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Key+Special+Triangle+Segment+Packet3

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School
William B Travis H S**We aren't endorsed by this school
Course
MATH GEOMETRY
Subject
Mathematics
Date
Dec 17, 2024
Pages
16
Uploaded by BrigadierDonkey4594
Special Segments in Triangles Name Learning Intentions 3 Apply Algebraic reasoning and verify conjectures about segments and angles in triangles by applying properties associated with special Segments. Apply algebraic reasoning along with properties of special segments to find unknown lengths and measures. Verify conjectures about segments and angles in triangles by applying the properties associated with special segments. Success Criteria — | WILL... 3 3 3 concurrency. triangles. Apply algebraic reasoning to solve problems. Apply Theorems and proofs specifically to isosceles, equilateral and right triangles Use a variety of tools to investigate special segments of triangles and their points of Investigate and make conjectures about the properties of special segments in Apply algebraic reasoning along with properties of special segments to find unknown lengths and measures. Nov 21 New Packet: Introduce Project and Special Segments [Nov 22 | Special Segments. 25 Thanksgiving Break 26 Thanksgiving Break 27 Thanksgiving Break 28 Thanksgiving Break 29 Thanksgiving Break | Dec 2 End of PR4 Triangle Angle Sum Theorem 3 Isosceles & Equilateral Triangles 4 Triangle Inequalities 5 Project Work Day 6 | | Project Work Day ; | 9 English | STAAR Retest Finish Presentations Project Presentations 10 Biology & US History STAARRetest e lrz Mectn Project Presentations 11 English Il STAAR Retest « Semester Review 12 Algebral STAAR Retest Tk Semester Review 13 Semester Review 16 Semester Review DB Daily Glode 16 (7 6" Period Semester Exam 16 2" & 4" Semester Exam 3 & 5" Semester Exam 116" End of 2" 9 weeks 1" & 7" Semester Exam — Early Dismissal
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Day 1: Special Segments in Triangles Four special types of segments are associated with triangles. > Amedian is a segment that connects a vertex of a triangle to the midpoint of the opposite side. » Analtitude is a segment that has one endpoint at a vertex of a triangle and the other endpoint on the line containing the opposite side so that the altitude is perpendicular to that line. > Anangle bisector of a triangle is a segment that bisects an angle of the triangle and has one endpoint at the vertex of that angle and the other endpoint on the side opposite that vertex. > A perpendicular bisector is a segment or line that passes through the midpoint of a side and is perpendicular to that side. For each triangle below, draw the median from A, the Altitude from A, and the perpendicular bisector of AB. P Complete using the figure at the right. [5F 1. 1FAB =BG, then __is a median of AAPC. A B [2 D 1 PCisa perpendicular bisector of __then BC = DC. legs @f e R e . If ZAPD is a right angle, then __and __ are altitudes of AAPD. ole . 1f PC is a median of APBD, then _=__. IfBC=CDand PC L BD, then __isa perpendicular bisector of __. . 1f PC and 4C are both altitudes of APCA, then Z__is a right angle.
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Complete using the figure at the right. As _ i zeex %¢ 7. 1f BX bisects ZABC,then /__=/__andDX=__ (L] XF ,thenED=__andXE=__. 8. If DX is the perpendicular bisector of £ B (&3 X6 _ i 9. IfXB = XF, then __is the perpendicular bisector of BF", and ZXBF = A_F >3 8 - 10. If XD = XG, then _"is the bisector of Z_". b6 Complete the statement. S K 11. If X is on the bisector of ZSKN, then X is equidistant from __ and __ SN 4 &N 12. If X is on the bisector of ZSNK, then X is equidistant from __ N and _. R L , ~ — — e 13. IfXis equidistant from SK and SV | then X lies on the __. ZL a2 fp 77 14. If O is on the perpendicular bisector of LA, then 0 F is equidistant from __and __. 0 LA ael” 15. If O is on the perpendicular bisector of AF ,then 0 is equidistant from __and __. (ot Sho . A 16. If 0 is equidistant from L and F, then O lies on the 17. Find 4B if BD is a median of AABC. 18, Find BC if 4D is an altitude of AABC. 43 = 2= A - PO x-7 20 =X
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2 At T o | 19. Find mZABC if BD is an angle bisector of AABC. mZABC=(4x-6)° In problems 20-23, A(2,5), B(12,-1), and C(-6,8) are the vertices of AABC. v 20. What are the coordinates of K if (K is a median of AABC? r Doxo Fels 5> 417 5 Boad mid Pornd of- AB 4—2 2 ‘?:l ) I (=, 2 =N P [ R { 21. What is the slope of the perpendicular bisector of 48?7 = . x E 5 = pey Feect FiaJ Slope AR bo-2 ™ Bz ks | biser 2 fe ‘1 — b'5 > 22. What is the slope of €1< if CL is the altitude from point C? / ¢ 1 c wouldh be L 3¢ Nne ' ( 23. Point N on BC has coordinates \ 2 / - Slope NP Slppe HE = 24. RT is amedian in ARLB with points R(3,8), T(12,3), and B(9,12) O v\ a. What are the coordinates of L? T ‘-va‘»f' De Midpoias ot LB bisector of RB
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Centers of Triangles Gircumcenter | | Incenter | [ Gentroid || Orthocenter | B A\ mportant Facts: mportant Facts: rmportant FacTs: mportant Facts: The Cirgnegentes The : 4 A 3 is created An O/ is equidistant from each is equidistant from each bya Y2777 connected is created by a vertex * / of the triangle. of the triangle. tothe 1 e per 77 of connected to the opposite the opposite side. side so that it is Drex A ' o that side.
