𝐴𝐵𝐶 is an isosceles triangle. Line 𝐴𝐵𝐷 is a straight line. Determine whether triangle 𝐴𝐵𝐶 is a right-angled triangle or not. You must give a reason for each step of your working.
Now because 𝐴𝐵𝐶 is an isosceles triangle, we know that two of the sides are equal in length. And we sometimes use the word congruent to describe that and their opposite angles sometimes known as the base angles of the isosceles triangle are also equal. And we’re also told that line 𝐴𝐵𝐷 is a straight line.
So how we’re gonna use this information to determine whether triangle 𝐴𝐵𝐶 is a right-angled triangle? Well, let’s start with the fact that line 𝐴𝐵𝐷 is a straight line. Well, we know that angles on a straight line sum to 180 degrees. And this means that angle 𝐴𝐵𝐶 plus angle 𝐶𝐵𝐷 must equal 180 degrees. And we know that angle 𝐶𝐵𝐷 is 133 degrees. So angle 𝐴𝐵𝐶 plus 133 degrees is 180 degrees. And subtracting 133 degrees from each side of that equation tells us that angle 𝐴𝐵𝐶 is 47 degrees.
So let’s mark that on our triangle. Now, we know that base angles in an isosceles triangle are equal. So that means that angle 𝐴𝐶𝐵 is equal to angle 𝐴𝐵𝐶. And we know that angle 𝐴𝐵𝐶 is 47 degrees. So that tells us that angle 𝐴𝐶𝐵 is also 47 degrees. And lastly, we know that angles in a triangle add up to 180 degrees. So angle 𝐵𝐴𝐶 plus angle 𝐴𝐵𝐶 plus angle 𝐴𝐶𝐵 is 180 degrees.
And we just worked out that angles 𝐴𝐵𝐶 and 𝐴𝐶𝐵 are both 47 degrees. So if we take 47 degrees away from both sides of the equation twice, we’ve got angle 𝐵𝐴𝐶 is equal to 180 degrees minus 47 degrees minus another 47 degrees. Now, that means that angle 𝐵𝐴𝐶 is 86 degrees.
So the three angles in triangle 𝐴𝐵𝐶 are 86 degrees, 47 degrees, and 47 degrees. And since none of the angles in the triangle are 90 degrees, it’s not a right-angled triangle.