Question

1. in the rhombus, ( mangle1 = 170 ). what are ( mangle2 ) and ( mangle3 )? the diagram is not to scale.

a. ( mangle2 = 170 ), ( mangle3 = 85 ) c. ( mangle2 = 170 ), ( mangle3 = 5 )
b. ( mangle2 = 10 ), ( mangle3 = 5 ) d. ( mangle2 = 10 ), ( mangle3 = 85 )

1. find the measure of the numbered angles in the rhombus. the diagram is not to scale.

a. ( mangle1 = 90 ), ( mangle2 = 19 ), and ( mangle3 = 19 ) c. ( mangle1 = 90 ), ( mangle2 = 19 ), and ( mangle3 = 71 )
b. ( mangle1 = 90 ), ( mangle2 = 19 ), and ( mangle3 = 80.5 ) d. ( mangle1 = 90 ), ( mangle2 = 71 ), and ( mangle3 = 19 )

1. defg is a rectangle. ( df = 2x - 4 ) and ( eg = x + 8 ). find the value of ( x ) and the length of each diagonal.

a. ( x = 6 ), ( df = 14 ), ( eg = 14 ) c. ( x = 12 ), ( df = 20 ), ( eg = 24 )
b. ( x = 12 ), ( df = 14 ), ( eg = 14 ) d. ( x = 12 ), ( df = 20 ), ( eg = 20 )

1. find the values of ( a ) and ( b ). the diagram is not to scale.

a. ( a = 115 ), ( b = 65 ) c. ( a = 100 ), ( b = 65 )
b. ( a = 115 ), ( b = 80 ) d. ( a = 100 ), ( b = 80 )

1. ( overline{lm} ) is the midsegment of ( parallelogram abcd ). ( ab = x + 8 ), ( lm = 4x + 3 ), and ( dc = 173 ). what is the value of ( x )?

Explanation:

Step1: Solve Question 9

In a rhombus, opposite angles are equal, and consecutive angles are supplementary. Also, the diagonal bisects the angles.

1. $m\angle2 = 180^\circ - m\angle1 = 180^\circ - 170^\circ = 10^\circ$
2. $m\angle3 = \frac{1}{2}(180^\circ - m\angle1) = \frac{1}{2}(10^\circ) = 5^\circ$

Step2: Solve Question 10

In a rhombus, diagonals are perpendicular, so $m\angle1=90^\circ$. Opposite angles are equal, and alternate interior angles are equal, so $m\angle2=19^\circ$. The other angle in the right triangle is $m\angle3=90^\circ - 19^\circ=71^\circ$.

Step3: Solve Question 11

In a rectangle, diagonals are equal, so set $DF=EG$:
$2x-4 = x+8$
$2x - x = 8 + 4$
$x=12$
Length of diagonal: $DF=2(12)-4=20$, $EG=12+8=20$

Step4: Solve Question 12

This is an isosceles trapezoid, so base angles are equal, and consecutive angles between the bases are supplementary.

1. $a^\circ = 180^\circ - 65^\circ = 115^\circ$
2. $b^\circ = 180^\circ - 100^\circ = 80^\circ$

Answer:

1. B. $m\angle2=10,\ m\angle3 = 5$
2. C. $m\angle1 = 90,\ m\angle2=19,\ \text{and}\ m\angle3 = 71$
3. D. $x = 12, DF = 20, EG = 20$
4. B. $a = 115, b = 80$