I Theorem 4.1 Triangle Sum Theorem

Transcription

1 NAME For use with pages _ DATE Classify triangles by their sides and angles and find angie measures in triangles A triangle is a figure formed by three segments joining three noncollinear points. An equilateral triangle has three congruent sides. An isosceles triangle has at least two congruent sides. A scalene triangle has no congruent sides. An acute triangle has three acute angles. An equiangular triangle has three congruent angles. A right triangle has one right angle. An obtuse triangle has one obtuse angle.. The three angles of a triangle are the interior angles. When the sides of a triangle are extended, the angles that are adjacent to the interior angles are exterior angles. Theorem 4.1 Triangle Sum Theorem The sum of the measures of the interior angles of a triangle is 180. Theorem 4.2 Exterior Angle Theorem, The measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles. Corollary to the Triangle Sum Theorem The acute angles of a right triangle are complementary. Practice Workbook Geometll"Y vvit:h Examples

2 -, LESSON NAME for use with pages _ DATE \ ~ C/ass,_ifl.:;..y'_"n.;;.9_Tf:_iB_fl..::;:y_16_S """""""' _ Classify the triangles by their sides and angles. a. L b. z J~";;";"----'-~ 16 XL...J :..::..~y 45 4 SOLUTiON a. D.JKL has one obtuse angle and no congruent sides. t is an obtuse scalene triangle. b. LlXYZ has one right angle and two congruent sides. t is a fight isosceles triangle..~~.'!!.~~~i!.~..~~.~. ~l!.i!.!!.!.~'!..! '". Classify the triangie by its sides and angles Geomet!'\f Practice Workbook with Examples Copyright McDougal Littell nc..allrights reserved.

3 LESSON NAME DATE Practice vith Examples For use with pages MUBJ.6'~ Measures a. Find the value of x. (4x - 5) b. Find the value of y. 2VO f " i (3x+ 11)" a. From the Corollary to the Triangle Sum Theorem, you can write and solve an equation to find the value of x. x = The acute angles of a right triangle are complementary. Solve for x. b. You can apply the Exterior Angle Theorem to write and solve an equation that will allow you to find the value of y = 2yo Apply the Exterior Angle Theorem. y = 70 Solve for y.!..;:.~!.~~~i!.~.!.~.~.~lj.?l!!l!.~i!..?. Find the value of x Copyright McDougal Littell nc.

4 NAME ~ DATE For use with pages identify congruent figures and corresponding parts VOCABULARY When two geometric figures are congruent, there is a correspondence: between their angles and sides such that corresponding angles are congruent and corresponding sides are congruent. Theorem 4.3 Third Angles Theorem f two angles of one triangle are congruent to two angles of another triangle, then the third. angles are also congruent. _ Using Properties of Congruent Figures n the diagram, ABCDE == FGHJ a. Find the value of x. b. Find the value of y. A B (3x+ 4) F J E D G J; SOLUTiON a. You know that AE == Fl. SoAE = Fl. 10 = 3x + 4 x=2 b. You know that LD == L. So, mld = ml. 4r = (8y - 9) 56 = 8y y=7 )

5 ) LESSON NAME _ DATE For use with pages Exercises for Example 1... n Exercises 1 and 2, for each pair of figures find (a) the value of x and (0) the value of y. 1. MBC-=MEF B C F A (5y - 3) 0 2. ABCDEF -= GHJKL G (4x - 5)" A E L j' ). F Using the Third Angles Theorem Find the value of x. c K o F A ~ f..lj B E SOLUTiON l n the diagram, LA = LD and LB = LE. From the Third Angles Theorem, you know that LC -= LF. So, mlc = mlf. From the Triangle Sum Theorem, mlc = ~ 40. mlc = mlf 40 = x.third Angles Theorem Substitute. J! t, -!

6 LESSON NAME For use with pages ~--- DATE _.~~.~!.?!.~f!.~.!.~l~1!.~!!.!.~f!. ~ ; : ""'." Find the value of x

7 . sides NAME _ DATE For use with pages =l Prove that triangles are congruent using the SSS and SAS Congruence Postulates Postulate 19 Side-Side-Side (SSS) Congruence Postulate. f three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent.. Postulate 20 Side-Angle-Side (SAS) Congruence Postulate ~ f two sides and the included angle of one triangle are congruent to two and the included angle of a second triangle, then the two triangles are congruent...,e» Using the SAS Congruence Postulate Prove that!:abc == 6DEF. B F o c E A S«:U.UTRON The marks on the diagram show that AB == DE, BC == EF, and LB == LE. So, by the SAS Congruence Postulate, you know that 6ABC~ 6DEF.!.~.~.~~~!.~~~~.~..~1!.~!!p./f!..!. State the congruence postulate you would use to prove that the two triangles are congruent Copyright McDougal Littetllnc. Georne1:ry

8 2 LESSON NAME DATE ~----:-_---, For use with pages Congruent Triangles in a Coordinate Plane. ~.'~-. WJ< ~. Use the SSS Congruence that MBC == 6CDE. Postuiate to show SOLUTON. Use the distance formula to show that corresponding lengths, d =.)(x 2 - X )2 + (h - Yl)2 AB =.)(-3- (-4))2 + (-3-2)2 =.) 12 + (- 5)2 =.J26 So, AB = CD, and hence AB == ro.. BC =.J( -1 - (- 3))2 + (0 - (- 3))2 sides are the same length. For all CD =.J(4- (-1))2 + (1-0)2 =.J =.J26 DE =.J(1-4)2 + (3-1)2,--; =.J = J3 So, BC =.DE, and hence BE == DE. CA =.J( -:-4 -' (-1))2 + (2-0)2 =.J(-3) = J3 So, CA = EC, and hence So, by the SSS Congruence CA == Ee. =.J(-3)2+2 2 = ffi EC =.)(-1-1)2 + (0-3)2 =.J(-2)2 + (-3)2 =.J3 Postulate, you know that!::,abc == 6CDE. Practice Workbook vvith Examples

9 ~' LESSON NAME ~ _ DATE For use with pages Exercise for Example 2... ~""""""""""" "".,," "" "" "", " " " " " " ". " "" '"'' "'" "". 3. Prove that MBC == 6.DEF. ';,

10 9 NAME ~ ~ DATE For use with pages Prove that triangles are congruent using the ASA Congruence Postulate and the AAS Congruence Theorem.,, t t \ \ \ 1 1 Postulate 21 Angle-Side-Angle (ASA)'Congruence Postulate f two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the two triangles are congruent., Theorem 4.5 Angle-Angle-Side (AAS) Congruence Theorem f two angles and a nonincluded side of one triangle are congruent to two! angles and the corresponding nonincluded side of a second triangle, then the two triangles are congruent. _ Proving Triangles are Congruent Using the ASA Congruence Postulate ~ ~ = em ms Given: Prove: BC == EC, LB == LE!::ABC == 6DEC o SOLUTON Statements 1. Be == Ee 2. LB== LE 3. LACB == LDCE 4.!::ABC == 6DEC Reasons 1. Given 2.. Given 3. Vertical Angles Theorem 4. ASA Congruence Postulate 8 i --~

11 -"""'::- LESSON r, :..1;;.. Foruse with pages ',. _"--4f NAME ~ D~E Exercises lor Example t... n Exercises 1 and 2, use the given information the triangles are congruent. 1. Given: MC == AC LNMC and LBAC are right angles. Prove: 6NMC == 6BAC N to prove that 2. Given: AE == DE, LA == LD Prove: 6BAE == 6CDE o "'-' " C B ~ Proving Triangles ere Congruent Using AAS Congruence Theorem!Sf ~ W~. ~wnr lzi.l&\!:wz ::::a:swt::::=i:i :::eu!'!'jw~.= Given: AD == AE, LB == L C Prove: MBD == MCE c B A Statements 1. AD==AE 2. LB == LC 3. LA == LA 4. MBD==MCE Reasons 1. Given 2. Given 3. Reflexive Property of Congruence 4. AAS Congruence Theorem.

