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Geometry 6.1 Inequalities Property Of Inequality (P.O.I.) If a > b and c ≥ d, then a + c > b + d If big > small and big/equal ≥ small/equal, then big + big/equal > small + small/equal If x > 7 and 2 ≥ 1, then… x+2>7+1 If AC > BC B and CE ≥ CD, then AE > BD by POI E C A D Property Of Inequality (P.O.I.) If a > b and b > c, then… a>c If big > medium and medium > small, then… big > small If x > y and y > 7, then… x>7 Property Of Inequality (P.O.I.) If a > b and c > 0, then ac > bc and a > b c c If big > small and a positive > 0, then big × positive > small × positive and big small pos.> pos. If x > 7 and 2 > 0, then… 2x > 14 and… x/2 > 7/2 Property Of Inequality (P.O.I.) 7 D 7 If a = b + c and c is positive, then a>b If big = number + small positive, then… big > number. If 10 = 6 + 4, then… 10 > 6 and 10 > 4 m 1= m D+m 7 E Why? Ext angle in a triangle = to sum of 2 remote int angles. Therefore, 7 7 7 7 E 1 F m 1 > m D and m 1 > m E by POI Exterior Angle Inequality Theorem The measure of the exterior angle of a triangle is greater than the measure of either remote interior angle. C 1 B 7 7 7 m 1 > m A and m 1 > m C by Ext. Ineq. Thm 7 A 7 Using the properties of inequality, classify each conditional as true or false. If the variables are confusing for you, try to replace them with numbers. 1. If 8 = 3x + 2, then 8 > 3x. 2. If x + y = z and a > y, then x + a > z. 3. If a < b and c < b, then a < c. 4. If p > q and r > q, then p > r. 5. If 4y < 16, then y < 4. 6. If d > e and f > g, then d + e > f + g. 7. If 5a > 25, then a > 125. 8. If 3x > 48 and 4 > 2, then 3x + 4 > 50. 9. If x > 10, then x + 3 > 13. 10. If c<d and –9 = -9, then c – 9 > d – 9. 11. If (1/2)y > 7, then y > 14. 12. If d > e and f > g, then d + f > e + g. 13. If p > 10, then p > 12. 14. If p > 12, then p > 10. Some information about the diagram is given. Tell whether the other statements can be deduced from what is given. W Given: XW = YW; Z is the midpoint of XW Z 1 X Y a. m< X = m< XYW b. m<1 > m<W c. m<W > m<X d. XW > ZW e. XY < XW f. 2XZ = XW And finally what we have all been waiting for…proofs. Yeah! 4. Given: AB > CD Prove: AC > BD A 5. B C D Given: mPQR mSRQ m2 m3 m1 m4 Prove: S P 4 1 Q 2 3 R HW P. 205-207 CE(1-20) WE(1-9, 11) A Few Together from the HW P. 205 #5, #10 P. 206 #4 P. 207 #9