Question

name: hour:
why are cows so good at math?
(angle relationships practice)
directions: solve each problem to find the value of x or the missing part of the figure, and find its matching answer in the answer box. decode the answer to the riddle using the letter of the answer associated with each question. show all of your work.

answer:
\\(\frac{b}{1}\\) \\(\overline{12}\\) \\(\overline{8}\\) \\(\overline{11}\\) \\(\overline{4}\\) \\(\overline{7}\\) \\(\overline{12}\\) \\(\quad\\) \\(\overline{2}\\) \\(\overline{13}\\) \\(\overline{12}\\) \\(\overline{5}\\) \\(\quad\\) \\(\overline{6}\\) \\(\overline{15}\\) \\(\overline{3}\\) \\(\overline{12}\\)
\\(\overline{2}\\) \\(\overline{15}\\) \\(\quad\\) \\(\overline{8}\\) \\(\overline{15}\\) \\(\overline{9}\\) \\(\overline{14}\\) \\(\overline{2}\\) \\(\overline{10}\\)

1. \\(m\angle dbc = \\)

(figure 1: line ac with ray bd. \\(m\angle abd = (5x + 47)^\circ\\), \\(m\angle dbc = (2x + 63)^\circ\\))

1. \\(m\angle aeb = \\)

(figure 2: intersecting lines ac and bd at e. \\(m\angle aeb = (8x - 17)^\circ\\), \\(m\angle dec = (2x + 49)^\circ\\))

1. \\(m\angle dbc = \\)

(figure 3: angle abc is a right angle, split by ray bd. \\(m\angle abd = (9x - 31)^\circ\\), \\(m\angle dbc = (2x + 22)^\circ\\))

1. \\(m\angle dbc = \\)

(figure 4: angle abc is a right angle, split by ray bd. \\(m\angle abd = (5x - 6)^\circ\\), \\(m\angle dbc = (3x + 40)^\circ\\))

1. \\(m\angle abd = \\)

(figure 5: line ac with ray bd. \\(m\angle abd = (14x + 80)^\circ\\), \\(m\angle dbc = (4x + 82)^\circ\\))

1. \\(m\angle dec = \\)

(figure 6: intersecting lines ac and bd at e. \\(m\angle aeb = (6x + 12)^\circ\\), \\(m\angle dec = (2x + 60)^\circ\\))

Explanation:

Set up equations using angle relationships

We analyze the geometric relationships shown in each of the six problems:

• Question 1: Angles \(\angle ABD\) and \(\angle DBC\) form a linear pair on line \(AC\). Using the Linear Pair Postulate, their sum is \(180^\circ\):

$$(5x + 47) + (2x + 63) = 180$$

• Question 2: Angles \(\angle AEB\) and \(\angle DEC\) are vertical angles. Using the Vertical Angles Theorem, they are equal:

$$8x - 17 = 2x + 49$$

• Question 3: Angles \(\angle ABD\) and \(\angle DBC\) form a right angle (\(90^\circ\)), as indicated by the square angle symbol at vertex \(B\). Using the definition of Complementary Angles, their sum is \(90^\circ\):

$$(9x - 31) + (2x + 22) = 90$$

• Question 4: Angles \(\angle ABD\) and \(\angle DBC\) form a right angle (\(90^\circ\)). Using Complementary Angles, their sum is \(90^\circ\):

$$(5x - 6) + (3x + 40) = 90$$

• Question 5: Angles \(\angle ABD\) and \(\angle DBC\) form a linear pair on line \(AC\). Using the Linear Pair Postulate, their sum is \(180^\circ\):

$$(14x + 80) + (4x + 82) = 180$$

• Question 6: Angles \(\angle AEB\) and \(\angle DEC\) are vertical angles. Using the Vertical Angles Theorem, they are equal:

$$6x + 12 = 2x + 60$$

Solve for x in each equation

We solve each linear equation to find the value of \(x\):

• Question 1:

$$7x + 110 = 180 \implies 7x = 70 \implies x = 10$$

• Question 2:

$$6x - 17 = 49 \implies 6x = 66 \implies x = 11$$

• Question 3:

$$11x - 9 = 90 \implies 11x = 99 \implies x = 9$$

• Question 4:

$$8x + 34 = 90 \implies 8x = 56 \implies x = 7$$

• Question 5:

$$18x + 162 = 180 \implies 18x = 18 \implies x = 1$$

• Question 6:

$$4x + 12 = 60 \implies 4x = 48 \implies x = 12$$

Calculate the requested angle measures

We substitute the value of \(x\) back into the expression for the target angle:

• Question 1: Find \(\text{m}\angle DBC = 2x + 63\):

$$\text{m}\angle DBC = 2(10) + 63 = 83^\circ$$

• Question 2: Find \(\text{m}\angle AEB = 8x - 17\):

$$\text{m}\angle AEB = 8(11) - 17 = 71^\circ$$

• Question 3: Find \(\text{m}\angle DBC = 2x + 22\):

$$\text{m}\angle DBC = 2(9) + 22 = 40^\circ$$

• Question 4: Find \(\text{m}\angle DBC = 3x + 40\):

$$\text{m}\angle DBC = 3(7) + 40 = 61^\circ$$

• Question 5: Find \(\text{m}\angle ABD = 14x + 80\):

$$\text{m}\angle ABD = 14(1) + 80 = 94^\circ$$

• Question 6: Find \(\text{m}\angle DEC = 2x + 60\):

$$\text{m}\angle DEC = 2(12) + 60 = 84^\circ$$

Answer:

No.	Answer
2	\(71^\circ\)
3	\(40^\circ\)
4	\(61^\circ\)
5	\(94^\circ\)
6	\(84^\circ\)