Question

5.03 swbat build fluency using parallelogram properties to solve for measures
directions: complete the following questions by showing all work and annotations. keep work organized and box any final answer. all work must be shown in order to receive full credit.
**#1.) rectangle jose has side lengths jo = (35 - 2x)yd, je = (4x - 19)yd and m∠s = (7x + 13)°. determine the area and perimeter.

#2.) given rhombus tuvw below, determine the measure of ∠3.
a. 62°
b. 28°
c. 90°
d. 78°
image of rhombus tuvw with angle markings: ∠wtv is 28°, ∠1 at w, ∠2 at t, ∠3 at u, ∠4 at the intersection of diagonals

Explanation:

Problem #1:

Step1: Recall rectangle properties

In a rectangle, all angles are \(90^\circ\), so \(m\angle S=(7x + 13)^\circ=90^\circ\). Also, opposite sides are equal, so \(JO = JE\)? Wait, no, in a rectangle, opposite sides are equal, so \(JO\) and \(SE\) are equal, \(JE\) and \(OS\) are equal. Wait, actually, in rectangle \(JOSE\), \(JO\) and \(JE\) are adjacent sides? Wait, no, maybe \(JO\) and \(JE\) are adjacent sides, so in a rectangle, adjacent sides are not necessarily equal, but angles are \(90^\circ\). Wait, first, let's solve for \(x\) using the angle.

Set \(7x + 13 = 90\)
\(7x=90 - 13\)
\(7x = 77\)
\(x = 11\)

Step2: Find side lengths

Now, substitute \(x = 11\) into \(JO=(35 - 2x)\) and \(JE=(4x - 19)\)

For \(JO\): \(35-2(11)=35 - 22 = 13\) yd

For \(JE\): \(4(11)-19=44 - 19 = 25\) yd

Step3: Calculate area and perimeter

Area of rectangle \(A = \text{length} \times \text{width}=JO\times JE = 13\times25 = 325\) square yards

Perimeter of rectangle \(P = 2(\text{length}+\text{width})=2(13 + 25)=2\times38 = 76\) yards

Step1: Recall rhombus properties

In a rhombus, the diagonals bisect the angles, and the diagonals are perpendicular? Wait, no, in a rhombus, the diagonals bisect the vertex angles, and opposite sides are parallel. Also, in triangle \(WVU\) or something? Wait, looking at the rhombus \(TUVW\), angle at \(V\) is \(28^\circ\), and we know that in a rhombus, \(WU\parallel TV\)? Wait, no, let's look at the angles.

Wait, in a rhombus, the diagonals bisect the angles, and also, the triangles formed by the diagonals are congruent? Wait, angle \(1\) and the \(28^\circ\) angle: wait, in rhombus \(TUVW\), \(WV = UV\) (sides of rhombus are equal), and the diagonal \(WV\) (wait, no, diagonal is \(TV\) and \(WU\)). Wait, angle at \(W\): angle \(1\) and the \(28^\circ\) angle? Wait, maybe triangle \(WVU\) and \(WV T\)? Wait, no, let's see: in a rhombus, adjacent angles are supplementary, but also, the diagonals bisect the angles. Wait, the angle at \(V\) is \(28^\circ\), and angle \(3\): let's see, in triangle \(UV V\)? Wait, no, the rhombus has sides \(TU = UV = VW = WT\). The diagonal \(TV\) bisects angle \(T\) and angle \(V\)? Wait, no, diagonal \(WU\) and \(TV\) intersect at point \(4\), which is a right angle? Wait, in a rhombus, diagonals are perpendicular, so angle \(4\) is \(90^\circ\). Wait, but we have angle \(28^\circ\) at \(V\), and angle \(3\): let's see, in triangle \(UV V\) (wait, \(U\), \(V\), and the intersection point). Wait, maybe angle \(3\) is equal to angle \(1\)? No, wait, in a rhombus, the diagonals bisect the angles, so angle at \(V\) is \(28^\circ\), so the other part of the angle at \(V\) is also \(28^\circ\)? Wait, no, maybe angle \(3\) is \(62^\circ\)? Wait, no, let's think again.

Wait, in a rhombus, the diagonals are perpendicular, so angle \(4\) is \(90^\circ\). Then in triangle \(UV V\) (wait, \(U\), \(V\), and the intersection), we have angle \(28^\circ\) and angle \(4 = 90^\circ\), so angle \(3 = 180 - 90 - 28 = 62^\circ\)? No, that's not right. Wait, maybe angle \(3\) is \(62^\circ\)? Wait, the options are \(62^\circ\), \(28^\circ\), \(90^\circ\), \(78^\circ\). Wait, in a rhombus, opposite angles are equal, and adjacent angles are supplementary. Also, the diagonals bisect the angles. So if one angle at \(V\) is \(28^\circ\), then the angle adjacent to it would be \(180 - 28\times2 = 124^\circ\)? No, maybe not. Wait, looking at the diagram, angle at \(V\) is \(28^\circ\), and angle \(3\): let's see, in triangle \(UV V\), angle at \(V\) is \(28^\circ\), angle at \(4\) is \(90^\circ\) (since diagonals in rhombus are perpendicular), so angle \(3 = 180 - 90 - 28 = 62^\circ\)? Wait, but option a is \(62^\circ\). Wait, maybe that's it.

Wait, another approach: in a rhombus, the diagonals bisect the angles, so angle \(3\) and the \(28^\circ\) angle: wait, no, maybe angle \(3\) is \(62^\circ\) because \(90 - 28 = 62\)? Wait, no, let's check the triangle. If angle \(4\) is \(90^\circ\) (diagonals perpendicular), angle at \(V\) is \(28^\circ\), then angle \(3 = 180 - 90 - 28 = 62^\circ\). So angle \(3\) is \(62^\circ\), which is option a.

Step2: Confirm with rhombus properties

In a rhombus, diagonals are perpendicular (so \(\angle 4 = 90^\circ\)). In \(\triangle UVV\) (wait, \(\triangle U V\) and the intersection), we have angles: \(\angle V = 28^\circ\), \(\angle 4 = 90^\circ\), so \(\angle 3 = 180^\circ - 90^\circ - 28^\circ = 62^\circ\).

Answer:

Area: \(\boxed{325}\) square yards, Perimeter: \(\boxed{76}\) yards

Problem #2: