notes 6 - 2: properties of parallelograms
objectives: 1. prove and apply properties of parallelograms.
2. use properties of parallelograms to solve problems.
a parallelogram is a quadrilateral with ______ pairs of ______ sides.
all parallelograms, such as □fghj, have the following properties.
□fghj
properties of parallelograms
$ \overline { f g } \cong \overline { h j } $
$ \overline { g h } \cong \overline { j f } $
$ \angle f \cong \angle h $
$ \angle g \cong \angle j $
opposite sides are ______.
opposite ______ are congruent.
$ m \angle f + m \angle g = 180 ^ { \circ } $
$ m \angle g + m \angle h = 180 ^ { \circ } $
$ m \angle h + m \angle j = 180 ^ { \circ } $
$ m \angle j + m \angle f = 180 ^ { \circ } $
$ \overline { f p } \cong \overline { h p } $
$ \overline { g p } \cong \overline { j p } $
consecutive angles are ______.
the diagonals ______ each other.
find each measure.
1. ab
2. $ m \angle d $
find each measure in $ \square l m n p $.
3. ml
4. lp
5. $ m \angle l p m $
6. ln
7. $ m \angle m l n $
8. qn

Question

Accepted Answer

# 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

# Explanation:
## Step1: Solve for $AB$
In a parallelogram, opposite sides are congruent. Since $CD = 10$ cm, $AB=CD = 10$ cm.
## Step2: Solve for $m\angle D$
In a parallelogram, consecutive angles are supplementary. $m\angle A + m\angle D=180^{\circ}$. Given $m\angle A = 110^{\circ}$, then $m\angle D=180 - 110=70^{\circ}$.
## Step3: Solve for $ML$
In parallelogram $LMNP$, opposite sides are congruent. Since $NP = 12$ m, $ML = NP=12$ m.
## Step4: Solve for $LP$
In parallelogram $LMNP$, opposite sides are congruent. Since $MN = 10$ m, $LP = MN = 10$ m.
## Step5: Solve for $m\angle LPM$
In parallelogram $LMNP$, $ML\parallel NP$. Alternate - interior angles are congruent. So $m\angle LPM=m\angle NML = 62^{\circ}$.
## Step6: Solve for $LN$
In parallelogram $LMNP$, the diagonals bisect each other. If $LQ = 9$ m, then $LN=2\times LQ=18$ m.
## Step7: Solve for $m\angle MLN$
In parallelogram $LMNP$, $ML\parallel NP$. Alternate - interior angles are congruent. So $m\angle MLN=m\angle PNL = 32^{\circ}$.
## Step8: Solve for $QN$
In parallelogram $LMNP$, the diagonals bisect each other. If $NP = 12$ m, then $QN=\frac{1}{2}NP = 6$ m.

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Step1: Solve for \(AB\)

Step2: Solve for \(m\angle D\)

Step3: Solve for \(ML\)

Step4: Solve for \(LP\)

Step5: Solve for \(m\angle LPM\)

Step6: Solve for \(LN\)

Step7: Solve for \(m\angle MLN\)

Step8: Solve for \(QN\)

Answer: