The sum of the measures of the angles of a triangle is 180. The sum of the measures of the second and third angles is three times the measure of the first angle. The third angle is 29 more than the second. Let x, y, and z represent the measures of the first, second, and third angles, respectively. Find the measures of the three angles.

x = 45°, y = 53°, z = 82°

Step-by-step explanation:

x is the first angle, y is the second, and z is the third.

The sum of the second and third, which is denoted by y + z, is 3 times the measure of the first, which is just x. So, we have:

y + z = 3 * x

Additionally, the third angle, z, is 29 more than the second, y, so:

z = 29 + y

We also know that the sum of the three is 180, so:

x + y + z = 180

Let's substitute y + z in the last equation with 3 * x:

x + y + z = 180

x + 3x = 180

4x = 180

x = 45

Now, we know that y + z = 3 * 45 = 135. We also know that z = 29 + y, so substitute 29 + y in for z in y + z = 135:

y + z = 135

y + (29 + y) = 135

2y + 29 = 135

2y = 106

y = 53

Finally, use this value of y to solve for z:

z = 29 + y

z = 29 + 53 = 82

Thus, the angles are x = 45°, y = 53°, and z = 82°.

~ an aesthetics lover

we have three similar triangles, because each has a right angle and shares an angle.   let's write the angles in order: opposite to short leg, long leg, hypotenuse.

cab similar to bad similar to cbd

or as ratios,

ca: ab: cb = ba: ad: bd = cb: bd: cd

we also know

ac = ad + cd

(ad+cd): ab: cb = ba: ad: bd = cb: bd: cd

(ad+cd)/cb=cb/cd

we have cb=6, ad=5 and seek x=cd.

we reject the negative root and conclude x=4

answer: 4