# Triangle XYZ is isosceles. The base angles, angle X and angle Y, are four times the measure of the vertex angle, angle Z. What is the measure of angle X?

Nov 14, 2015

Set up two equations with two unknowns
You will find X and Y = 30 degrees, Z = 120 degrees

#### Explanation:

You know that $X = Y$, that means that you can substitute $Y$ by $X$ or vice versa.

You can work out two equations:
Since there are 180 degrees in a triangle, that means:
$1 : X + Y + Z = 180$

Substitute $Y$ by $X$:
$1 : X + X + Z = 180$
$1 : 2 X + Z = 180$

We can also make another equation based on that angle $Z$ is 4 times bigger than angle $X$:

$2 : Z = 4 X$

Now, let's put equation 2 into equation 1 by substituting $Z$ by $4 x$:

$2 X + 4 X = 180$
$6 X = 180$
$X = 30$

Insert this value of X into either the first or the second equation (let's do number 2):

$Z = 4 X$
$Z = 4 \cdot 30$
$Z = 120$

$X = Y \to X = 30$ and $Y = 30$