sides of a triangle. 1. 8, 9, 17 2. 11, 12, 15 3. 10, 16, 8 4. 5.7, 9.2, 4.3 Determine Possible Side Lengths Find the range for the measure of the third side of a triangle given the measures of two sides. Subtract the two given numbers Add the two given numbers Write as a range: (subtracted amount) < n < (added amount) 1. 7, 8 2. 12, 9 3. 6, 11 4. 45, 78 Review for Quiz (Targets 5a – 5b) Perpendicular Bisectors Find DL if DK = 26 and LK = 10 Angle Bisectors Find UD if PC = 4 and UP = 7 Medians Find BG if XB = x – 1 and XG = x Altitudes Find x if mBEA = 7x + 6 Homework: 5c, page 251 from 5.2 in text (10 – 15), AND page 264 from 5.4 in text (14 – 37) (30 problems) Target 5d Solve problems using inequalities involving one and two triangles. SAS Inequality / Hinge Theorem – If ______ sides of a triangle are congruent to two __________ of another triangle and the __________________ angle in one triangle has a _________________ measure than the included angle in the other, then the third side of the first triangle is __________________ than the third side of the second triangle. Directions: Write an inequality relating the given pair of segment measures. a) and b) and c) and Directions: Write an inequality to describe the possible values of x. Hint: When writing your inequality, start with the largest x value on the left. a) b) SSS Inequality Theorem – If two ___________ of a triangle are congruent to _________ sides of another triangle and the _______________ side one triangle is _________________ than the third side in the other then the ________________ between the pair of congruent sides in the first triangle is ____________________ than the corresponding angel in the second triangle. Directions: Write an inequality relating the given pair of angle measures. a) WZV and UZV b) PNS and MNS c) GJH and XZY Directions: Write an inequality to describe the possible values of x. Hint: When writing your inequality, start with the largest x value on the left. a) RPA = 4x + 16 and BPA = 7x – 5 b) ACD = 9x + 2 and ACB = 3x + 14 Homework: 5d, page 270 – 271 from 5.5, (3 – 4, 10 – 17) (10 problems)