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?

Question

Mason Adams · Accepted Answer

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

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: 2X + Z = 180# We can also make another equation based on that angle #Z# is 4 times bigger than angle #X#: #2: Z = 4X# Now, let's put equation 2 into equation 1 by substituting #Z# by #4x#: #2X + 4X = 180# #6X = 180# # X = 30# Insert this value of X into either the first or the second equation (let's do number 2): #Z = 4X# #Z = 4 * 30# #Z = 120# #X = Y to X=30# and #Y = 30#

James Atkins · Answer

undefined Let the measure of angle Z be $x$.

Since angles X and Y are each four times the measure of angle Z in an isosceles triangle:

Angle X = Angle Y = $4x$.

In a triangle, the sum of all angles is $180^\circ$.

So, we have the equation:

$$x + 4x + 4x = 180^\circ$$

Solving for $x$:

$$9x = 180^\circ$$

$$x = \frac{180^\circ}{9} = 20^\circ$$

Thus, angle X is:

$$4x = 4(20^\circ) = 80^\circ$$