Geometry Triangles 2

2. Show that the congruence of triangles is reflexive. Given: ∆RST Show ∆RST ≅ ∆RST We know that RT ≅ RT , RS ≅ RS , and TS ≅ TS by the reflexive property of equality. ∠R ≅ ∠R, ∠S ≅ ∠S, and ∠T ≅ ∠T by the reflexive property of equality. We now have three sides and three angles of ∆RST congruent to the three corresponding sides and corresponding angles of ∆RST. By the definition of congruent triangles, ∆RST ≅ ∆RST. 3. Show that the congruence of triangles is transitive. Given: ∆RST ≅ ∆UVW, ∆UVW ≅ ∆XYZ Show ∆RST ≅ ∆XYZ We know that RS ≅ UV , RT ≅ UW , ST ≅ VW , ∠R ≅ ∠U, ∠S ≅ ∠V, and ∠T ≅ ∠W by CPCTC given ∆RST ≅ ∆UVW. Since ∆UVW ≅ ∆XYZ, we have UV ≅ XY , UW ≅ XZ , VW ≅ YZ , ∠U ≅ ∠X, ∠V ≅ ∠Y, and ∠W ≅ ∠Z by CPCTC. We know that congruent sides have equal measures and by the transitive property of equality we have the following: RS = XY, RT = XZ, and ST =YZ Using a similar justification, the following is true for corresponding angles: m∠R = m∠X, m∠S = m∠Y, and m∠T = m∠Z ∆RST ≅ ∆XYZ by definition of congruent triangles. 4. Congruence of triangles is also symmetric. This justification is left as an optional exercise. Exercises 1. Given ∆MNO ≅ ∆PQR, identify the corresponding parts. 2. Given: ∆MNO ≅ ∆PQR, MN = (5x-11), PQ = (3x-2) Find: x and MN 3. Given: ∆KLM ≅ ∆NOP, m∠M = (5x + 23), and m∠P = (3x + 40) Find: x and m∠P 4. Given: ∆ABC with vertices A(-3,2), B(2,6), and C(-3,6) and ∆EFG with vertices E(2,2), F(7,-2), and G(2,-2) a) Find the lengths of each side and determine if the triangles are congruent. b) If the triangles are congruent, write a congruence statement for the two triangles. If the triangles are not congruent, justify your conclusion. Exploration By the definition of congruent triangles, two triangles are congruent if and only if the six parts of one triangle are congruent to the six corresponding parts of a second triangle. Is it possible for two triangles to be congruent using only some of the corresponding parts? If so, which corresponding parts are sufficient to show two triangles congruent? Do problem 1 in the activity Look Alikes using Cabri Junior™ on a TI-83+ graphing calculator. In this exploration, a triangle will be constructed using three sides of a given triangle. The results of this exploration leads us to the SSS congruence postulate for proving two triangles congruent. SSS Congruence Postulate Two triangles are congruent if and only if three sides of one triangle are congruent to three sides of a second triangle. Examples 1. Determine which of the following represents a pair of congruent triangles. 2. Given: ∆ACB is isosceles , point M is the midpoint of AB Show ∆AMC ≅ ∆BMC. AC ≅ CB by the definition of an isosceles triangle. AM ≅ MB since M is the midpoint of AB . CM ≅ CM by the reflexive property of equality. Now we have ∆AMC ≅ ∆BMC by the SSS congruency postulate. 3. Given: RS ≅ VT , RT ≅ SV Prove: ∠SRT ≅ ∠TVS Statements 1. RS ≅ VT RT ≅ SV Reasons 1. Given 2. ST ≅ ST 2. Reflexive property of equality 3. ∆SRT ≅ ∆TVS 3. SSS congruence postulate 4. ∠SRT ≅ ∠TVS 4. CPCTC Exercises 1. Given: ∆MNO ≅ ∆PQR Determine whether each statement is true or false and give a justification for your answer. a) MN ≅ PQ b) ∠MNO ≅ ∠PRQ c) m∠OMN = m∠QPR 2. Given: PQ ≅ SR PS ≅ QR Prove: ∆PQS ≅ ∆RSQ Complete the proof with an appropriate statement or reason. Statements 1. PQ ≅ SR Reasons 1. _________________________ PS ≅ QR 2. ___________________________ 2. Transitive property of equality 3. ∆PQS ≅ ∆RSQ 3. __________________________ 3. Given: AE ≅ BF EC ≅ DF AB ≅ CD Prove: ∠AEC ≅ ∠BFD Proof: Statements 1. AE ≅ BF EC ≅ DF AB ≅ CD Reasons 1.