Question

Kevin is asked to draw a triangle with the following specifications:

at least one angle measuring 52°
two angles with equal measure

Which of the following statements about this triangle is true?
A.
More than one triangle exists with the given conditions, and all instances must be scalene triangles.

B.
Exactly one triangle exists with the given conditions, and it must be an isosceles triangle.

C.
More than one triangle exists with the given conditions, and all instances must be isosceles triangles.

D.
No triangle exists with the given conditions.

Answer 1

We are given the following conditions:

a) Atleast one angle must measure 52 degrees
b) Two angles must be of the same measure

Following cases are possible.

Case 1:
One angle is 52 degrees. The rest two angles are same.
The sum of all 3 angles of a triangle must be 180 degrees. If each other angle measures x degrees, we can write:

52 + x + x = 180
⇒ x = 64 degrees
So, the angles of the triangle will be 52, 64, 64

Case 2:
Two angles measure 52 degrees. Let the third angle be x. So we can write:

52 + 52 + x = 180
⇒ x = 76 degrees.
So, the angles of the triangle will be 52, 52, 76

Thus, with the given conditions two triangle can be formed and both will be isosceles triangles as they have two angles with same measure.

Therefore, the answer to this question is option C

Answer 2

Let
`x` be the measure of one angle in our triangle; since we have two equal angles in our triangle, their measure will be
`2x`.
We know from our problem that at least one angle of our triangle measure 52°; since the sum o the interior angles of a triangle is 180°, we can use an equation to relate the quantities and solve for
`x` to find the measure of the tow equal angles:

`2x+52=180`

`2x=128`

`x= (128)/(2)`

`x=64`

Now how know that the measure of the angles of our triangle are 52°, 64°, and 64°. Since we have tow equal angles in our triangle, we can conclude that our triangle is isosceles. Notice that we don't have any given side, so the sides of our isosceles triangle can vary in length.

We can conclude that the correct answer is: C. More than one triangle exists with the given conditions, and all instances must be isosceles triangles.