SOLUTION

1. Opposite angles are congruent: So we will equate the given angles and solve for x.

• < L = < V
• x+30 = 60°
• x = 60°-30°
• x = 30

2. Adjacent angles are supplementary: So we will equate the given angles to 180 and solve for y:

• < E + < L = 180°
• 120° + 2y-40 = 180°
• 80°+2y = 180°
• 2y = 180-80
• 2y = 100
• y = 50°

3. Opposite angles are congruent: So we will equate the given angles and solve for x. Then finding the measure of angle L.

• < O = < E
• 4x-40 = 2x+80
• 4x-2x = 80+40
• 2x = 120
• x = 60

Substitute the value of x to the angles

• < O = 4(60)-40 = 240-40 = 200

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Now , < L and < O are supplementary

1. < L + < O = 180°
2. < L + 200° = 180°
3. < L = 180°-200°
4. < L = -20°

Negative angles are meaningless , Hence < L is undefined.

4. Adjacent angles are supplementary: So we will equate the given angles to 180 and solve for x: Then solve for the angles:

• < O + < V = 180°
• (3x+25) + (2x+15) = 180°
• 3x+2x+25+15 = 180°
• 5x+40 = 180°
• 5x = 180°-40°
• 5x = 140°
• x = 28

Substitute the value of x to get the angles:

• < O = 3x+25 = 3(28)+25 = 109°
• < V = 2x+15 = 2(28)+15 = 71°

5. Given the angles , i noticed that the question is looking for AB but AB is not in the figure so , we will not solve for it.

6. The bisected diagonals are equal. LR = 2x-7 , LV = 5x+18.

• LR + VR = LV
• (2x-7)+VR= 5x+18
• VR = 5x+18 -2x+7
• VR = 3x+25

Equating the value of VR and LR to get x:

• 2x-7 = 3x+25
• 2x-3x = 25+7
• -x = 32
• x = -32

Now , Substitute the value of x to get the Lengths:

• VR = 3(-32)+25
• VR = -96+25

VR = -71 , negative lengths are meaningless , Hence VR is undefined.

7. The bisected diagonals are equal.

• OR = ER
• 3x-9 = 2x+10
• 3x-2x = 10+9
• x = 19

Substitute the value of x to get the length:

OR = 3x-9 = 3(19)-9 = 48

8. The bisected diagonals are equal.

• OR = ER
• 4x-9 = 2x+13
• 4x-2x = 13+9
• 2x = 22
• x = 11

Substitute the value of x to get the lengths:

• OR = 4x-9 = 4(11)-9 = 44-9 = 35
• ER = 2x+13 = 2(11)+13 = 22+13 = 35
• Now , OR + ER = OE

• OE = 35+35
• OE = 70

ANSWER

1. 30
2. 50

• < L is undefined
• < O = 109° , < V = 71°
• undefined
• undefined
• OR = 48
• OE = 70