Question

what are the values of x and y?

x =

y =

Explanation:

Step1: Identify the triangle type

Since two sides of triangle \(DEF\) are equal (marked with the same tick), it is an isosceles triangle. In an isosceles triangle, the angles opposite the equal sides are equal. So, \(\angle D=\angle E\) is incorrect. Wait, no, in \(\triangle DEF\), if \(DE = DF\) (the two sides with tick marks), then \(\angle F=\angle E\) is wrong. Wait, no, if two sides are equal (say \(DE = DF\)), then the base - angles (the angles opposite the equal sides) are equal. Wait, no, looking at the side - marking, if two sides (say \(DE\) and \(DF\)) are equal, then \(\angle F=\angle E\) is wrong. Wait, actually, if two sides of a triangle are equal (the triangle is isosceles), the angles opposite those sides are equal. Here, if \(DE = DF\) (the two sides with tick marks), then \(\angle F=\angle E\) is wrong. Wait, no, in \(\triangle DEF\), if \(DE = DF\) (the two sides with tick marks), then \(\angle F=\angle E\) is wrong. Wait, actually, in \(\triangle DEF\), if \(DE = DF\) (the two sides with tick marks), then \(\angle F=\angle E\) is wrong. Wait, no, using the angle - sum property of a triangle (\(\angle D+\angle E+\angle F = 180^{\circ}\)). Since \(DE = DF\) (the two sides with tick marks), \(\angle E=\angle F=x\).

Step2: Apply the angle - sum formula

We know that \(\angle D = 69^{\circ}\), and \(\angle E=\angle F=x\). Using the formula \(\angle D+\angle E+\angle F = 180^{\circ}\), we substitute the values: \(69^{\circ}+x + x=180^{\circ}\).
Simplify the equation: \(69^{\circ}+2x=180^{\circ}\).
Subtract \(69^{\circ}\) from both sides: \(2x=180^{\circ}- 69^{\circ}=111^{\circ}\).
Divide both sides by 2: \(x=\frac{111^{\circ}}{2}=55.5^{\circ}\).
Then \(y = x = 55.5^{\circ}\) (because \(\angle E=\angle F\) as the triangle \(DEF\) is isosceles with \(DE = DF\)).

Answer:

\(x = 55.5\), \(y = 55.5\)