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Additional resources for The foundations of mathematics: A study in the philosophy of science

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61 9. 62, m(LCAB) = m(LCBA). In triangle ABD, m(LDAB) = m(LDBA). Prove that m( L CAD) = m( L CBD). 62 0. The lines LAB and LBc are cut by the line Lps at points Q and R. Assume m(Lb) = m(Lc). Prove that m(La) = m(Ld). 63 36 DISTANCE AND ANGLES [1, §5] 11. 64. Furthermore: m(Lx) = m(Ly), BK is the bisector of L ABC, CK is the bisector of LACB. Prove that m(LKBC) = m(LKCB). 64 1, §5. RIGHT ANGLES AND PERPENDICULARITY We define two lines to be perpendicular if they intersect, and if the angle between the lines is a right angle.

Let m(LQOP) = X What is the measure of LROP in terms of x? The angle between the bisectors is the sum oft of LROP plus t of LQOR. Express this sum algebraically and simplify. 0 • Look over your answers to the questions above and write out a complete argument in paragraph form "proving" your conclusion for each of the two exercises, just as a lawyer presents evidence to support his case in court. Use full English sentences and correct mathematical statements. In the next section,. you will find a statement and proof, but don't look them up without first trying it out for yourself.

Example. 54 m(LSTQ) = m(LPTR). Prove that m(LPTQ) = m(LSTR). Proof. We know that m(LSTQ) = m(LPTR). Adding m(LSTP) to both sides of the equation, we have: m(LSTQ) + m(LSTP) = m(LPTR) + m(LSTP). Since m(LSTQ) + m(LSTP) = m(LPTQ) m(LPTR) + m(LSTP) = m(LSTR), and substitute in the previous equation to get: m(LPTQ) = m(LSTR). This proves what we wanted. Example. 55 assume that m(LAOB) Prove that m(LAOC) = = m(LBOD). m(LCOD). [1, §4] 33 PROOFS Proof. Angles LAOC and LCOD together form LAOD. Hence m(LAOC) + m(LCOD) = m(LAOD).

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