In Triangle XYZ, the measure of angle X is 5/8 as large as the measure of angle Y. If the measure of angle Z is 1/4 as large as the measure of the angle Y, what's the measure of all three angles.

I've tried making up numbers to fit in the problem, but nothing is working. My teacher gave a hint that there is an equations that would figure this out.

You should of course know that the sum of the 3 angles is 180°

how about angle Y = x
then angle X = (5/8)x
and angle Z = (1/4)x

so 5x/8 + x/4 + x = 180

I suggest you multiply each term by 8, the LCD
and it is a piece of cake after that.

Well, it seems like Triangle XYZ is having a little party with angles X, Y, and Z. Let's call the measure of angle Y "y" because we don't want to hurt its feelings by giving it an identity crisis.

According to the problem, the measure of angle X is 5/8 as large as the measure of angle Y. So, we can say that X = (5/8)y.

Similarly, the measure of angle Z is 1/4 as large as the measure of angle Y. So, Z = (1/4)y.

Now, we know that the sum of all angles in a triangle is always 180 degrees. So, we can write an equation:

X + Y + Z = 180.

Substituting the values we found earlier, we get:
(5/8)y + y + (1/4)y = 180.

To avoid any fractions, let's multiply everything by 8:
5y + 8y + 2y = 1440.

Now, we can combine like terms:
15y = 1440.

Divide both sides by 15:
y = 96.

So, the measure of angle Y is 96 degrees.

Now, we can plug this value back into the equations for X and Z:
X = (5/8)(96) = 60 degrees,
Z = (1/4)(96) = 24 degrees.

Therefore, the measure of angle X is 60 degrees, the measure of angle Y is 96 degrees, and the measure of angle Z is 24 degrees. It looks like Triangle XYZ has quite the angle party going on!

To solve this problem, let's assign a variable to the measure of angle Y.

Let Y = the measure of angle Y.

According to the problem:
Angle X = 5/8 * Angle Y, and
Angle Z = 1/4 * Angle Y.

We know that the sum of all angles in a triangle is 180 degrees.
Therefore, we can write the equation:
Angle X + Angle Y + Angle Z = 180.

Substituting the given expressions for Angle X and Angle Z, we have:
(5/8 * Angle Y) + Angle Y + (1/4 * Angle Y) = 180

To simplify the equation, we can find a common denominator for the fractions:

[(5/8) + 1 + (1/4)] * Angle Y = 180
[(5/8) + (8/8) + (2/8)] * Angle Y = 180
(15/8) * Angle Y = 180

To isolate Angle Y, we can multiply both sides of the equation by the reciprocal of (15/8).

[(8/15) * (15/8)] * Angle Y = 180 * (8/15)
Angle Y = 144

Now, we can substitute the value of Angle Y into the expressions for Angle X and Angle Z:
Angle X = (5/8) * 144 = 90
Angle Z = (1/4) * 144 = 36

Therefore, the measure of Angle X is 90 degrees, the measure of Angle Y is 144 degrees, and the measure of Angle Z is 36 degrees.

To solve this problem, we can use algebraic equations. Let's denote the measures of angles X, Y, and Z with x, y, and z respectively.

According to the problem, we know the following relationships:
1. The measure of angle X is 5/8 as large as the measure of angle Y.
This can be written as x = (5/8)y.

2. The measure of angle Z is 1/4 as large as the measure of angle Y.
This can be written as z = (1/4)y.

We also know that the measures of angles in a triangle add up to 180 degrees. So, we can write the equation:

x + y + z = 180.

Now, we can substitute the values of x and z from the first two equations into the third equation:

(5/8)y + y + (1/4)y = 180.

To simplify this equation, let's get rid of the fractions. We can do this by multiplying all terms by the least common multiple (LCM) of the denominators, which is 8 in this case. Then we get:

(5/8)y * 8 + y * 8 + (1/4)y * 8 = 180 * 8.

Simplifying further:

5y + 8y + 2y = 1440.

Combining like terms:

15y = 1440.

Now we can solve for y by dividing both sides of the equation by 15:

y = 1440 / 15.

Calculating this, we find that y = 96.

Using this value, we can substitute it back into the first and second equations to find the measures of angles x and z:

x = (5/8)y = (5/8) * 96 = 60.

z = (1/4)y = (1/4) * 96 = 24.

Therefore, the measures of angles X, Y, and Z are 60, 96, and 24 degrees respectively.