12 LESSON 4.4 NAME '~~~~~~~~~~~~~~~~~~~~~ DATE ~ _ For use with pages ~~.f!!.~~~f!.~.!.c!!..~l!.~!!p/f!..?. n Exercises 3 and 4, use the given information to prove that the triangles are congruent, 3. Given: LG == LB, CB GA 4. Given: LOMN == LONM, LLMO== LNO Prove: 6GCA == 6BAC Prove: 6MN == 6NLM A" \"78 'M L G J Practice Workbook vvith Examples

13 NAME ~ _ DATE For use with pages Use congruent triangles to plan and write proofs Given: Prove: PR == PQ, SR == TQ QS == RT Plan for Proof: QS and RT are corresponding parts of 6PQS and 6PRT and also of 6RQS and 6QRT. The first set of triangles is easier to prove congruent than the second set. Then use the fact that corresponding parts of congruent triangles are congruent. SOU..!TON R p Q Statements 1. PR == PQ 2. PR = PQ 3. PR = PS + SR 4. PQ = PT + TQ 5. PS + SR = PT + TQ 6. SR == TQ 7. SR = TQ 8. PS = PT 9. PS== PT 10. LP == LP 11. 6PQS == 6PRT 12. QS == RT. Reasons 1. Given 2. Definition of congruence 3. Segment Addition Postulate 4. Segment Addition Postulate 5. Subsitution 6. Given 7. Definition of congruence 8. Subtraction property of equality 9. Definition of congruence 10. Reflexive Property of Congruence 11. SAS Congruence Postulate 12. Corresponding parts of congruent triangles are congruent..

14 \ ~ LESSON NAME _ DATE For use with pages Exercises for Example Use the given information to prove the desired statement. 1. Given: RT==AS, RS==AT 2. Given: Ll==L2==L3,L4==L5,ES==DT Prove: L TSA == LSTR Prove: HE == HD H T t>?},"j R A E o s, Using More than One Pair of Triangles Given: Ll == L2, L5 == L6 Prove: AC.lBD Plan for Proof: t can be helpful to reason backward from what is to be proved. You can show that AC.lBD if you can show L3 == L4. Notice that L3 and L4 are corresponding parts of MBO and MbO. You can prove MBO == bado by SAS if you first prove AB == AD. AB and AD are. - corresponding parts of MBC and MDC. You can prove MBC == MDC by ASA. A~----::------':-t-"---=---7C B o l-./.;

15 LESSON NAME _ DATE For use with pages Statements 1. Ll ~ L2,L5 ~ L6 2. Ae~AC 3. MBe~MDe 4. AB~AD 5. AO ~AO 6. MBO==MDO 7. L3 ~ L4 Reasons 1. Given 2. Reflexive Property of Congruence 3. ASA Congruence Postulate 4. Corresponding parts of congruent triangles are congruent. 5. Reflexive Property of Congruence 6. SAS Congruence Postulate. 7. Corresponding parts of congruent triangles are congruent. 8. f 2 lines intersect to form a linear pair of congruent angles, then the lines are 1.. ~;:.~!.'?/~~~. t~.~.~l!:?'!!l!.~~.?. n Exercises :3and 4, use the given information desired statement. 3. Given: PA ~ KA, LA ~ NA Prove: AX ~ AY N to prove the 4. Given: LDAL ~ LBCM, LeDL~ DC~BA Prove: AL == em LABM rc+r+: -1- -; C L A B Copvriqht McDougal Littstl lnc, All rights reserved, PracticeWorkbook with Examples

16 NAME ~ ~ _. For use with pages DATE Use properties Of isosceles, equilateral, and right triangles VOCABULARY f an isosceles triangle has exactly two congruent sides, the two angles adjacent to the base are base angles. f an isosceles triangle has exactly two congruent sides, the angle opposite the base is the vertex angle. Theorem 4.6 Base Angles Theorem f two sides of a triangle are congruent, then the angles opposite them are congruent Theorem 4.7 Converse of the Base Angles Theorem f two angles of a triangle are congruent, then the sides opposite them are congruent. Corollary to Theorem 4.6 f a triangle is equilateral, then it is equiangular. Corollary to Theorem 4.7, f a triangle is equiangular, then it is equilateral. Theorem 4.8 Hypotenuse-Leg (HL) Congruence Theorem f the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of a second right triangle, then the two triangles are congruent.. 1- i~ Using Properties.of Right Triangles Given that LA and LD are Tight angles and AB == DC, show that MBC == l',.dcb. SC)l.UTON A~ ~ ~C ~-, - ~ Paragraph proof You are given thatla and LD are right angles. By definition, MBC and l',.dcb are right triangles. You are also given that a leg of MBC, AB, is B congruent to a leg of ~DCB, DC. You know that the hypotenuses of these two triangles, BC for both triangles, are congruent because BC == BC by the Reflexive Property of Congruence. Thus, by the Hypotenuse-Leg Congruence Theorem, MBC == /'::,DCB. D Geomet!'y