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Directions: Based on the markings, classify each centeras a circumcenter, incenter, centroid, or orthocenter. 9. fifi ntroid — Medlars o e wree e L Bogect Crr Meda. 7 Tneeater = 7 O04+hoe e < AlsAtudes a Ao = PNy 2 oz _ L. RisecTors
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(el N 8 X S el # C‘(’.-. Day 2: Special Segments on the Coordinate Plane : Write the equation of each line in point-slope form. ‘ 1) Triangle: A(4,5), B(-4,6),C(2,—6) 2) Triangle: D(-2,3), E(6,-3), F(~4,-3) Write the equation of the altitude Wrte the equation of the perpendicular through A. bisector of FE A \ (ol ' : ) neg ., " cep * /'.‘f:_,-_'l -3 =3 e et i }' = Y " oo\ - » e 5; +L."(/'(’[l¢( ™M .—" | \/”j_“— v 5 - pe J , < - fjoe s G-iproty R pr. \ /. Li y—S=7z (-4 : =V, x \ \/,7 - '/{l./ 3 3) Triangle: G(—3,5), H(3,—8),I(-5,—2) 4) Trangle: J(2,5), K(6,—7),L(~6,1) Write the equation of the median through I. Write the equation of the perpendicular bisector of LK . e 1ok N : ,‘.l‘r.' ,.f’ s G f"’ o C e I~ I | /'/””I ” a | 11 G- i ] ) -~ ¢ S i 1 T e 1] 1] e MOyl P L Slope TM 2= TIE W o e e ) w— . _f g bt —4 SR RS S ) S e = e (45 | el ] R I & e 1 —— 2 FeHue {2 4% need i, Tuss 5) Triangle. M (—4,5), N(G,Z),P(l,—5) 6) Tnangle. Q(-3,5), R(5,5),S8(1,-17) Wnte the equation of the altitude through M. Write the equation of the median through Q. Need olope o MP e ol e gk e 3 . ’ | e 32 ot i BN S B0 ! 558 B R B i SR | LA T 405t - T PR 21 N 0 31 1 651 ) o V1 S R - O O 3 5 7 4 ol U 0 S R S ; v i o e - 8.. - 5 B O BL:I “T( aneale t—+4 ?“.\: 8._, ; Ll btc o .’Q:.— ’BL 4] ./4&, i \ » b3 ‘ - — + A - -_ —— 4 3 - + . - HAEER SN EEN IR LI YT XIHTITTL N/ 3 SRt mm e x\“:;\.\\,,x - 14L. Mg /"" ’ P+ Sleopc oI EEEEE N NI REE SEESNNERLEENE AREan Y ' = O N S 1O S B A A 30 S O S A A 0 BN A T v NN PR Y R Y O (R AR A ) ] SV £ O K O O -S~2 _ o1 TolalelaoN T T g T Ian 0B lelaleU 2 Yy g gl pl = | '.2[_ AT T }9[ /NN — i e~ 50 R s o i 5 MY B R i = & % }G.Lj: ! 1 \ ¢-’\<g”/ #:” //'/’_ ‘-y( | "8"" | 1 T TTTT11 '78" S | | ‘ / s - S \/\N S e ) S SN A | - — ettt —t | S T S - - S NS N T - s - , —y A legL1 1 1 ! n ,-}IB-: 1 I S S A \ RY . ;‘/'_’ /'//‘J i A A . C il 7 Oops, | made the Median for S instead of Q
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7) Tnangle U(14), V(2-7), T(-7.2) 8) Triangle: W(0,3), X(6,~1),Y(~6,-5) Wnite the equation of the alitude Whrite the equation of the perpendicular through U. . k : e ?,7;;70‘——‘ 4TV bisector of YX . Y4 57 9) Trangle: A(-4,5), B(6,4),C(2,-6) 10) Triangle J(2,8), K(~5.2),L(~6,~1) Write the equation of the median through A Write the equation of the perpendicular bisector of LK . Y-Yo= Va(x+55) 11) Trangle. D(0,5), E(3,-2), F(-2,0) 12) Triangle. G(~3,5), H(5,4),1(1,~2) Write the equation of the altitude through D. Write the equation of the median and altitude through G. P ) Ts oscele wi Fh Ve Per (:
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Day 3: Triangle Angle Sum Theorem, Exterior Angles Theorem Find the value of x. o 232 (103-x)° 03-% +2%
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Using what you know about angles in triangles, parallel lines, and other angle relationships, find the missing angles in each of the following problems. *Hint - look at ALL the angles in the diagram.* 6 =_55 h=_¥% k= 55 10
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11
