Question

the problem contains multiple geometric figures with angle expressions and requires determining angle relationships, writing equations, and solving for x (and y in one case). each figure has a diagram, a relationship blank, an equation blank, and a blank for the value of x (or y). for example, one figure has angles (7y = 105)°, (45 - y)°, another has (x + 5)°, (2x - 5)°, etc. the figures include intersecting lines, right angles, and straight angles with angle measures expressed in terms of variables x or y.

Explanation:

---

Problem 1 (Top Left)

Step1: Identify vertical angles

Vertical angles are equal: $7p = 15p - 32$

Step2: Solve for $p$

$15p - 7p = 32$
$8p = 32$
$p = \frac{32}{8} = 4$

---

Problem 2 (Top Middle)

Step1: Identify complementary angles

Angles sum to $90^\circ$: $(x+6)+(2x-9)=90$

Step2: Simplify and solve for $x$

$3x - 3 = 90$
$3x = 93$
$x = \frac{93}{3} = 31$

---

Problem 3 (Top Right)

Step1: Identify vertical angles

Vertical angles are equal: $4x+20 = x-10$

Step2: Solve for $x$

$4x - x = -10 -20$
$3x = -30$
$x = \frac{-30}{3} = -10$

---

Problem 4 (Middle Left)

Step1: Identify supplementary angles

Angles sum to $180^\circ$: $(15x-12)+(5x-18)=180$

Step2: Simplify and solve for $x$

$20x - 30 = 180$
$20x = 210$
$x = \frac{210}{20} = 10.5$

---

Problem 5 (Middle Middle)

Step1: Identify vertical angles

Vertical angles are equal: $x+16 = 4x-5$

Step2: Solve for $x$

$4x - x = 16 +5$
$3x = 21$
$x = \frac{21}{3} = 7$

---

Problem 6 (Middle Right)

Step1: Identify angle addition (straight line)

Angles sum to $180^\circ$: $(5x+4)+(x+2)+(3x+7)=180$

Step2: Simplify and solve for $x$

$9x +13 = 180$
$9x = 167$
$x = \frac{167}{9} \approx 18.56$

---

Problem 7 (Bottom Left)

Step1: Identify supplementary angles

Angles sum to $180^\circ$: $(7x-14)+(2x+5)=180$

Step2: Simplify and solve for $x$

$9x -9 = 180$
$9x = 189$
$x = \frac{189}{9} = 21$

---

Problem 8 (Bottom Middle)

Step1: Identify complementary angles

Angles sum to $90^\circ$: $(3x-10)+(2x+10)=90$

Step2: Simplify and solve for $x$

$5x = 90$
$x = \frac{90}{5} = 18$

---

Problem 9 (Bottom Right, Top)

Step1: Identify angle addition (straight line)

Angles sum to $180^\circ$: $(6x+12)+(16x+4)+(18x+4)=180$

Step2: Simplify and solve for $x$

$40x +20 = 180$
$40x = 160$
$x = \frac{160}{40} = 4$

---

Problem 10 (Bottom Right, Bottom)

Step1: Identify complementary angles

Angles sum to $90^\circ$: $63 + (2x+17)=90$

Step2: Simplify and solve for $x$

$2x +80 = 90$
$2x = 10$
$x = \frac{10}{2} = 5$

---

Problem 11 (Bottom Middle Left)

Step1: Identify complementary angles

Angles sum to $90^\circ$: $(x+1)+(8x+1)=90$

Step2: Simplify and solve for $x$

$9x +2 = 90$
$9x = 88$
$x = \frac{88}{9} \approx 9.78$

---

Problem 12 (Bottom Middle Right)

Step1: Identify vertical angles pair

Sum of vertical angle pairs is $360^\circ$, so opposite pairs equal: $2x+3 = 3x+1$

Step2: Solve for $x$

$3x -2x = 3-1$
$x = 2$

Answer:

1. Relationship: Vertical Angles, Equation: $7p=15p-32$, $p=4$
2. Relationship: Complementary Angles, Equation: $(x+6)+(2x-9)=90$, $x=31$
3. Relationship: Vertical Angles, Equation: $4x+20=x-10$, $x=-10$
4. Relationship: Supplementary Angles, Equation: $(15x-12)+(5x-18)=180$, $x=10.5$
5. Relationship: Vertical Angles, Equation: $x+16=4x-5$, $x=7$
6. Relationship: Angle Addition (Straight Line), Equation: $(5x+4)+(x+2)+(3x+7)=180$, $x=\frac{167}{9}\approx18.56$
7. Relationship: Supplementary Angles, Equation: $(7x-14)+(2x+5)=180$, $x=21$
8. Relationship: Complementary Angles, Equation: $(3x-10)+(2x+10)=90$, $x=18$
9. Relationship: Angle Addition (Straight Line), Equation: $(6x+12)+(16x+4)+(18x+4)=180$, $x=4$
10. Relationship: Complementary Angles, Equation: $63+(2x+17)=90$, $x=5$
11. Relationship: Complementary Angles, Equation: $(x+1)+(8x+1)=90$, $x=\frac{88}{9}\approx9.78$
12. Relationship: Vertical Angles, Equation: $2x+3=3x+1$, $x=2$