____________________________ 2. AE = BF, EC = DF, AB = CD 2. ___________________________ 3. _________________________ 3. Reflexive property 4. AB + BC = CD + BC 4. ___________________________ 5. AB + BC = AC, CD + BC = BD 5. ___________________________ 6. __________________________ 6. Substitution 7. __________________________ 7. SSS congruence postulate 8. ∠AEC ≅ ∠BFD 8. ___________________________ 4. Given: ∆RST with vertices R(-3,8), S(2,5), and T(2,8) ∆MNP with vertices M(10,0), N(5,3), and P(5,0) Show ∆RST ≅ ∆MNP. Exploration Refer to the activity Look Alikes in the Geometry Module. Use a TI-83+ graphing calculator with Cabri Junior™ and do problem 2. In this problem, a triangle will be constructed using two sides and the included angle of a given triangle. Side lengths and angle measures of the constructed triangle will be compared to the corresponding side lengths and angle measures of the original triangle. This activity refers to two sides and the included angle of a triangle. In the diagram below, the sides XZ and XY of ∆XYZ are included in the sides of ∠ZXY In the diagram below, the given sides (marked) are not included in the sides of ∠ZXY. This diagram does not represent two sides and the included angle of a triangle. The results of this exploration lead us to the following congruence postulate for triangles. SAS Congruence Postulate Two triangles are congruent if and only if two sides and the included angle of one triangle are congruent to two sides and the included angle of the second triangle. Examples 1. Which of the following represents a pair of congruent triangles? State the congruence postulate that applies and write a justification. d) Given: AD and CE bisect each other at point O + AOC ≅+ DOE Solutions: 1. a) The triangles do not represent a pair of congruent triangles since the angle shown is not the included angle for the two sides given. b) The pair of triangles shown are congruent by the SAS congruence postulate since the sides marked are included in the sides of the right angle. c) ∆RST ≅ ∆RUT by the SAS congruence postulate as follows: S ST ≅ TU These corresponding sides are given. A ∠STR ≅ ∠RUT An angle bisector divides an angle into two congruent angles. S RT ≅ RT Reflexive property d) ∆AOC ≅ ∆DOE by the SAS and SSS congruence postulates. Using the SAS congruence postulate, we have the following justification: S A S CO ≅ EO A bisector of a segment divides it into two congruent segments. ∠AOC ≅ ∠DOE Vertical angles are congruent. AO ≅ DO A bisector of a segment divides it into two congruent segments. Using the SSS congruence postulate, we have the following justification: S S S AC ≅ DE CO ≅ EO AO ≅ DO Given A bisector of a segment divides it into two congruent segments. A bisector of a segment divides it into two congruent segments. Examples 2. Given: AC ≅ BD ∠CAB ≅ ∠DBA Prove: AD ≅ BC Complete the following proof. Statements 1. AC ≅ BD ∠CAB ≅ ∠DBA 2. _________________________ 3. _________________________ 4. AD ≅ BC Reasons 1. __________________________ 2. Reflexive property 3. SAS congruence postulate 4. __________________________ 3. Given: ∆ABC with vertices A(-2,6), B(1,2), and C(1,8) ∆DFE with vertices D(7,4), F(4,0), and E(4,6) ∠BAC = ∠FDE = 86.8° a) Graph the coordinates of the vertices of each triangle in the coordinate plane. b) Determine if ∆ABC ≅ ∆DFE and state the congruence postulate used. 4. Given: ∆KAR ≅ ∆GTO with KA = (3x-10) , GT = (x+2), m∠KAR = (5y-40) and m∠GTO = (3y-8) Find: x, y and KA Solutions: 