Question

quadrilateral abcd is a square. if ab=25 and ac=36, find each measure indicated below.

1. m∠1 =

m∠2 =
m∠3 =
m∠abc =

1. ae =

eb =
bc =
dc =
bd =

1. if m∠1 = 13x + 6, find x.
2. if m∠3 = 14y - 8, find y.

Explanation:

Step1: Properties of a square

In square \(ABCD\), all angles are \(90^{\circ}\), so \(m\angle ABC = 90^{\circ}\). The diagonals of a square are perpendicular bisectors of each other and bisect the vertex angles.

Step2: Angle measures

Since the diagonal bisects the vertex angle of the square (\(90^{\circ}\)), \(m\angle1=\frac{90^{\circ}}{2} = 45^{\circ}\). In \(\triangle AEB\), \(\angle AEB = 90^{\circ}\) (diagonals of a square are perpendicular), so \(m\angle2=45^{\circ}\) (because \(\angle1 + \angle2+\angle AEB=180^{\circ}\), \(45^{\circ}+\angle2 + 90^{\circ}=180^{\circ}\), \(\angle2=45^{\circ}\)). And \(m\angle3 = 90^{\circ}\) (diagonals are perpendicular).

Step3: Side lengths

All sides of a square are equal, so \(BC = AB=25\), \(DC = AB = 25\).

Step4: Diagonal lengths

The diagonals of a square are equal. Using the Pythagorean theorem for right - triangle \(ABC\) (\(AC^{2}=AB^{2}+BC^{2}\), \(AC = BD\)). Given \(AC = 36\), so \(BD = 36\). Since the diagonals bisect each other, \(AE=\frac{AC}{2}=\frac{36}{2}=18\), \(EB=\frac{BD}{2}=\frac{36}{2}=18\)

Step5: Solving for \(x\) and \(y\)

Since \(m\angle1 = 45^{\circ}\), if \(m\angle1=13x + 6\), then \(13x+6=45\), \(13x=45 - 6=39\), \(x = 3\). Since \(m\angle3=90^{\circ}\), if \(m\angle3=14y-8\), then \(14y-8 = 90\), \(14y=90 + 8=98\), \(y = 7\)

Answer:

\(m\angle1 = 45^{\circ}\), \(m\angle2 = 45^{\circ}\), \(m\angle3 = 90^{\circ}\), \(m\angle ABC=90^{\circ}\), \(BC = 25\), \(DC = 25\), \(AE = 18\), \(EB = 18\), \(BD = 36\), \(x = 3\), \(y = 7\)