Question

11/7 - isosceles and equilateral practice
use the diagram to solve for the following values:
what is the value of y?
what is the value of x?
what is the m∠e? degrees

Explanation:

Step1: Find \( x \)

In an isosceles triangle \( \triangle DEF \) (since \( DE = DF \)), the sum of interior angles is \( 180^{\circ} \).
The angles are \( \angle E=4x^{\circ} \), \( \angle D = 2x^{\circ} \), \( \angle F=2x^{\circ} \).
So, \( 4x + 2x+2x=180 \).
Combine like - terms: \( 8x = 180 \).
Divide both sides by 8: \( x=\frac{180}{8}=22.5 \).

Step2: Find \( y \)

Since \( DE = DF \), and \( DF = 21 \), \( DE=2y + 5 \).
Set \( 2y+5=21 \).
Subtract 5 from both sides: \( 2y=21 - 5=16 \).
Divide both sides by 2: \( y = 8 \).

Step3: Find \( m\angle E \)

Since \( x = 22.5 \), and \( \angle E=4x^{\circ} \).
Substitute \( x \) into \( 4x \): \( m\angle E=4\times22.5^{\circ}=90^{\circ} \).

Answer:

The value of \( y \) is \( 8 \), the value of \( x \) is \( 22.5 \), and \( m\angle E = 90 \) degrees.