Question

find the value of each variable.

find each unknown measure and describe how you determined that was the measure for the angle.
a. ( mangle ytz )
b. ( mangle ztw )
c. ( mangle twz )
d. ( mangle ywx )
e. ( mangle wyx )

Explanation:

First, we solve for the variables in the top triangle, then solve the bottom quadrilateral problem:

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Top Triangle (Solve for $x$ and $y$)

Step1: Use exterior angle theorem

The exterior angle $115^\circ$ equals the sum of the two remote interior angles.
$115 = 2x + 2x$

Step2: Simplify and solve for $x$

Combine like terms, then isolate $x$.
$115 = 4x \implies x = \frac{115}{4} = 28.75$

Step3: Use isosceles triangle property

The sides with single ticks are equal, so their opposite angles are equal.
$2x = 4y - 5$

Step4: Substitute $x$ and solve for $y$

Plug $x=28.75$ into the equation, then isolate $y$.
$2(28.75) = 4y - 5 \implies 57.5 = 4y - 5 \implies 62.5 = 4y \implies y = \frac{62.5}{4} = 15.625$

Step1: Solve for $x$ using rectangle angle sum

The sum of angles at $T$ and $W$ is $180^\circ$ (consecutive angles in a rectangle are supplementary).
$(10x+2) + (8x-2) = 180$

Step2: Simplify and solve for $x$

Combine like terms, then isolate $x$.
$18x = 180 \implies x = 10$

Step3: Solve for $y$ using vertical angles

Vertical angles formed by intersecting lines are equal.
$7y + 3 = 6y - 4 + 52$

Step4: Simplify and solve for $y$

Combine like terms, then isolate $y$.
$7y + 3 = 6y + 48 \implies y = 45$

Step5: Calculate $\angle YTZ$ (a)

Use right triangle angle sum: $\angle YTZ + 52^\circ + 90^\circ = 180^\circ$
$m\angle YTZ = 180 - 90 - 52 = 38^\circ$

Step6: Calculate $\angle ZTW$ (b)

Substitute $x=10$ into $(10x+2)$, then subtract $\angle YTZ$.
$m\angle ZTW = (10(10)+2) - 38 = 102 - 38 = 64^\circ$

Step7: Calculate $\angle TWZ$ (c)

Substitute $x=10$ into $(8x-2)$.
$m\angle TWZ = 8(10)-2 = 78^\circ$

Step8: Calculate $\angle YWX$ (d)

Substitute $y=45$ into $(7y+3)$.
$m\angle YWX = 7(45)+3 = 318^\circ$ (reflex angle, or the smaller angle is $42^\circ$, but the expression gives the reflex measure)

Step9: Calculate $\angle WYX$ (e)

Sum of angles in rectangle corner: $\angle WYX = 52^\circ + (6y-4)$
$m\angle WYX = 52 + (6(45)-4) = 52 + 266 = 318^\circ$ (reflex angle, or smaller angle is $42^\circ$)

Answer:

(Top Triangle):
$x = 28.75$, $y = 15.625$

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Bottom Quadrilateral (Solve for angles a-e)