Question

consider kite abcd. what are the values of x and y? 2x + 7 c b 79° (5y)° 4x - 3 61° a d x = 2,y = 22 x = 2,y = 44 x = 5,y = 22 x = 5,y = 44

Explanation:

Step1: Use property of kite - adjacent sides are equal

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

Step2: Use angle - sum property of a quadrilateral

The sum of the interior angles of a quadrilateral is \(360^{\circ}\).
We know three angles: \(79^{\circ}\), \(61^{\circ}\) and \((5y)^{\circ}\). Let the fourth - angle be \(z\).
Since a kite has one pair of non - vertex angles equal, assume the non - vertex angles are \(79^{\circ}\) and \(z\).
The sum of the angles in the kite: \(79^{\circ}+79^{\circ}+61^{\circ}+5y^{\circ}=360^{\circ}\).
First, add the known angles: \(79 + 79+61=219\).
The equation becomes \(219+5y = 360\).
Subtract 219 from both sides: \(5y=360 - 219\), so \(5y = 141\). This is wrong.
The correct way is to use the fact that the sum of all four angles \(79^{\circ}+(5y)^{\circ}+61^{\circ}+(360-(79 + 5y+61))^{\circ}=360^{\circ}\).
The sum of the non - congruent angles and the other two angles: \(79+5y + 61+(180)=360\).
\(79+61+5y+180 = 360\).
\(320+5y=360\).
Subtract 320 from both sides: \(5y=40\).
Divide both sides by 5: \(y = 8\). But this is wrong.
The correct angle - sum: \(79^{\circ}+(5y)^{\circ}+61^{\circ}+(180-(5y))^{\circ}=360^{\circ}\).
The sum of the four angles of the kite: \(79 + 5y+61+(180 - 5y)=360\) (the sum of interior angles of a quadrilateral).
We know that the sum of the four angles of a quadrilateral \(A + B+C + D=360^{\circ}\).
\(79+5y + 61+(180 - 5y)=360\) holds for all \(y\). But if we consider the non - angle related property of sides first and then angle sum.
Since the sum of the interior angles of a quadrilateral \(ABCD\) is \(360^{\circ}\), we have \(79+61 + 5y+(180 - 5y)=360\).
Let's use the fact that in a kite, the non - vertex angles are equal in pairs.
The sum of the known non - congruent angles is \(79+61 = 140\).
The sum of the other two congruent angles is \(360-140 = 220\).
Since one of the congruent angles is \(5y\), then \(5y = 110\).
Divide both sides by 5: \(y = 22\).

Answer:

C. \(x = 5,y = 22\)