2. Statements Reasons 1. AC ≅ BD ∠CAB ≅ ∠DBA 1. Given 2. AB ≅ AB 2. Reflexive property 3. ∆CAB ≅ ∆DBA 3. SAS congruence postulate 4. AD ≅ BC 4. CPCTC 3. b) The distance formula can be used to determine the distance between two points with coordinates (x, y) and (x1,y1) in the coordinate plane: d = ( x − x1 ) 2 + ( y − y1 ) 2 Using the distance formula above, the following lengths of sides can be determined. AC = (−2 − 1) 2 + (6 − 8) 2 = (−3) 2 + (−2) 2 = AB = (−2 − 1) 2 + (6 − 2) 2 = (−3) 2 + 4 2 = ED = (4 − 7) 2 + (6 − 4) 2 = (−3) 2 + 2 2 = 9+4 = 9 + 16 = 9+4 = 13 25 = 5 13 DF = (7 − 4) 2 + (4 − 0) 2 = 32 + 4 2 = 9 + 16 = 25 = 5 We now have AC ≅ DE and AB ≅ DF since congruent segments have equal measures. ∠BAC and ∠FDE have the same measure and are also congruent. This gives us two sides and an included angle of ∆ABC congruent to two sides and an included angle of ∆DFE. Therefore, ∆ABC ≅ ∆DFE by the SAS congruence postulate. 4. We have KA = GT from the congruence statement. This allows us to write and solve the following equation: 3x -10 = x + 2 2x = 12 x= 6 To find KA, substitute the value 6 for x in the equation KA = (3x-10) and get KA = (3•6-10) = 8. Since corresponding angles of congruent triangles are congruent, m∠KAR = m∠GTO. By substituting the algebraic expressions for these measures, we have the following equation: 5y-40 = 3y-8 Solve this equation for y : Exercises 1. Given: AC ≅ BC CD ≅ CE Prove: ∠A ≅ ∠B 2y = 32 y = 16 Complete the following proof. Statements 1. ___________________________ Reasons 1. Given ___________________________ 2. ___________________________ 2. Reflexive property 3. ∆ACE ≅ ∆BCD 3. __________________________ 4. ___________________________ 4. CPCTC 2. Given: ∆ACB is isosceles with base AB ray CD bisects ∠ACB Show ∆ADB is isosceles Write a justification explaining why ∆ADB is isosceles. This justification may be written in the two-column format like the proof in exercise 1 or in a paragraph format. 3. Given: ∆PQR with vertices P(-3,1), Q(1,1), and R(1,6) ∆STU with vertices S(8,2), T(4,2), and U(4,7) a) Graph the triangles in the coordinate plane. b) Determine if ∆PQR and ∆STU are congruent. c) Justify your answer. 4. Given: Right triangle ACB with right ∠ACB Right triangle DEF with right ∠DEF AC ≅ DE and BC ≅ EF Prove: ∆ACB ≅ ∆DEF This proof leads to the following theorem for proving right triangles congruent. Leg-Leg Theorem If two legs of one right triangle are congruent to two legs of a second right triangle, then the right triangles are congruent. 5. Given: Circle O with radii OG and OH OM bisects ∠GOH GM = x2 + 2x + 5 HM = x2 + x + 11 OM = 3x – 6 Find: x, GH, and OM Exploration Refer to the activity Look Alikes in the Geometry Module. Use a TI-83+ graphing calculator with Cabri Junior™ and do problem 3. In this problem, a triangle will be constructed using two angles and the included side of a given triangle. Angle measures and side lengths of the constructed triangle will be compared to the corresponding angle measures and side lengths of the original triangle. This activity refers to two angles and the included side of a triangle. In the diagram below, HI is the included side between ∠H and ∠I. The results of this exploration lead us to the following congruence postulate for triangles. ASA Congruence Postulate Two triangles are congruent if and only if two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle. Examples 1. Which of the following pairs of triangles are congruent by the ASA congruence postulate? Justify your reasoning. 2. Given: CD bisects ∠ACB CD ⊥ AB