congruence-conds-in-proofs

# congruence-conds-in-proofs - 16:14 FAX An Example Proof...

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Unformatted text preview: 09/10/2000 16:14 FAX An Example Proof Using the Triangle Gongruence Conditions Note that. before you can apply the SAS Congruence Condition. You must ﬁrst explicitly establish within the proof the two congmences of corresponding segments and the congruence of the corresponding included angles. Only then are you permitted to conclude that the two triangles are congruent by SAS. - The same ls true for the other congruence conditions, ASA, AAS, 553. and HL: that is. you must explicitly establish the required congruences before applying the principle. Once you have concluded that two triangles are congruent, you will often want to conclude that some corresponding parts (corresponding sides or corresponding angles) are congruent. When you do this. give as a reason "by GPGF“. GPGF stands for "Qorrespondlng Earls of Qongruent figures are congruent." Problem: It is Given that quadrilateral ABGD Is a square (has four right angles and all sides congruent). To Prove: (1lThe diagonals "RE and ﬁ have equal length, and (2) mmeoc) .- 45° Proof: (1) [First show that AADB _ ABBA] [Put comments in brackets] XII; e E5 because AEGD ls asqusre. ZDAB g AGBA because both are right angles. B a BA because asegmentls congruentto Itself. D Therfore(.‘.), AADB ; ABGA by as -. E; E by cPcF. '. ED :1 AG by deﬁnition ofcongruence. (QED for (1)) (2) [Next,show that ABDA E ABDG] E s E because asegmentis congruenttol’lself. A 35 2 EE because ABGD lsasquare. E s E because ABGDIsasqusre. ABDA a oeoc by 555. ‘. ABBA E AEDG by GPCF. -. m(zBDA) = mMBDc) by deﬁnition ofcongruence. memos) a: mason) + mueoc) FMAADC) ll mt£BDC) 4' mMBDG) by EHbﬁﬂtl-ltlﬁl'l- 2m(.£BDC) ll MAADG) 3' 90°. -. m(£BDG) = 45°. QED I001 ...
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