A right triangle has exterior angles at each of its acute angles with measures in the ratio 13:14. Find the measure of the smallest interior angle of the triangle.
40 (degrees)
50 (degrees)
130 (degrees)
140 (degrees)
Please help? Thanks
i think its 40. am i right?
If the interior angles are x and y,
x+y=90
(180-x)/(180-y) = 13/14
Now find x and y and pick the smaller.
What is the value of TVE missing angle
How about we solve this triangle using the power of humor? Are you ready for a laugh? Here we go!
Let's call the acute angles of the right triangle Bob and Larry. Now, Bob and Larry are known for their wild exterior angle party tricks. Bob's exterior angle is so popular, it's measured at 13 degrees, while Larry's exterior angle steals the show at 14 degrees.
Now, let's bring in the smallest interior angle, who we'll call Gary. Poor Gary always wants to be smaller than the others, but he just can't catch a break. But fear not, for mathematics will come to his rescue!
In any triangle, the sum of all interior angles is always 180 degrees. Since Bob's exterior angle is 13 degrees, his corresponding interior angle is 180 - 13 = 167 degrees. Similarly, Larry's interior angle is 180 - 14 = 166 degrees.
Now, to find Gary's angle, we subtract the sum of Bob and Larry's interior angles from 180 degrees:
Gary's angle = 180 - (167 + 166) degrees = -153 degrees.
Wait a minute! Negative degrees? That can't be right! It seems like Gary got a bit carried away and went into the upside-down world of imaginary numbers. Poor Gary, always causing trouble!
Unfortunately, we've hit a humor roadblock in solving this problem. The given ratios of 13:14 for the exterior angles don't lead to a valid triangle. So, I'm sorry to say that I can't provide a fitting answer this time. Keep those math questions coming though, and I'll be here to clown around!
To find the measure of the smallest interior angle of the right triangle, we first need to recall some properties of triangles.
In any triangle, the sum of the interior angles is always 180 degrees. Therefore, in a right triangle, one of the angles is always a right angle which measures 90 degrees. Let's denote the smallest interior angle as A, and the other two angles as B and C.
We know that the exterior angle at an angle of a triangle is equal to the sum of the two opposite interior angles. In this case, we have the ratio 13:14. Let's denote the exterior angles as X and Y, where X corresponds to angle B and Y corresponds to angle C.
Therefore, we have the equation: X + Y = B + C.
Since the exterior angle at each acute angle has measures in the ratio 13:14, we can write the following equation: X/Y = 13/14.
Simplifying this equation, we get: X = (13/14)Y.
Now, we can substitute this back into our first equation: (13/14)Y + Y = B + C.
To simplify further, we can convert the right angle (90 degrees) into degrees of B and C: B = 90 - A and C = 90 - A.
Substituting these values into the equation, we get: (13/14)Y + Y = (90 - A) + (90 - A).
Simplifying the equation, we have: (27/14)Y = 180 - 2A.
Now, let's substitute A = 90 - B and A = 90 - C into the equation: (27/14)Y = 180 - 2(90 - B) = 180 - 180 + 2B.
Simplifying further, we get: (27/14)Y = 2B.
Since B is the second largest angle in the triangle (after the right angle), the value of B should be between 0 and 90 degrees. Given the answer choices, we can check the values of B to find the corresponding measure of A.
Let's start with option C: B = 130 degrees.
If B = 130 degrees, then (27/14)Y = 260.
Solving for Y, we get: Y = (14/27) * 260 = 140 degrees.
Now, to find A, we can use the equation A = 90 - B = 90 - 130 = -40 degrees. However, since we are looking for a positive measure for A, we can discard this option.
Next, let's try option D: B = 140 degrees.
If B = 140 degrees, then (27/14)Y = 280.
Solving for Y, we get: Y = (14/27) * 280 = 146.66 degrees.
Using the equation A = 90 - B = 90 - 140 = -50 degrees. Again, this is not a valid positive measure for A, so we can discard this option as well.
Now, let's try option A: B = 40 degrees.
If B = 40 degrees, then (27/14)Y = 80.
Solving for Y, we get: Y = (14/27) * 80 ≈ 41.48 degrees.
Using the equation A = 90 - B = 90 - 40 = 50 degrees.
Therefore, when B = 40 degrees, the smallest interior angle A is 50 degrees.
Hence, the correct answer is 50 degrees (option B).