Week 3

Example 1. One of the angles of an isosceles triangle is 100°. Is this a base angle or a vertical angle? Calculate all the angles of the triangle.

Example 2. Resolve 99899 into factors.

Of course, you can find the solution just below, but it is highly recommended that, you first try to solve it on your own.

Just remember the words of Paul Halmos, who says “the only way to learn mathematics is to do mathematics”.

 Solution 1. Recall that, in an isosceles triangle any two angles are of equal measure. These equal angles are known as the base angles. The other angle is called the vertical angle.

Now consider the given question. Since one angle is 100°, either this is a base angle or a vertical angle. Therefore, we need to consider the following two cases:

Case 1. Let this is a base angle. By definition, there must be another angle of measure 100° in the triangle and a vertical angle of some measure.

But the sum of base angles = 100° + 100° = 200°, a contradiction to the fact that the sum of all the interior angles of a triangle if 180°.

Otherwise, if we call the vertical angle as x, then by the angle sum property x + 100° + 100° = 180° giving x = −20°, an obvious impossibility.

Therefore, it cannot be a base angle.

Case 2. Let this be a vertical angle. Recall that, the sum of all the interior angles of a triangle if 180°.

Therefore, sum of the two base angles = 180° − 100° = 80°.

Since each of the base angles are equal, measure of each of them = 80° ÷ 2 = 40°.

Otherwise, if we call the base angles as x, then by the angle sum property

x + x + 100° = 180° giving x = 40°, as above.

So, an isosceles triangle with the given angle measures is possible.

Therefore, it can be a vertical angle.

Solution 2. Resolving 99899 into factors is same as finding prime numbers a, b, . . . such that 99899 = a^mbⁿ…, where 1 < a, b, . . . < 99899 and m, n, . . .  are whole numbers. Recall that 2, 3, 5, 7, 11, . . . are first few primes and let us divide the number by the primes to get some prime factors.

By trial and error, we find that the number 99899 is not divisible by the primes, less than 100. But how long we can go on dividing? This method of dividing by primes will not work (it becomes laborious) if the number is divisible by a big prime.

If we take the square root of 99899 using a calculator, it will show us as below:

Yes, you got it correctly.  We will use the Sieve method to find prime factors of this number (as it is confirmed from the question). We can see that the number is divisible by 283 and our search ends. Then the other prime factor is 353.

Hence, we have 99899 = 283 × 353.

Of course, this is not a practical method to solve. Below we will present an algebraic solution to this problem.