Video: GCSE Mathematics Foundation Tier Pack 5 β€’ Paper 2 β€’ Question 7

GCSE Mathematics Foundation Tier Pack 5 β€’ Paper 2 β€’ Question 7

02:20

Video Transcript

𝐴𝐡𝐢 is an isosceles triangle. Line 𝐴𝐡𝐷 is a straight line. Determine whether triangle 𝐴𝐡𝐢 is a right-angled triangle or not. You must give a reason for each step of your working.

Now because 𝐴𝐡𝐢 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 𝐴𝐡𝐷 is a straight line.

So how we’re gonna use this information to determine whether triangle 𝐴𝐡𝐢 is a right-angled triangle? Well, let’s start with the fact that line 𝐴𝐡𝐷 is a straight line. Well, we know that angles on a straight line sum to 180 degrees. And this means that angle 𝐴𝐡𝐢 plus angle 𝐢𝐡𝐷 must equal 180 degrees. And we know that angle 𝐢𝐡𝐷 is 133 degrees. So angle 𝐴𝐡𝐢 plus 133 degrees is 180 degrees. And subtracting 133 degrees from each side of that equation tells us that angle 𝐴𝐡𝐢 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 𝐴𝐢𝐡 is equal to angle 𝐴𝐡𝐢. And we know that angle 𝐴𝐡𝐢 is 47 degrees. So that tells us that angle 𝐴𝐢𝐡 is also 47 degrees. And lastly, we know that angles in a triangle add up to 180 degrees. So angle 𝐡𝐴𝐢 plus angle 𝐴𝐡𝐢 plus angle 𝐴𝐢𝐡 is 180 degrees.

And we just worked out that angles 𝐴𝐡𝐢 and 𝐴𝐢𝐡 are both 47 degrees. So if we take 47 degrees away from both sides of the equation twice, we’ve got angle 𝐡𝐴𝐢 is equal to 180 degrees minus 47 degrees minus another 47 degrees. Now, that means that angle 𝐡𝐴𝐢 is 86 degrees.

So the three angles in triangle 𝐴𝐡𝐢 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.

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