Geometry Triangles 3

Prove: ∆ACB is isosceles using a flow-chart proof. 3. Given: ∠ADB ≅ ∠CBD ∠ABD ≅ ∠CDB m ∠A = 3x + 15 m ∠C = 8x – 20 Find: x and m ∠A Solutions: 1. a) ∆CAM ≅ ∆DBM by the ASA congruence postulate. A ∠A ≅ ∠B (Right angles are congruent.) S AM ≅ MB (A midpoint divides a segment into two congruent segments.) A ∠AMC ≅ ∠BMD (Given) b) ∆WOX ≅ ∆ZOY by the SAS congruence postulate. The ASA congruence postulate does not apply to this problem. S XO ≅ YO (A bisector divides a segment into two congruent segments.) A ∠XOW ≅ ∠YOZ S WO ≅ OZ 2. (Vertical angles are congruent.) (A bisector divides a segment into two congruent segments.) A flow-chart proof shows the logical development of a proof using statements with supporting justification(s) in a flow-chart format. The statements and reasons are shown in rectangles with arrows indicating how they connect to other information in rectangles. The flow of information should lead to the conclusion. The proof that follows models one format that can be used. CD bisects ∠ACB Given CD ≅ CD Reflexive CD ⊥ AB Given ∠ACD≅∠BCD Definition of an angle bisector ∆ADC≅∆BDC ASA postulate ∠ADC and∠BDC are right angles ⊥s form rt. ∠s AC ≅ BC CPCTC ∆ACB is isosceles Definition of an isosceles ∆ ∠ADC≅∠BDC All right ∠s≅ 3. ∆ADB ≅ ∆CBD by the ASA congruence postulate. A ∠ADB ≅ ∠CBD (Given) S DB ≅ DB (Reflexive property) A ∠ABD ≅ ∠CDB (Given) ∠A ≅ ∠C (CPCTC) m ∠A = m ∠C (Congruent angles have equal measures.) 3x+15 = 8x-20 (Substitution) 35 = 5x (Addition and subtraction properties of equality) 7=x (Division property of equality) m ∠A = 3x + 15 = 3•7 + 15 = 21 + 15 = 36 Exercises 1. Given: AD bisects BC AB ⊥ BC DC ⊥ BC Prove: BC bisects AD Proof: Complete the proof below. Statements Reasons 1. AD bisects BC AB ⊥ BC DC ⊥ BC 1. __________________________ 2. ∠B and ∠C are right angles 2. __________________________ 3. _________________________ 3. All right angles are congruent. 4. BE ≅ BC 4. __________________________ 5. ∠AEB ≅ ∠DEC 5. __________________________ 6. _________________________ 6. ASA congruence postulate 7. _________________________ 7. CPCTC 8. BC bisects AD 8. __________________________ 2. Given: ∆YVW is isosceles with base VW ∠UYV ≅ ∠XYW ∠UVY ≅ ∠XWY Prove ∠U ≅ ∠X using a flow-chart proof. 3. Given: ∠BAC ≅ ∠DCA AE ≅ EC DC = (x2 + 2x – 1) cm AB = (x2 + 5x – 7) cm Find: x and AB 4. Given: ∠M ≅ ∠R MI ≅ RI m ∠RPI = (9x – 10) m ∠MQI = (4x + 45) Find: x and m ∠MQI We have explored the SSS, SAS, and ASA congruence postulates and LL theorem (right triangles only). Are there other ways to show two triangles congruent? We will consider the possibility of using any two angles and a side (not included) of one triangle (AAS) congruent to the corresponding parts of another triangle in the following proof. Given: ∆HIJ and ∆KLM ∠H ≅ ∠K, ∠J ≅ ∠M HI ≅ KL Prove: ∆HIJ ≅ ∆KLM Proof: ∆HIJ and ∆KLM Given ∠H ≅ ∠K Given ∠J ≅ ∠M Given ∆HIJ ≅ ∆KLM ∠I ≅ ∠L If 2 ∠s of one ∆ ≅ 2 ∠s of a second ∆, 3rd ∠s are ≅. ASA congruence postulate HI ≅ KL Given This proof leads to the following theorem for showing two triangles congruent. AAS Congruence Theorem If two angles and a side (not included) are congruent to the corresponding parts of a second triangle, then the two triangles are congruent. Is it possible for two triangles to be congruent using SSA where the angle is not included between the two sides? Make a conjecture: ____________________________________ _______________________________________________________________________. This conjecture will be tested in the