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Foundations of Math III Unit 4: Logical Reasoning (Part 2) in Geometry 1.12 Practice Making Mini-Proofs Fill in the missing pieces for each mini-proof. Μ Μ Μ Μ M is the midpoint of π΄π΅ A Μ Μ Μ Μ bisects Μ Μ Μ Μ πΈπ· π΄π΅ M C Definition of Segment Bisector Μ Μ Μ Μ Μ β Μ Μ Μ Μ Μ If π΄π ππΆ Definition of Congruent Segments then ________________ Μ Μ Μ Μ β Μ Μ Μ Μ π΄πΆ πΆπ΅ 1 ________ β₯ ________ Definition of Perpendicular Lines β πΉπ»πΌ and β πΌπ»πΊ are supplementary angles β πΉπ»πΌ and β πΌπ»πΊ are supplementary angles β 1 β β 2 β β 3 β β 4 2 β‘ππ is the perpendicular bisector of Μ Μ Μ π½πΎ ππ bisects β πππ Definition of Midpoint πβ _________ = πβ _________ JL = LK 3 Μ Μ Μ Μ is the perpendicular πΆπΊ bisector of Μ Μ Μ Μ π·πΈ Definition of Segment Bisector Μ Μ Μ Μ πΊ is the midpoint of π·πΈ π΅ is the midpoint of Μ Μ Μ Μ Μ ππ΄ _______________ β _____________ _______________ = _____________ 4 1.13 Homework - Triangle Congruence Theorems Determine which theorem can be used to prove that the triangles are congruent. If it is not possible to prove that they are congruent, write not possible. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 5 Flow Proof Examples 1) C is the midpoint of Μ Μ Μ Μ π΄π· οA ο οD ο1 ο ο2 Μ Μ Μ Μ π΄πΆ ο Μ Μ Μ Μ π·πΆ οABC ο ___________ 2) Μ Μ Μ Μ Μ ππ β₯ Μ Μ Μ Μ ππ ο3 ο ο4 Μ Μ Μ Μ ππ ο Μ Μ Μ Μ ππ ο1 ο ο2 οMNO ο ___________ 3) Μ Μ Μ Μ bisects π΅π· οABC Μ Μ Μ Μ π΅π· ο Μ Μ Μ Μ π΅π· Μ Μ Μ Μ π΄π΅ ο Μ Μ Μ Μ πΆπ΅ οABD ο οCBD οABD ο ___________ 6 4) E is the midpoint of Μ Μ Μ Μ π΄π· οA ο οD οB ο οC Μ Μ Μ Μ Μ Μ Μ Μ π΄πΈ ο π·πΈ οABE ο ___________ 5) Μ Μ Μ Μ bisects π΄π Μ Μ Μ Μ Μ ππ Reflexive Property of Congruence Given οMAS ο ___________ 6) Given Definition of Angle Bisector οABD ο ___________ 7 1.14 Homework - Triangle Congruence Theorems Determine which theorem can be used to prove that the triangles are congruent. If it is not possible to prove that they are congruent, write not possible. 1. 2. 3. State what additional pair of angles or sides need to be congruent order to prove that the triangles congruent using the given theorem. 4. ASA 5. SAS 6. SSS 7. Fill in the blanks in the following flow proof. Given Μ Μ Μ Μ Μ Μ Μ π π β Μ ππ οRVS ο __________ 8 Reflexive Property 1.15 Examples β Flow Proofs using CPCTC Example 1 Hutchins Lake is a long, narrow lake. Its length is represented by Μ Μ Μ Μ π΄π΅ in the diagram shown below. Dmitri Μ Μ Μ Μ . designed the following method to determine its length. First, he paced off and measured Μ Μ Μ Μ π΄πΆ and π΅πΆ Then, using a transit, he made πβ ππΆπ΄ = πβ π΄πΆπ΅. He then marked point D on πΆπ so that π·πΆ = π΅πΆ, and Μ Μ Μ Μ . Dmitri claims that π΄π΅ = π΄π·. Is he correct? Justify your answer. he measured π΄π· Example 2 To measure the width of a sinkhole on his property, Harry marked off congruent triangles as shown in Μ Μ Μ Μ . How does he know that the the diagram. He then measures the length of Μ Μ Μ Μ Μ Μ π΄β²π΅β² to find the length of π΄π΅ Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ lengths of π΄β²π΅β² and π΄π΅ are equal? Justify your answer. Example 3 Μ Μ β Μ Μ Μ Μ Given: Ξπ»π½πΎ is an isosceles triangle with Μ Μ π»π½ π»πΎ . An auxiliary line is drawn from vertex H to the Μ Μ Μ to form two triangles. midpoint N of π½πΎ Prove: β π½ β β πΎ 9 1.15 Homework β Flow Proofs using CPCTC 1) According to legend, on of Napoleonβs officers used congruent triangles to estimate the width of a river. On the riverbank, the officer stood up straight and lowered the visor of his cap until the farthest thing he could see was the edge of the opposite bank (Point F). He then turned and noted the spot on his side of the river that was in line with his eye and the tip of his visor (Point G). The officer then paced off the distance along the riverbank from where he was standing to the spot he sighted (EG). He declared that distance to be the same as the width of the river. Prove that he was correct. Given: β π·πΈπΊ and β πΉπΈπΊ are right angles (We assume that the officer was standing perpendicular to the ground) β πΈπ·πΊ β β πΈπ·πΉ (because he sighted both points F and G from the same angle) Prove: Μ Μ Μ Μ πΈπΉ β Μ Μ Μ Μ πΈπΊ Fill in the empty spots in the flow proof below. β π·πΈπΊ and β πΉπΈπΊ are right angles Given Μ Μ Μ Μ π·πΈ ο Μ Μ Μ Μ π·πΈ ο________ ο ο________ οFDE ο ___________ Μ Μ Μ Μ πΈπΉ β Μ Μ Μ Μ πΈπΊ 10 Fill in the missing reasons or statements in each proof. 1. 2. 11 1.16 Warm Up 12 1.16 Review for Quiz For each example, determine which triangle congruence postulate is needed to prove the two triangles are congruent. Then write the congruence statement. Complete the following flow proof. 13 1.17 Warm Up β Missing Parts 14