Question

If each quadrilateral below is a parallelogram, find the missing measures.

Answer 1

Step-by-step explanation:

We will use parallelogram properties to answer our given problems.

(1) Since we know that opposite sides of parallelogram are congruent, therefore measure of side MN and side KN will be equal to the measure of their opposite sides.

LM=KN   and   KL=MN

Upon substituting our given values we will get,

45=KN   and   31=MN

Property: The opposite angles of parallelogram are congruent and consecutive angles are supplementary.

We have been given $m\angle N=119^{o}$, therefore $m\angle L=119^{o}$.

We will find measure of angle K by using supplementary angles.

$m\angle N+m\angle M=180^{o}$

$119^{o}+m\angle M=180^{o}$

$m\angle M=(180-119)^{o}$

$m\angle M=61^{o}$

By opposite angles of parallelogram $m\angle K=61^{o}$.

(2) Opposite sides of parallelogram are equal. Therefore, CF=DE and FE=CD.

Upon substituting our given values we will get,

CF=10  and FE=15.

Property: Diagonals of parallelogram bisect each other.

So CE will be 2 times CG and GD will be half the length of FD.

$CE=2\times 7=14$

$GD=\frac{1}{2}\times 22=11$

(3) Property: Opposite sides of parallelogram are congruent.

$SR=PQ$   and   $QR=PS$

Upon substituting our given values we will get,

$SR=24$   and   $QR=19$

Property: Diagonals of parallelogram bisect each other.

So SQ will be 2 times TQ and PT will be half the length of PR.

$SQ=2\times 10=20$

$PT=\frac{1}{2}\times 42=21$

Property: The opposite angles of parallelogram are congruent and consecutive angles are supplementary.

$m\angle PQR=106^{o}$

$m\angle PQS=m\angle QSR$  Alternate interior angles.

$m\angle PQS=49^{o}$

$m\angle QRS=m\angle QPS$ Opposite angles of parallelogram.

$m\angle PQR+m\angle QPS=180^{o}$

$106^{o}+m\angle QPS=180^{o}$

$m\angle QPS=180^{o}-106^{o}$

$m\angle QPS=74^{o}$

Therefore, measure of angle QRS will 74 degrees.

$m\angle PRS=m\angle QPR$ Alternate interior angles.

$35^{o}=m\angle QPR$

$m\angle RPS=74^{o}-m\angle QPR$

$m\angle RPS=74^{o}-35^{o}$

$m\angle RPS=39^{o}$

Now let us find measure of angle PSQ.

$m\angle PSQ=m\angle PQR-m\angle QSR$

$m\angle PSQ=106^{o}-49^{o}$

$m\angle PSQ=57^{o}$

The image is not clear, so I am unable to answer the last two problems.

Answer 2

The properties of a parallelogram simplify the steps required to find the

missing sides and angles.

The correct responses are;

(1) MN = 31

KN = 45

m∠K = 61°

m∠L = 119°

m∠M = 61°

(2) CF = 10

FE = 15

CE = 14

GD = 11;

(3) QR = 19

SR = 24

PT = 21

SQ = 20

m∠QRS = 74°

m∠PQS = 49°

m∠RPS = 39°

m∠PSQ = 57°

4. Parts not added

5. x = 9

Reasons:

(1) MN = KL = 31

KN = ML = 45

$\displaystyle m\angle K = \frac{360 - 2 \times 119}{2} = 61$ sum of angles in a parallelogram = 360°

m∠K = 61°

m∠L = m∠N = 119° Opposite angles of a parallelogram are equal

m∠M = m∠K = 61° Opposite angles of a parallelogram are equal

(2) CF = DE = 10 Opposite sides of a parallelogram are equal

FE = CD = 15

CE = 2 × CG = 2 × 7 = 14

GD = 0.5 × FD = 0.5 × 22 = 11; Diagonals of a parallelogram bisect each other

(3) QR = PS = 19

SR = PQ = 24

PT = 0.5 × PR = 0.5 × 42 = 21

SQ = 2 × TQ = 2 × 10 = 20

m∠QRS = 180 - m∠PQR = 180° - 106° = 74° Adjacent angles of a parallelogram are supplementary

m∠PQS = m∠QSR = 49°; Alternate angles theorem

m∠RPS = m∠PRQ = m∠QRS - m∠PRS = 74° - 35° = 39°

m∠PSQ = m∠SQR = m∠PQR - m∠PQS = 106° - 49° = 57°

4. Parts not added

5. AC = 2 × EC

Therefore;

8·x - 14 = 2 × (2·x + 11)

8·x - 4·x = 22 + 14 = 36

4·x = 36

$x = \dfrac{36}{4} = 9$

x = 9

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