following exploration. Exploration Given: Two segments and a non-included angle Patty paper, a compass, and straight edge will be needed for this exploration. Follow the procedure outlined below. a) Trace the longer segment given ( AB ) on a sheet of patty paper. b) Place the patty paper over the given angle and align the vertex of the angle with endpoint A of the longer segment. Align one of the sides of the angle with AB . c) Trace the angle and extend the side not aligned with the segment as shown below. d) Construct circle B with a radius equal to the length of the shorter segment given. ( The construction below is not drawn to scale.) Circle B intersects ray AD in two points, point C and point D. There are two possible triangles that can be constructed with the given ∠CAB, AB , and the shorter segment represented by the radii BC and BD . ∆ABC and ∆ABD are the two triangles shown in the figure above. Therefore, it is not possible to construct a unique triangle given two sides and a non-included angle. Examples 1. Given: ∆CED is isosceles with base CD GB ⊥ AD CF ⊥ GE CH = ED CE = AG ∠A ≅ ∠FEC ∠GBF ≅ ∠CFB Identify all pairs of congruent triangles in the figure given and explain the congruence postulate or theorem used. 2. Given: ∠1 ≅ ∠2 DB ⊥ AC Prove: DB bisects ∠ADC Write a paragraph, two-column, or flow-chart proof. 3. Given: ∆DBE is isosceles with base DE ∠A ≅ ∠C ∠ABE ≅ ∠CBD 1 AB = ( x + 1) cm 2 3 BC = ( x – 5) cm 4 Find: x, AB, and BC Solutions: 1. ∆ABG ≅ ∆EFC by the AAS congruence theorem A ∠A ≅ ∠FEC (Given) A ∠ABG ≅ ∠EFC (⊥s form rt. ∠s and all rt. ∠s ≅) S AG ≅ CE (CE=ED=AG; ≅ seg. have = meas.) ∆CDE ≅ ∆CDH by the SSS congruence postulate S CH ≅ DE (Given; ≅ seg. have = meas.) S CE ≅ DH (CE=ED=CH=DH; ≅ seg. have = meas.) S CD ≅ CD (Reflexive property) ∆GBF ≅ ∆CFB by the SAS congruence postulate S GB ≅ CF (∆ABG ≅ ∆EFC; CPCTC) A S ∠GBF ≅ ∠CFB BF ≅ BF (Given) (Reflexive property) 2. Triangle ABD will be shown congruent to triangle CBD using the AAS congruence theorem in a paragraph-style proof as follows. Angle 1 is congruent to ∠DAB and ∠2 is congruent to ∠DCB because vertical angles are congruent. Angle DBA and ∠DBC are right angles since DB ⊥ AC . It follows that ∠DBA ≅ ∠DBC because all right angles are congruent. Side DB is congruent to itself by the reflexive property. We now have ∆ADB ≅ ∆CDB by the AAS congruence theorem. Angle ADB is congruent to ∠CDB by CPCTC. Therefore, DB bisects ∠ADC. 3. ∆ABE ≅ ∆CBD by AAS A ∠A ≅ ∠C A ∠ABE ≅ ∠CBD S BE ≅ BD AB ≅ BC AB = BC 1 3 x+1= x–5 2 4 (Given) (Given) (An isosceles triangle has 2 congruent sides.) (CPCTC) (Congruent segments have equal measures.) (Substitution) 1 x 4 24 = x 6= AB = 1 •24 + 1 = 13 cm 2 CD = AB = 13 cm Exercises 1. Given: ∠BAE ≅ ∠DBA DB ≅ AE a) Identify all pairs of congruent triangles in the given figure. b) Explain the congruence postulate or theorem used. 2. Given: ∠RVS ≅ ∠USV ∠RSV ≅ ∠UVS ∠WRV ≅ ∠TUS ∠W ≅ ∠T 2 VW = ( x + 3) in 3 5 ST = ( x – 1) in 6 1 RS = ( x + 5) in 2 3. Given: Isosceles ∆ABC with base AB CD bisects the vertex angle Prove: CD is a median to the base. Write a paragraph, two-column, or flow-chart proof. This proof leads to the following theorem for isosceles triangles. Isosceles Triangle Theorem The bisector of the vertex angle of an isosceles triangle is the same segment as the median to the base. 4. Given: ∆FRD with vertices F(-4,1), R(-1,3), and D(-3,5) ∆TSU with vertices T(1,1), S(4,3), and U(2,5) ∠F ≅ ∠T and ∠D≅ ∠U a) Graph each triangle in the coordinate plane. b) Show ∆FRD ≅ ∆TSU.