Question

mark on the figures the measurements for the missing angles.

1. fghj is a rectangle
2. abcd is a rhombus
3. rstu is a square

find the measurements for the missing segments and sides.

1. fghj is a rectangle
2. abcd is a rhombus
3. rstu is a square

Explanation:

Step1: Properties of a rectangle

In rectangle FGHJ, the diagonals are equal and bisect each other. Also, each angle of a rectangle is 90°. In the first - rectangle figure:

• $\angle3 = 90^{\circ}-\angle7$. Since $\angle7 = 27^{\circ}$, then $\angle3=90 - 27=63^{\circ}$.
• $\angle1=\angle7 = 27^{\circ}$ (alternate interior angles for parallel sides of the rectangle and the diagonal as a transversal).
• $\angle2=\angle1 = 27^{\circ}$ (diagonals of a rectangle bisect each other).
• $\angle4 = 180^{\circ}-126^{\circ}=54^{\circ}$ (linear - pair).
• $\angle5 = 126^{\circ}$ (vertical angles).
• $\angle6 = 90^{\circ}-\angle1=63^{\circ}$.

In the second - rectangle figure:

• Using the Pythagorean theorem in right - triangle FJG, if $FJ = 5$ and $JH = 12$, then $GJ=\sqrt{5^{2}+12^{2}}=\sqrt{25 + 144}=\sqrt{169}=13$. Since the diagonals of a rectangle are equal and bisect each other, $FM=\frac{1}{2}FH=\frac{1}{2}GJ = 6.5$, $JM=\frac{1}{2}GJ = 6.5$, $FG = 12$, $GH = 5$.

Step2: Properties of a rhombus

In rhombus ABCD, the diagonals are perpendicular bisectors of each other and bisect the angles of the rhombus.

• In the first - rhombus figure:
• $\angle8 = 90^{\circ}-53^{\circ}=37^{\circ}$.
• $\angle9 = 53^{\circ}$ (diagonals of a rhombus bisect the angles).
• $\angle10 = 90^{\circ}$ (diagonals of a rhombus are perpendicular).
• $\angle11=\angle8 = 37^{\circ}$.
• $\angle12 = 90^{\circ}-\angle11 = 53^{\circ}$.

In the second - rhombus figure:

• Since all sides of a rhombus are equal, $CD = AB = 17$, $AD = 17$.
• The diagonals of a rhombus bisect each other. If $AM = 8$ and $DM = 15$, then $AC = 2AM=16$, $BD = 2DM = 30$, $MC = AM = 8$, $BM = DM = 15$.

Step3: Properties of a square

In square RSTU, all sides are equal, all angles are 90°, and the diagonals are equal, perpendicular, and bisect each other.

• In the first - square figure:
• $\angle13=\angle14=\angle15=\angle16 = 45^{\circ}$ (diagonals of a square bisect the angles and are perpendicular).

In the second - square figure:

• Since $RS = 17$, then $RU = 17$, $UT = 17$, $RT=\sqrt{17^{2}+17^{2}}=\sqrt{2\times17^{2}} = 17\sqrt{2}$, $MS=\frac{1}{2}RT=\frac{17\sqrt{2}}{2}$.

Answer:

For rectangle FGHJ (angles in the first figure):
$\angle1 = 27^{\circ}$, $\angle2 = 27^{\circ}$, $\angle3 = 63^{\circ}$, $\angle4 = 54^{\circ}$, $\angle5 = 126^{\circ}$, $\angle6 = 63^{\circ}$
For rectangle FGHJ (sides in the second figure):
$FG = 12$, $GH = 5$, $JM = 6.5$, $GJ = 13$, $FM = 6.5$, $FH = 13$
For rhombus ABCD (angles in the first figure):
$\angle8 = 37^{\circ}$, $\angle9 = 53^{\circ}$, $\angle10 = 90^{\circ}$, $\angle11 = 37^{\circ}$, $\angle12 = 53^{\circ}$
For rhombus ABCD (sides in the second figure):
$CD = 17$, $AC = 16$, $MC = 8$, $BD = 30$, $BM = 15$, $AD = 17$
For square RSTU (angles in the first figure):
$\angle13 = 45^{\circ}$, $\angle14 = 45^{\circ}$, $\angle15 = 45^{\circ}$, $\angle16 = 45^{\circ}$
For square RSTU (sides in the second figure):
$RU = 17$, $MS=\frac{17\sqrt{2}}{2}$, $RT = 17\sqrt{2}$, $UT = 17$