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Day 4: Isosceles and Equilateral Triangle Theorems Definition of Isosceles Triangle - An isosceles triangle has at least two congruent sides. Isosceles Triangle Theorem - If two sides of a triangle are congruent, then the angles opposite those sides are congruent, Corollary - A triangle is equilateral if and only if it is equiangular. Converse of the Isosceles Triangle Theorem - If two angles of a triangle are congruent, then the sides opposite those angles are congruent, B B R A C y C Find x. 1x=__ 7~ 2% 2. x= 3 3.x=_111 23 - ; 33 - 2x 12 3x A ax_o R L 6. x=_- 1.80“' ¥ B9 (& /) - el :’/ >3 Y 2 - Y=o 60 a2 L " et -6 ops 4x = 720 8x - .= ///, - . ‘ 3x-7 4x-8 X" 56 2x 12 120
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10.x= 11 x= 12.x=_1 X -2 I\t — v zs 3x-5 x+1 2x+9) R -x-2, 2x x+1 18B.x=_ ¢ 3+l ux=_19 Ly 15.x= / 1 =5 DR ed: 1 192 s A= 4x-2 5x 25 ¢ =L 14 3x+16 X Examples Check your answer: The angles of QUAD should add up to . BD bisects ZABC 2. Given: €D pisects £BCA m/BAC = 80 Find: m/ABC=_20 m/DCB = m/BDC = p: mZABD =_"~ 3. The angles of a triangle are 3x - 10, 3x - 5, and 2x + 3. Findx. and angles. 4. Find x. X +3x-3 Q 10 a A B 0 U A 4 > and the angles of the triangle—Classify the triangle by sides 13
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Day 5: Inequalities in Triangles Comparison Property of Inequality -Ifa=b + cand ¢ >0,thena>banda>c. Recall that the exterior angle of a triangle is equal to the sum of the remote interior angles. m<4=m<1+m<2 Using the Comparison Property of Inequality, we get the following: Corollary to the Triangle Exterior Angle Theorem: mZ4>m < _| _andm<4>m< z Ex. Explainwhy m£4>m £5. P i «, o E . A D 4 Theorem: In a triangle, the larger angle lies opposite the longer side. /,»fi\ Ex. In ARGY, RG = 14, GY = 12, and RY =20. List the angles from largest to smallest. \\ v £ o 26 &Y, <f K=o 7 Theorem: In a triangle, the longer side lies opposite the larger angle. Ex. List the sides of ASTU in order from shortest to longest. The figure at the right is not drawn to scale. (a) Name the angle with the least measure in ALMN. 2 /\/ (b) Which angle in AMOT has the greatest measure? 27T (c) Name the greatest of the six angles in the two triangles, LMN and MOT. , , M The figure at the right is not drawn to scale. (a) What is the longest segment in ACED? £ E (b) What is the longest segment in ABCE? {2 i — B <4 () Find the longest segment in the figure. BE Q 2
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(d) Find the shortest segment in the figure. C Triangle Inequality Theorem - The sum of the lengths of any two sides of a triangle is greater than the length of the third side. Ex. Atriangle has side lengths 7 and 12. What are the possible lengths for the third side? P12ox amd TEX For Exercises 3-6, list the angles of each triangle in order from smallest to largest. 15
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Can a triangle have sides with the given lengths? Explain. 15.8cm,7cm,9cm e 16.7 1, 13 1,6 fi /10 17.201in, 18 in, 16in. e < 18.3m,11m7m ~/O The lengths of two sides of a triangle arc given. Describe the possible lengths for the third side. ) o > 7 e 19.5.11 s+ Zx 4 Lo o & =X 7 b - ' DL XL 20.12. 12 4 . AT e ’ 2 n¢ 23 21.25, 10 e Gud |oyx D25 ' 2 2 = / + > > F i - £.5¢ ~ 22.6,8 b4 B> ¥ ik di 2 2 16
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