Question

quadrilateral j k l m is a square. answers and information from one problem does not carry to the next problem.

1. if ( m j = 12 ), then ( m l = )__ & ( l k = )__.
2. if ( m x = 8 ), then ( x j = )____.
3. if ( j l = 18 ), then ( m k = )__, ( j x = ) & ( x k = )__.
4. ( m angle m j k = )__ & ( m angle m x j = )__
5. the numbered angles are all congruent. each angle measures __. 20.__

Explanation:

Step1: Properties of a square

In a square \(JKLM\), all sides are equal (\(MJ = ML=LK = KJ\)), and the diagonals bisect each other at right - angles. The diagonals of a square are equal and bisect each other.

Step16: Side lengths

Since \(MJ = 12\), and in a square \(MJ=ML = LK\). So \(ML = 12\) and \(LK=12\)

Step17: Diagonal bisecting

The diagonals of a square bisect each other. If \(MX = 8\), then \(XJ=MX = 8\) (because the diagonals of a square bisect each other)

Step18: Diagonal length

The diagonals of a square bisect each other. If \(JL = 18\), then \(MK=JL = 18\) (diagonals of a square are equal). Also, \(JX=\frac{JL}{2}=\frac{18}{2}=9\) and \(XK = 9\) (diagonals bisect each other)

Step19: Angles in a square

The diagonals of a square are perpendicular to each other. So \(m\angle MJK = 45^{\circ}\) (diagonal of a square bisects the vertex angles) and \(m\angle MXJ=90^{\circ}\) (diagonals of a square are perpendicular)

Step20: Angle measure in a square

All the numbered angles are congruent. In a square, each angle is \(90^{\circ}\), and the diagonals bisect the vertex angles. The sum of angles around a point formed by the diagonals: Each of the numbered angles (since they are congruent) \(m\angle1=m\angle2=\cdots=m\angle8 = 45^{\circ}\)

Answer:

1. \(ML = 12\) \(\&\) \(LK = 12\)
2. \(XJ=8\)
3. \(MK = 18\), \(JX = 9\) \(\&\) \(XK = 9\)
4. \(m\angle MJK=45^{\circ}\) \(\&\) \(m\angle MXJ = 90^{\circ}\)
5. \(45^{\circ}\)