17 .! l ~- LESSON NAME ~ _ DATE For use with pages ~~~!.~~~f!.~.!.~.~.~lf.f!.!!.e.~l!.. Write a paragraph proof.!. 1. Given: BCl.AD, AB == DB 2. Given:mLKL = mlmlk = 90, Prove: MBC == L.DBC L == MK B Prove: K==ML J ~ A C 0 L M Using Equilateral and s,!sceles.., T;iangles., Find the values of x and y. SOLUTiON Notice that tiabc is an equilateral triangle. By the Corollary to Theorem 4.6, /':,.ABCis also an equiangular triangle. Thus mla = mlabc = mlacb = 60. So, x = 60. Notice also that f:..dbc is an isosceles triangle, and thus by the Base Angles Theorem, mldbc = mldcb. Now, since mlabc = mlabd + mldbc, mldbc = = 300. Thus, y = 30 by substitution. A Copyright McDougal Littell nc.

18 LESSON NAME ~ ~ ~-- _ DATE For use with pages Exercises for Example Find the values of x and y. 3. «~

19 NAME ~ ~ DATE For use with pages Place geometric figures in a coordinate plane and write a coordinate proof VOCABULARY A coordinate proof involves placing geometric figures in a coordinate plane and then using the Distance Formula and the Midpoint Formula, as well as postulates and theorems, to prove statements about the figures. t., _ Using the Distance formula A right triangle has legs of 6 units and 8 units. Place the triangle in a coordinate plane. Label the coordinates of the vertices and find the length of the hypotenuse..\- One possible placement of the triangle is shown. Points on the same vertical segment have the same x-coordinate, and points on the same horizontal segment have the same y-coordinate. Use the Distance Formula to find d. = --1(8-0)2 + (6-0)2 = -fioo = 10.~~.f!!.l?,~~~.~.!.C!.~.~~c:.'!!P..~f! Use a coordinate the question. Distance Formula Substitute. Simplify and evaluate square root. y!!, i! i(8,6)! Y i r i V /'! dla i!! 1/ i - 1 l/,! A -1! i!! (O,O)! :3! 1(8,0) xl i i i..! _. plane and the Distance Formula to answer ' 1. A rectangle has sides of length 8 units and 2 units. What is the length of one of the diagonals? Practice Workbook vvith Examples

20 LESSON NAME _ DATE For use with pages n the diagram, 6ABD == 6CBD. Find the coordinates of point B. y SOLUY ON. Because the triangles are congruent, AB == CB. So, point B is the midpoint of AC: This means you can use the Midpoint Formula to find the coordinates of point B. ) _ (Xl + X') )'1 + Y2) Midpoint B(x, )) - 2' 2 Formula ~- ( , T2 '7) -_ (-22,2 2 ) Substitute and Simplify..~;:.~!f?~~f!.~.!.l!.~.~l!.l!.i!!.?~f!..?. 2. n the diagram, 6FGH == 6JH. Find the coordinates of H. PracticeWorkbook with Examples

21 LESSON NAME ~ ~ _ for use with pages DATE WO'itillll1l a Coordinate Write a plan to prove that 6,DEF == 6,DGF. Given: Coordinates of figure DEFG. Prove: 6DEF == 6DGF Plan for Proof Use the Distance Formula to show that segments EF and GF have equal lengths and that segments DE and DG have equal lengths. Use the Reflexive Property of Congruence to show that DF 2:: DF. Then use SSS Congruence Postulate to conclude that 6DEF 2:: 6DGF.!.~.~ y E(b, c) Gib, -c) 'F(a,O).~~.~~~~~l!.~.~l!.C?r!!.!.~~.?. Describe a pian for the proof. 3. Given: Coordinates of figure ABeD. Prove: MBC == 6CDA A(O,O) y ~~------~ ~~ x x D(2a, -b), Practice Workbook with. Examples