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Transcript
4.7 Theorems On Perpendicular Lines Theorem 4.7: If two lines intersect to form a linear pair of congruent angles, then the lines are ______________. perpendicu lar If 1 2, then g ____ h. g 1 2 h 4.7 Theorems On Perpendicular Lines Theorem 4.8: If two lines are perpendicular, then they intersect to form four ______________. right angles If a b, then 1, 2, 3, and 4 and 4 are __________ ____. right angles b 1 2 3 4 a 4.7 Theorems On Perpendicular Lines Theorem 4.9: If two sides of two adjacent acute angles are perpendicular, then the angles are ________________. complement ary If BA BC, then 1 and 2 are __________ complement____. ary A 1 2 B C 4.7 Theorems On Perpendicular Lines Theorem 4.10: Perpendicular Transversal Theorem If a transversal is perpendicular to one of two parallel lines, then it is _____________ perpendicu lar to the other. If h k and j h, then j ___ k. j h k 4.7 Theorems On Perpendicular Lines Theorem 4.11: Lines Perpendicular to a Transversal Theorem In a plane, if two lines are perpendicular to the same line, then they are _________ parallel to each other. If m p and n p, then m ___ n. m n p 4.7 Theorems On Perpendicular Lines g 2 K 1 2 h N J 1 13 L 13 2 M 4.7 Theorems On Perpendicular Lines Example 1 Application of the Theorems Find the value of x. a. a b l b. xo xo 55o p Solution perpendicular all four angles a. Because a and b are _______________, formed are right angles by _______________. Theorem 4.8 90 By definition of a right angle, x = ____. b. Because l and p are perpendicular, all four angles Theorem 4.8 formed are right angles by ____________. Theorem 4.9 the 55o angle and the xo angle are By ___________, ______________. complementary Thus x + 55 = 90, so x = ____. 35 4.7 Theorems On Perpendicular Lines Checkpoint. Find the value of x. 1. 2 x x 90 3x 90 x 30 o 2x o x o o By Theorem 4.8, all angles are right angles. o 4.7 Theorems On Perpendicular Lines Checkpoint. Find the value of x. 2. 3x x 90 4 x 90 x 22.5 o xo 3x o o By Theorem 4.8, all angles are right angles. o 4.7 Theorems On Perpendicular Lines Example 2 Find the distance between a point and a line What is the distance from point B to line q? Solution You need to find the slope of line q. Using the points (3, 2) and (6, 5), the slope of line q is 5 2 1 . m ___ 6 3 B 3, 8 q 6, 5 3, 2 1 1 The distance from point B to line q is the length of the perpendicular segment from point B to line q. The slope of the perpendicular segment from point B to line q is the negative 1 reciprocal of ___, 1 1 or _____ = ____. 1 4.7 Theorems On Perpendicular Lines Example 2 Find the distance between a point and a line What is the distance from point B to line q? Solution The segment from (6, 5) to (3, 8) has a slope of ____. 1 So, the segment is perpendicular to line q. Find the distance between (6, 5) and (3, 8). d 3 __ 6 __ 5 __ 8 __ 2 2 B 3, 8 q 6, 5 3, 2 1 1 4.2 ____ The distance from point B to line q is about ____ 4.2 units. 4.7 Theorems On Perpendicular Lines 5.2 Use Perpendicular Bisectors Theorem 5.3: Converse of the Perpendicular Bisector Theorem In a plane, if a point is equidistant from the endpoints of a segment, then it is on the A ____________ perpendicu lar _________ bisector of the segment. C P B D If DA DB, then D lies on the ________ bisector of AB. 5.2 Use Perpendicular Bisectors Example 1 Use the Perpendicular Bisector Theorem AC is the perpendicular bisector of BD. Find AD. Solution 7 x7x– 66 B C A AB AD ____ D ___ ______ 6 ___ 3x ___ x ___ 2 42 ___ 4 x ____ 8 . AD ____ 44xx 5.2 Use Perpendicular Bisectors Example 2 Use perpendicular bisectors In the diagram, KN is the perpendicular bisector of JL. a. What segment lengths in the diagram are equal? b. Is M on KN? Solution K N J 13 L 13 a. KN bisects JL, so ____ ____. NJ = NL Because K is on the perpendicular bisector M of JL, ____ KJ = ____ KL by the Theorem 5.2. The diagram shows that ____ MJ = ____ ML= 13. equidistant from J and L. b. Because MJ = ML, M is ____________ So, by the Converse __________________________________________, of the Perpendicular Bisector Theorem M is on the perpendicular bisector of JL, which is KN. 5.2 Use Perpendicular Bisectors Checkpoint. In the diagram, JK is the perpendicular bisector of GH. F 1. What segment lengths are equal? KG KH JG JH FG FH 2. Find GH. x 1 4.1 4.1 J G K 2x x 1 2x x+1 GH 2 x x 1 21 1 1 4 H 5.2 Use Perpendicular Bisectors Theorem 5.4: Concurrency of Perpendicular Bisector of a Triangle B The perpendicular bisector of a triangle intersects at a point that is equidistant from the vertices of the triangle. D A If PD, PE, and PF are perpendicular bisectors, then PA = _____ PC PB = _____. P F E C 5.2 Use Perpendicular Bisectors Example 3 Use the concurrency of perpendicular bisectors N The perpendicular bisectors of MNO meet at point S. Find SN. 7 Q P S 9 Solution 2 R M Using ______________, you know that point S is Theorem 5.4 _____________ equidistant from the vertices of the triangle. So, _____ SO SN = _____. SM = _____ _____ SM SN = _____ 9 _____ SN = _____ Theorem 5.4. Substitute. O 5.2 Use Perpendicular Bisectors Checkpoint. Complete the following exercise. 3. The perpendicular bisector of ABC meet at point G. Find GC. B 12 By ______________, Theorem 5.4 D _____ GC GB = _____. GA = _____ GC GA GC 15 E G 15 A 6 F C 5.2 Use Perpendicular Bisectors Pg. 284, 5.2 #1-10