Question

consider kite abcd. what are the values of x and y? x = 2, y = 22 x = 2, y = 44 x = 5, y = 22 x = 5, y = 44

Explanation:

Step1: Use the property of equal adjacent sides in a kite

In a kite, two pairs of adjacent sides are equal. So, \(4x - 3=2x + 7\).
Subtract \(2x\) from both sides: \(4x-2x - 3=2x-2x + 7\), which gives \(2x-3 = 7\).
Add \(3\) to both sides: \(2x-3 + 3=7 + 3\), so \(2x=10\).
Divide both sides by \(2\): \(x=\frac{10}{2}=5\).

Step2: Use the angle - sum property of a quadrilateral

The sum of interior angles of a quadrilateral is \(360^{\circ}\). In kite \(ABCD\), \(\angle B+\angle C+\angle D+\angle A = 360^{\circ}\). Since \(AB = AD\) and \(CB = CD\), \(\angle A=\angle C\) (not always, but for angle - sum here, we know \(\angle B = 79^{\circ}\), \(\angle D=61^{\circ}\)).
So, \(79^{\circ}+5y^{\circ}+61^{\circ}+5y^{\circ}=360^{\circ}\).
Combine like terms: \((79 + 61)+(5y+5y)=360\), \(140 + 10y=360\).
Subtract \(140\) from both sides: \(10y=360 - 140=220\).
Divide both sides by \(10\): \(y=\frac{220}{10}=22\).

Answer:

\(x = 5,y = 22\) (the third option \(x = 5,y = 22\))