Question

more practice: in each triangle below, solve for x and find the measures of all missing angles.
1)
triangle with angles: 41°, (8x)°, and an exterior angle (12x - 7)°
2)
triangle with an exterior angle 120° and interior angles (7x)°, (8x)°
3)
triangle with angles: 85°, (23x)°, and an exterior angle 131°
4)
triangle with angles: (13x)°, (65 + x)°, (9x)°

Explanation:

Problem 1:

Step1: Use linear pair property

The exterior angle $(12x - 7)^\circ$ and the adjacent interior angle sum to $180^\circ$, so the adjacent interior angle is $180 - (12x - 7) = (187 - 12x)^\circ$.

Step2: Apply triangle angle sum

$41 + 8x + (187 - 12x) = 180$
Simplify: $228 - 4x = 180$ → $4x = 48$ → $x = 12$.

Step3: Calculate missing angles

$8x = 96^\circ$, adjacent interior angle: $187 - 12×12 = 43^\circ$.

Problem 2:

Step1: Use supplementary angle property

The angle adjacent to $120^\circ$ is $180 - 120 = 60^\circ$.

Step2: Apply triangle angle sum

$60 + 7x + 8x = 180$ → $15x = 120$ → $x = 8$.

Step3: Calculate missing angles

$7x = 56^\circ$, $8x = 64^\circ$.

Problem 3:

Step1: Use linear pair property

The exterior angle $131^\circ$ and adjacent interior angle sum to $180^\circ$, so adjacent interior angle is $180 - 131 = 49^\circ$.

Step2: Apply triangle angle sum

$23x + 85 + 49 = 180$ → $23x + 134 = 180$ → $23x = 46$ → $x = 2$.

Step3: Calculate missing angle

$23x = 46^\circ$.

Problem 4:

Step1: Apply triangle angle sum

$13x + (65 + x) + 9x = 180$ → $23x + 65 = 180$ → $23x = 115$ → $x = 5$.

Step2: Calculate missing angles

$13x = 65^\circ$, $(65 + x) = 70^\circ$, $9x = 45^\circ$.

Answer:

1. $x=12$; Missing angles: $96^\circ$, $43^\circ$
2. $x=8$; Missing angles: $56^\circ$, $64^\circ$
3. $x=2$; Missing angle: $46^\circ$
4. $x=5$; Missing angles: $65^\circ$, $70^\circ$, $45^\circ$