Question

1. a triangle with angles 82°, 71°, and (12x + 3)°. 4) a triangle with angle 25°. 5) a triangle with angles 105°, 39°, and (11x + 3)°. 6) a right triangle with angle 45° and (10x + 5)°.

Explanation:

Let's solve each triangle problem one by one. We'll use the fact that the sum of the interior angles of a triangle is \(180^\circ\).

Problem 3:

We have a triangle with angles \(82^\circ\), \(71^\circ\), and \((12x + 3)^\circ\).

Step 1: Set up the equation

The sum of the angles in a triangle is \(180^\circ\), so:
\[
82 + 71 + (12x + 3) = 180
\]

Step 2: Simplify the left side

First, add the constant terms:
\[
82 + 71 + 3 + 12x = 180
\]
\[
156 + 12x = 180
\]

Step 3: Solve for \(x\)

Subtract \(156\) from both sides:
\[
12x = 180 - 156
\]
\[
12x = 24
\]
Divide both sides by \(12\):
\[
x = \frac{24}{12} = 2
\]

Problem 5:

We have a triangle with angles \(105^\circ\), \(39^\circ\), and \((11x + 3)^\circ\).

Step 1: Set up the equation

Using the sum of angles in a triangle:
\[
105 + 39 + (11x + 3) = 180
\]

Step 2: Simplify the left side

Add the constant terms:
\[
105 + 39 + 3 + 11x = 180
\]
\[
147 + 11x = 180
\]

Step 3: Solve for \(x\)

Subtract \(147\) from both sides:
\[
11x = 180 - 147
\]
\[
11x = 33
\]
Divide both sides by \(11\):
\[
x = \frac{33}{11} = 3
\]

Problem 6:

We have a right triangle (one angle is \(90^\circ\)) with angles \(45^\circ\), \(90^\circ\), and \((10x + 5)^\circ\).

Step 1: Set up the equation

The sum of angles in a triangle is \(180^\circ\), so:
\[
45 + 90 + (10x + 5) = 180
\]

Step 2: Simplify the left side

Add the constant terms:
\[
45 + 90 + 5 + 10x = 180
\]
\[
140 + 10x = 180
\]

Step 3: Solve for \(x\)

Subtract \(140\) from both sides:
\[
10x = 180 - 140
\]
\[
10x = 40
\]
Divide both sides by \(10\):
\[
x = \frac{40}{10} = 4
\]

Problem 4:

We have a triangle with angles \(25^\circ\), let's assume the third angle is \(y\) (but the diagram is cut off, so we'll assume it's a triangle with angles \(25^\circ\), \(y\), and another angle. Wait, maybe it's a triangle with angles \(25^\circ\), \(180 - 25 - \text{third angle}\), but since the diagram is incomplete, we'll focus on the ones we can solve.

For Problem 3: \(x = 2\)
For Problem 5: \(x = 3\)
For Problem 6: \(x = 4\)

If you need help with Problem 4, please provide the complete diagram or angle information.

Answer:

Let's solve each triangle problem one by one. We'll use the fact that the sum of the interior angles of a triangle is \(180^\circ\).

Problem 3:

We have a triangle with angles \(82^\circ\), \(71^\circ\), and \((12x + 3)^\circ\).

Step 1: Set up the equation

The sum of the angles in a triangle is \(180^\circ\), so:
\[
82 + 71 + (12x + 3) = 180
\]

Step 2: Simplify the left side

First, add the constant terms:
\[
82 + 71 + 3 + 12x = 180
\]
\[
156 + 12x = 180
\]

Step 3: Solve for \(x\)

Subtract \(156\) from both sides:
\[
12x = 180 - 156
\]
\[
12x = 24
\]
Divide both sides by \(12\):
\[
x = \frac{24}{12} = 2
\]

Problem 5:

We have a triangle with angles \(105^\circ\), \(39^\circ\), and \((11x + 3)^\circ\).

Step 1: Set up the equation

Using the sum of angles in a triangle:
\[
105 + 39 + (11x + 3) = 180
\]

Step 2: Simplify the left side

Add the constant terms:
\[
105 + 39 + 3 + 11x = 180
\]
\[
147 + 11x = 180
\]

Step 3: Solve for \(x\)

Subtract \(147\) from both sides:
\[
11x = 180 - 147
\]
\[
11x = 33
\]
Divide both sides by \(11\):
\[
x = \frac{33}{11} = 3
\]

Problem 6:

We have a right triangle (one angle is \(90^\circ\)) with angles \(45^\circ\), \(90^\circ\), and \((10x + 5)^\circ\).

Step 1: Set up the equation

The sum of angles in a triangle is \(180^\circ\), so:
\[
45 + 90 + (10x + 5) = 180
\]

Step 2: Simplify the left side

Add the constant terms:
\[
45 + 90 + 5 + 10x = 180
\]
\[
140 + 10x = 180
\]

Step 3: Solve for \(x\)

Subtract \(140\) from both sides:
\[
10x = 180 - 140
\]
\[
10x = 40
\]
Divide both sides by \(10\):
\[
x = \frac{40}{10} = 4
\]

Problem 4:

We have a triangle with angles \(25^\circ\), let's assume the third angle is \(y\) (but the diagram is cut off, so we'll assume it's a triangle with angles \(25^\circ\), \(y\), and another angle. Wait, maybe it's a triangle with angles \(25^\circ\), \(180 - 25 - \text{third angle}\), but since the diagram is incomplete, we'll focus on the ones we can solve.

For Problem 3: \(x = 2\)
For Problem 5: \(x = 3\)
For Problem 6: \(x = 4\)

If you need help with Problem 4, please provide the complete diagram or angle information.