Question

topic 2: parallelograms

1. if ebcd is a parallelogram, eb = 16, ed = 25, bf = 11, ec = 34, m∠bed = 55°, m∠cdb = 67°, and m∠bce = 24°, find each missing measure.

bc = ____ m∠edc = ____
bd = ____ m∠ebd = ____
fc = ____ m∠bec = ____
cd = ____ m∠dbc = ____

1. find m∠n.
2. find m∠r.
3. in parallelogram abcd, if ed = 7x - 13 and bd = 16x - 38, find bd.

topic 3: rectangles

1. if abcd is a rectangle, ad = 9, ac = 22, and m∠bca = 66°, find each missing measure.

bc = ____ m∠adc = ____
ab = ____ m∠bac = ____
bd = ____ m∠cdb = ____
ec = ____ m∠aeb = ____

1. if pqrs is a rectangle, pr = 9x + 1, and qs = 13x - 11, find tr.
2. if defg is a rectangle, m∠deg = (4x - 5)°, and m∠fge = (6x - 21)°, find m∠dge.

Explanation:

Problem 14

Step1: Use parallelogram side properties

In parallelogram $EBCD$, $BC=ED$, $CD=EB$.
$BC=25$, $CD=16$

Step2: Use diagonal bisect property

Diagonals bisect each other: $BD=2\times EB=2\times16=32$, $FC=EC-BF=34-11=23$

Step3: Find $\angle EDC$

$\angle EDC=\angle BED=55^\circ$ (alternate interior angles)

Step4: Find $\angle EBD$

In $\triangle BED$, $\angle EBD=180^\circ-55^\circ-67^\circ=58^\circ$

Step5: Find $\angle BEC$

In $\triangle BCE$, $\angle BEC=180^\circ-\angle EBD-\angle BCE=180^\circ-58^\circ-24^\circ=98^\circ$

Step6: Find $\angle DBC$

$\angle DBC=\angle BDC=67^\circ$ (alternate interior angles)

Step1: Set consecutive angles supplementary

In parallelogram, consecutive angles sum to $180^\circ$:
$(8x+17)+(12x-39)=180$

Step2: Solve for $x$

$20x-22=180 \implies 20x=202 \implies x=10.1$

Step3: Calculate $m\angle N$

$\angle N=\angle L=12x-39=12(10.1)-39=82.2^\circ$

Step1: Set alternate interior angles equal

$5x-6=14x-4$

Step2: Solve for $x$

$-9x=2 \implies x=-\frac{2}{9}$

Step3: Calculate $\angle U$

$\angle U=5(-\frac{2}{9})-6=-\frac{10}{9}-6=-\frac{64}{9}^\circ$ (Note: This indicates a typo, assuming $5x+6=14x-4$:
$9x=10 \implies x=\frac{10}{9}$, $\angle U=5(\frac{10}{9})+6=\frac{50}{9}+6=\frac{104}{9}\approx11.56^\circ$

Step4: Find $m\angle R$

$\angle R=180^\circ-\angle U=180-\frac{104}{9}=\frac{1516}{9}\approx168.44^\circ$

Answer:

$BC=25$, $m\angle EDC=55^\circ$
$BD=32$, $m\angle EBD=58^\circ$
$FC=23$, $m\angle BEC=98^\circ$
$CD=16$, $m\angle DBC=67^\circ$

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Problem 15