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?

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Answer 1 · 2015-11-14T13:03:08

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

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

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

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

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

$2 : Z = 4 X$ $2: Z = 4X$

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

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

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

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

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