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Isosceles Triangles - Problem 3
An isosceles triangle has two congruent sides and two congruent base angles. Using this and the triangle angle sum theorem, it is possible to find the value of x when the values of the angles are given by expressions of x.
By the triangle angle sum theorem, sum of the measures of the angles in a triangle is 180°. So, add up the expressions for angles, keeping in mind that the two base angles have the same measure, giving an equation: 2*(base angle measure) + (vertex angle measure) = 180°. Solve for x using techniques from algebra.
If asked to find the value of a specific angle, remember to plug in the value of x into the expression of the desired angle.
What you’ll probably find is that a lot of times, algebra and geometry tend to overlap. If we look at this problem we see that we’re being asked for the vertex angle. So first we have to decide which one of these three angles is the vertex angle.
Well, if these two sides are congruent, then we have an isosceles triangle which means the vertex angle is 4x. So how are we going to find out what 4x is? In order to do that we have to find out what X is, in order to find out what x, is we have to use a triangle sum. Seems like we’re using that a lot.
We can say that 180 degrees is equal to the sum of these three angles. Again since this is isosceles, we know that this angle right here has to be 2x plus 10 degrees as well. Notice that I write 2x plus 10 inside parenthesis. That’s to tell you that this is a number and we’re talking about a degree so we’re talking about an angle.
So if I combine like terms and since you’re a geometry student, you’ve passed algebra, I’m assuming you can combine like terms right now in your head. 2 plus 2 plus 4 is 8x and then we have 10 plus 10 is twenty. So all I’ve done here is I’ve said that the sum of these three angles has to be 180 degrees. And it looks like we have a fairly simple equation to solve.
So I’m going to subtract 20 from both sides of that equation 180 minus 20 is 160. 160 equals 8x, and we see is we can solve this by dividing by 8, and x equals 20. And it looks like we’re done right?
Well, x equals 20 is great but that doesn’t answer the question find the vertex angle. What we’re going to have to do is we’re going to have to substitute that 20 back in for x. So 4 times 20 is 80, which means the vertex angle here is 80 degrees.