Question

can you please please help me with these proofs

R Cand CoDNder the triangle AABC and M and NIwopaints such thatM PrOve that MEMAN kBA) M MBC)m ACRI.

Answer 1

h Pros-Claim -m(LMAN) m(BAC) + mcABC) + AASYw m(CACB)) DASIre Cansides ABC bizes ba Mand N are tuo Doints such that M-B-C Cd B-C-N 21 AMA M Here point M is BC as sudh as aond paint N is Loxtending line the On M-B-C He endieg line as sudh as Bc-N Then jaining Line seament MAM BC the paiuts m and A lay nd ja teget A md N by pe Paints Lile dawing e bnow that, BAC ABc and CACB art amgles af AABC ue three MAM tnaw that Surn of all angles ta triangle is 180

1 Hencl mBAC) m(ZABOm(CACB) 180 And consideing<MAN trom fgy we can cude at ZMAN Henca, MAN shauld nof loe at measut 1800 straight ge not mAN) 130 AAMB Tn cansidenioe engle ABM is on So MARB is an acuk gle AlsD considesingtuse angle ACN an acut argle CAN acute angle Sos CAN 1 MAN MAR+(BAO MAB and CAN ane mMAN S m(MAN 180 fram en & m(MAN)1 tes than 1go.

mmAN m(<BAC) mKAB) + m(LACB) m(ZMAN)< (mCBAd+ m/ABd+ mCACe) Hence proved Pros2Gven that het DABC be side AB side A mle nith et D be a Hiat A-C-D Atso m be Paint Hhat Paiut an line AC Such line BC sudh B-mrc A B Clairm:-m(LABM)m(LBMA>(mCmD (mpc)

Canstructian:-Praw perpendiculas lie fram pout B olint Ac Such that mBEA)- m(<BEC) = 96 eANL Consider ABEC As by cansthuctian glet angled D BEC iS a mBEC) + m(CECB) t mERC) c1800 As we enas mCBEC)e 3 m(ECB) tm(Z £BC) =g6 oN2 DA Arsunuing m(<ECB) KEBC) m(ECB)-4 we have B-M^C As m (९Ecnm)= 4o. kle bnas that LECM and mCD Sugplimentay opre ECM) (mc)180 mmcD) 1e-4

ge Na ate MCMCD) 1350 CansideAMCD tAS mmc) As dahue angled AMCD is In AMCD, AAD MMCD)mCmb)(Dm) 1 130 +MLCMD) + m ( CDM)1 MA MCMD) + m( CDm) 18d -135 450 mCCmD) (CDm) + YY As by conshuctian m(<BEA)40 In 4AEB By Arngu propesty mAEB) mCARE) mBAS) 1 Sum mABE) + m (BAE= e 90t m(ABE) m (ZRAE) = 9o° m BAE) AsaupasigE mCcRA)9 mCCBAF) =45°

CBAM But we need ay m BAM m(BAE 40 TmRAM) at Tn ARM; AMB)1 ARm) + mBAM) m m 521 18D 22.S ABM) m m (LARM) tmAm 157. YY ram en AS73 4 m{<ABm)+ m(BmA))mCm) Hence Aroved