Question

4 given $\triangle stw \cong \triangle bfn$, find each missing measure.
a) $bn = \underline{\quad}$
b) $tw = \underline{\quad}$
c) $bf = \underline{\quad}$
d) $m\angle w = \underline{\quad}$
e) $m\angle b = \underline{\quad}$
f) $m\angle f = \underline{\quad}$
5 given $\triangle ghj \cong \triangle xyz$, find each missing measure.
a) $gj = \underline{\quad}$
b) $xy = \underline{\quad}$
c) $zy = \underline{\quad}$
d) $m\angle h = \underline{\quad}$
e) $m\angle z = \underline{\quad}$
f) $m\angle j = \underline{\quad}$
6 given $\triangle uvw \cong \triangle tsr$, find the values of $x$, $y$, and $z$.
7 given $\triangle phs \cong \triangle cnf$, find the values of $x$, $y$, and $z$.
8 if $\triangle def \cong \triangle jkl$, $de = 18$, $ef = 23$, $df = 9x - 23$, $jl = 7x + 11$, and $jk = 3y - 21$, find the values of $x$ and $y$.

Explanation:

Ejercicio 4

Step1: Igualar lados congruentes

Por congruencia $\triangle STW \cong \triangle BFN$, lados correspondientes son iguales:

• $BN = SW = 9\ \text{cm}$
• $TW = FN = 14\ \text{cm}$
• $BF = ST = 17\ \text{cm}$

Step2: Igualar ángulos congruentes

Ángulos correspondientes son iguales, y suma ángulos triángulo es $180^\circ$:

• $m\angle W = m\angle N = 82^\circ$
• $m\angle B = m\angle S = 67^\circ$
• $m\angle F = 180^\circ - 82^\circ - 67^\circ = 31^\circ$

Ejercicio 5

Step1: Igualar lados congruentes

Por $\triangle GHJ \cong \triangle XYZ$, lados correspondientes:

• $GJ = XZ = 18\ \text{ft}$
• $XY = GH = 27\ \text{ft}$
• $ZY = GJ = 27\ \text{ft}$ (ya que $\triangle GHJ$ es isósceles)

Step2: Igualar ángulos congruentes

Ángulos correspondientes, suma ángulos triángulo es $180^\circ$:

• $m\angle H = m\angle Y = 38^\circ$
• $m\angle Z = m\angle G = \frac{180^\circ - 38^\circ}{2} = 71^\circ$
• $m\angle J = m\angle Z = 71^\circ$

Ejercicio 6

Step1: Resolver para $x$

Igualar lados congruentes: $12x - 7 = 53$
$$12x = 53 + 7 = 60 \implies x = \frac{60}{12} = 5$$

Step2: Resolver para $y$

Igualar lados congruentes: $5y - 33 = 57$
$$5y = 57 + 33 = 90 \implies y = \frac{90}{5} = 18$$

Step3: Resolver para $z$

Igualar lados congruentes: $3z + 14 = 50$
$$3z = 50 - 14 = 36 \implies z = \frac{36}{3} = 12$$

Ejercicio 7

Step1: Resolver para $x$

Igualar ángulos congruentes: $6x - 29 = 115$
$$6x = 115 + 29 = 144 \implies x = \frac{144}{6} = 24$$

Step2: Resolver para $y$

Igualar ángulos congruentes: $3y - 1 = 36$
$$3y = 36 + 1 = 37 \implies y = \frac{37}{3} \approx 12.33$$

Step3: Resolver para $z$

Igualar ángulos congruentes: $4z - 32 = 180^\circ - 115^\circ - 36^\circ = 29$
$$4z = 29 + 32 = 61 \implies z = \frac{61}{4} = 15.25$$

Ejercicio 8

Step1: Resolver para $x$

Igualar lados congruentes: $DF = JL \implies 9x - 23 = 7x - 11$
$$9x - 7x = 23 - 11 \implies 2x = 12 \implies x = 6$$

Step2: Resolver para $y$

Igualar lados congruentes: $JK = DE \implies 3y - 21 = 18$
$$3y = 18 + 21 = 39 \implies y = \frac{39}{3} = 13$$

Answer:

Ejercicio 4

a) $BN = 9\ \text{cm}$
b) $TW = 14\ \text{cm}$
c) $BF = 17\ \text{cm}$
d) $m\angle W = 82^\circ$
e) $m\angle B = 67^\circ$
f) $m\angle F = 31^\circ$

Ejercicio 5

a) $GJ = 18\ \text{ft}$
b) $XY = 27\ \text{ft}$
c) $ZY = 27\ \text{ft}$
d) $m\angle H = 38^\circ$
e) $m\angle Z = 71^\circ$
f) $m\angle J = 71^\circ$

Ejercicio 6

$x = 5$, $y = 18$, $z = 12$

Ejercicio 7

$x = 24$, $y = \frac{37}{3}$, $z = \frac{61}{4}$

Ejercicio 8

$x = 6$, $y = 13$