Question

find the value of x.
1.
$(5x - 2)^{circ}$
$(3x + 4)^{circ}$
$11x-2+9x+1=180-2.$
2.
$(9x - 8)^{circ}$
$(x + 10)^{circ}$
$(x + 2)^{circ}$
3.
$(11x - 2)^{circ}$
$(19x + 3)^{circ}$
$(9x + 1)^{circ}$
$11x-9+cx-2+12=180$
4.
$(17x - 23)^{circ}$
$(7x + 2)^{circ}$

1. find the values of $x$ and $y$ in the diagram below.

$(6x - 2)^{circ}$
$30^{circ}$
$72^{circ}$
$(11x - 9)^{circ}$
$(5y - 8)^{circ}$

1. find the values of $x$ and $y$ in the diagram below.

$86^{circ}$
$(4x - 7)^{circ}$
$(7y - 1)^{circ}$
$(9x + 4)^{circ}$

1. if $l || m$, find the values of $x$ and $y$ in the diagram below.

$(7x - 31)^{circ}$
$63^{circ}$
$(5x - 8)^{circ}$
$(4y + 27)^{circ}$

Explanation:

Problem 1

Step1: Sum angles to 180° (right triangle)

$90 + (5x-2) + (3x+4) = 180$

Step2: Simplify and solve for x

$8x + 92 = 180 \implies 8x=88 \implies x=11$

Problem 2

Step1: Sum angles to 180° (triangle)

$(x+2) + (9x-8) + (x+10) = 180$

Step2: Simplify and solve for x

$11x + 4 = 180 \implies 11x=176 \implies x=16$

Problem 3

Step1: Apply exterior angle theorem

$19x+3 = (11x-2) + (9x+1)$

Step2: Simplify and solve for x

$19x+3=20x-1 \implies x=4$

Problem 4

Step1: Apply exterior angle theorem

$17x-23 = 90 + (7x+2)$

Step2: Simplify and solve for x

$17x-23=7x+92 \implies 10x=115 \implies x=11.5$

Problem 5

Step1: Find x (vertical angles + triangle sum)

First, find the missing angle in the top triangle: $180 - (6x-2) -72 = 110-6x$. This equals $11x-9$ (vertical angles):
$11x-9=110-6x \implies 17x=119 \implies x=7$

Step2: Find y (triangle sum)

Substitute $x=7$: $11x-9=68$. Then:
$30 + 68 + (5y-8)=180 \implies 5y+90=180 \implies 5y=90 \implies y=18$

Problem 6

Step1: Find x (exterior angle theorem)

$9x+4 = 86 + (4x-7)$

Step2: Solve for x

$9x+4=4x+79 \implies 5x=75 \implies x=15$

Step3: Find y (triangle sum)

Substitute $x=15$: $4x-7=53$. Then:
$86 + 53 + (7y-1)=180 \implies 7y+138=180 \implies 7y=42 \implies y=6$

Problem 7

Step1: Find x (alternate interior angles)

$7x-31=63$

Step2: Solve for x

$7x=94 \implies x=\frac{94}{7} \approx13.43$

Step3: Find y (corresponding angles)

$4y+27=5x-8$. Substitute $x=\frac{94}{7}$:
$4y+27=5(\frac{94}{7})-8 = \frac{470}{7}-\frac{56}{7}=\frac{414}{7}$
$4y=\frac{414}{7}-27=\frac{414}{7}-\frac{189}{7}=\frac{225}{7}$
$y=\frac{225}{28} \approx8.04$

Answer:

1. $x=11$
2. $x=16$
3. $x=4$
4. $x=11.5$
5. $x=7$, $y=18$
6. $x=15$, $y=6$
7. $x=\frac{94}{7}$, $y=\frac{225}{28}$