Question

consecutive angles are
the diagonals each other.
find each measure.

1. ( a b )
2. ( m angle d )

find each measure in ( square l m n p ).

1. ( m l )
2. ( l p )
3. ( m angle l p m )
4. ( l n )
5. ( m angle m l n )
6. ( q n )

Explanation:

Step1: Properties of parallelogram

In a parallelogram, opposite sides are equal. For the first parallelogram \(ABCD\), \(AB = CD\). Given \(CD = 10\) cm, so \(AB=10\) cm.

Step2: Consecutive angles in parallelogram

In parallelogram \(DABO\), consecutive angles are supplementary. So \(m\angle A+m\angle D = 180^{\circ}\). Given \(m\angle A=110^{\circ}\), then \(m\angle D=180 - 110=70^{\circ}\)

Step3: Opposite sides of parallelogram

In parallelogram \(LMNP\), \(ML = NP\). Given \(NP = 12\) m, so \(ML = 12\) m

Step4: Opposite sides of parallelogram

In parallelogram \(LMNP\), \(LP = MN\). Given \(MN = 10\) m, so \(LP = 10\) m

Step5: Alternate - interior angles (using diagonal)

In parallelogram \(LMNP\), \(ML\parallel NP\). The diagonal \(MP\) acts as a transversal. So \(m\angle LPM=m\angle PML = 62^{\circ}\) (alternate - interior angles)

Step6: Diagonals bisect each other (part 1)

In parallelogram \(LMNP\), diagonals bisect each other. \(LQ=QN\) and \(MQ = QP\). Given \(LQ = 9\) m, then \(LN=2\times LQ=18\) m

Step7: Alternate - interior angles (using diagonal)

In parallelogram \(LMNP\), \(LM\parallel NP\). The diagonal \(LN\) acts as a transversal. So \(m\angle MLN=m\angle LNP = 32^{\circ}\) (alternate - interior angles)

Step8: Diagonals bisect each other (part 2)

In parallelogram \(LMNP\), diagonals bisect each other. Given \(NP = 12\) m, then \(QN=\frac{1}{2}NP = 6\) m

Answer:

1. \(10\) cm
2. \(70^{\circ}\)
3. \(12\) m
4. \(10\) m
5. \(62^{\circ}\)
6. \(18\) m
7. \(32^{\circ}\)
8. \(6\) m