 So here we have another isosceles triangle you can see the two legs are shown in red We have the base shown in blue The base angles are shown the vertex is shown and we have a theorem about isosceles triangles And that theorem says if we have two sides of a triangle congruent In other words if we have an isosceles triangle Then that isosceles triangle has congruent base angles So that means that these Angles will always be congruent to each other So in other words if you have an isosceles triangle, then the base angles are always going to be congruent Pull that up for you. So we can use that to solve some problems. Let's work on a quick actually Before we work on that example one thing about this theorem is it's it's converse is also true Remember the converse of an if-then statement takes the hypothesis and changes it around with the conclusion So the original theorem was if two sides are congruent then the base angles are congruent the converse would be If the base angles are congruent Then the sides are congruent in other words if the base angles are congruent then we know we have an isosceles triangle So the bioconditional says two sides are congruent if and only if The two the two angles are congruent. So let's use that to solve some problems here Here I have an isosceles triangle. The reason I know it's isosceles is there are these congruence marks on the legs So remember isosceles triangles if the side the legs are congruent then the base angles are congruent So here's the base of The isosceles triangle so that means these base These are the base angles and Those are congruent in other words x and y are equal Now using the triangle sum theorem all three of these angles need to add up to a hundred and eighty Which means x and y together? Well 180 minus 120 is 60 degrees and So angles x and y have to share 60 degrees equally So x and y are the same if we divide 60 by 2 then we get 30 degrees So each angle x and y Is equal to 30 degrees Let's take a look at another example Here I've got well, I know I have a right triangle, but I don't have any congruence markings to indicate that it's isosceles however Since both of these angles say x degrees that means those two angles must be congruent and just like we saw in the previous slide If the angles are congruent then the sides opposite them are congruent in other words If the base angles are congruent, then I know I have an isosceles triangle. So that means These two legs are congruent Well those congruence marks show me that y must also equal 3 Now how are we gonna find the red angles? Well, we'll do it in kind of a similar way that we did in the previous problem Since both of those angles must be congruent and This angle is 90 degrees The rest of the two angles must share 180 minus 90 and so both of those angles And that angle are 45 degrees Again because of the triangle sum theorem and also the theorem about angles in isosceles triangles We know we can just subtract from 180 and divide by 2