Maggie draws a triangle with a right angle. The other two angles have equal measures. What are the possible values of the exterior angles for Maggie’s triangle? Explain.
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Maggie draws a triangle with a right angle. The other two angles have equal measures. What are the possible values of the exterior angles for Maggie’s triangle? Explain.
In a triangle with a right angle, the two other angles must be equal. Let’s denote the right angle as 90 degrees and the other two equal angles as (x). Since the sum of all angles in a triangle is 180 degrees, we can set up the equation:
[
90 + x + x = 180
]
This simplifies to:
[
90 + 2x = 180
]
Subtracting 90 from both sides gives us:
[
2x = 90
]
Dividing by 2 leads to:
[
x = 45
]
So, in Maggie’s triangle, the two equal angles each measure 45 degrees.
Now, the exterior angle of a triangle at any vertex is equal to the sum of the measures of the opposite interior angles. Therefore, the exterior angles corresponding to the 45-degree angles are:
[
180 – 45 = 135 quad (text{for each 45-degree angle})
]
The exterior angle corresponding to the 90-degree angle is:
[
180 – 90 = 90
]
Thus, the possible values of the exterior angles for Maggie’s triangle are 135 degrees (for each of the two 45-degree angles) and 90 degrees (for the right angle).