



 
Con, For Q2, don't give up so easily. There is indeed a triangle ABC with D between A and C and E between A and B that satisfies all the conditions. The only tools needed to find that triangle are (1) the sum of the angles of a triangle equals 180 degrees, and (2) the base angles of an isosceles triangle are equal (where an isosceles triangle has two sides equal). If you are not good at algebra, guess either the angle at A or at C, then fill in all the other angles; then adjust your guess up or down as needed. For Q1 there are simple divisibility rules for 3, 4, and 5. It is easy to see that your number is correct, and almost as easy to check whether or not it is the only correct answer Chris Con wrote back
You're close, which is good when playing horseshoes, but not so good when playing math. The only way to see what's wrong is to check the angles: Draw a picture while you follow my computations. If you want the angle at A to be 50 degrees then in triangle AED, E would be 50 and D would be 80. (We're using angle A = angle E because DA = DE; moreover, the sum of all three angles must be 180.) Next, in triangle EBD (with ED = EB) E is 130 while B = D = 25. Next, the angles at D must sum to 180; we already have 80 and 25, so the angle at D in triangle DBC must be 180  105 = 75. Next, we use AB = AC to get the angle at C must be half of 180  50; that is, the angle at C must be 65. Now we're in trouble: We want BD = BC, whence in triangle BCD the angles at C and D should be equal; unfortunately one is 65 while the other is 75. We conclude that the guess that A = 50 was TOO BIG. Why not try A = 45 and go through the calculation once more. Chris  


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