Question

1. if pqrs is a rectangle, ( st = 12 ), and ( mangle prs = 23^{circ} ), find each measure.

a) ( sq=)

b) ( pr=)

c) ( mangle qpr=)

d) ( mangle psr=)

e) ( mangle sqr=)

f) ( mangle ptq=)

1. if jklm is a rhombus, ( mk = 30 ), ( nl = 13 ), and ( mangle mkl = 41^{circ} ), find each measure.

a) ( nk=)

b) ( jl=)

c) ( kl=)

d) ( mangle jkm=)

e) ( mangle jml=)

f) ( mangle mlk=)

g) ( mangle mnl=)

h) ( mangle kjl=)

1. if wxyz is a square with ( wz = 27 ), find each measure.

a) ( zy=)

b) ( wy=)

c) ( rx=)

d) ( mangle wrz=)

e) ( mangle xyz=)

f) ( mangle zwy=)

© gina wilson (all things algebra)

Explanation:

Problem 4: PQRS is a rectangle

Step1: Recall rectangle diagonal properties

In a rectangle, diagonals are equal and bisect each other. So $SQ=PR=2\times ST$.

Step2: Calculate SQ and PR

$ST=12$, so $SQ=PR=2\times12=24$.

Step3: Find $\angle PSR$

All angles in a rectangle are right angles, so $\angle PSR=90^\circ$.

Step4: Find $\angle QPR$

$\angle PRS=23^\circ$, and $PQ\parallel SR$, so alternate interior angles are equal: $\angle QPR=\angle PRS=23^\circ$.

Step5: Find $\angle SQR$

First, $\angle PRQ=\angle PRS=23^\circ$. In $\triangle QSR$, $\angle SQR=90^\circ - 23^\circ=67^\circ$.

Step6: Find $\angle PTQ$

$\angle QTR=180^\circ-2\times23^\circ=134^\circ$, and $\angle PTQ=\angle QTR=134^\circ$ (vertical angles).

Step1: Recall rhombus diagonal properties

Diagonals bisect each other at right angles. So $NK=\frac{1}{2}MK$, $JL=2\times NL$.

Step2: Calculate NK and JL

$MK=30$, so $NK=\frac{30}{2}=15$. $NL=13$, so $JL=2\times13=26$.

Step3: Calculate KL

Use Pythagoras in $\triangle KNL$: $KL=\sqrt{NK^2+NL^2}=\sqrt{15^2+13^2}=\sqrt{225+169}=\sqrt{394}$.

Step4: Find $\angle JKM$

Diagonals bisect angles, so $\angle JKM=\angle MKL=41^\circ$.

Step5: Find $\angle JML$

Opposite angles of a rhombus are equal, and $\angle MKL=41^\circ$, so $\angle JML=2\times41^\circ=82^\circ$.

Step6: Find $\angle MLK$

Consecutive angles are supplementary: $\angle MLK=180^\circ-82^\circ=98^\circ$.

Step7: Find $\angle MNL$

Diagonals intersect at right angles, so $\angle MNL=90^\circ$.

Step8: Find $\angle KJL$

$\angle KJL=\angle MKL=41^\circ$ (alternate interior angles, $JK\parallel ML$).

Step1: Recall square side properties

All sides of a square are equal, so $ZY=WZ=27$.

Step2: Calculate diagonal WY

Diagonal of a square is $side\times\sqrt{2}$, so $WY=27\sqrt{2}$.

Step3: Calculate RX

Diagonals bisect each other, so $RX=\frac{1}{2}WY=\frac{27\sqrt{2}}{2}$.

Step4: Find $\angle WRZ$

Diagonals of a square bisect right angles, so $\angle WRZ=45^\circ$.

Step5: Find $\angle XYZ$

All angles in a square are right angles, so $\angle XYZ=90^\circ$.

Step6: Find $\angle ZWY$

Diagonals bisect right angles, so $\angle ZWY=45^\circ$.

Answer:

a) $SQ=24$
b) $PR=24$
c) $m\angle QPR=23^\circ$
d) $m\angle PSR=90^\circ$
e) $m\angle SQR=67^\circ$
f) $m\angle PTQ=134^\circ$

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Problem 5: JKLM is a rhombus