Question

find the measure of the exterior angle shown.

1. triangle with angles 39°, (x + 34)°, and exterior angle 2x°
2. triangle with angles 71°, (x - 6)°, and exterior angle 2x°
3. triangle with angles x°, (2x + 48)°, and exterior angle (4x + 18)°
4. triangle with angles x°, (x + 85)°, (x - 7)°

lock 4 (screen will be blank until lock 3 is unlocked)
#1#2#3#4
73:a 130:b 138:c 34:d
35.6:f 30:g 146:h 65:i

Explanation:

Let's solve each problem one by one. We'll use the exterior angle theorem, which states that the exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles.

Problem 1

Step 1: Apply the exterior angle theorem

The exterior angle is \(2x^{\circ}\), and the two non - adjacent interior angles are \(39^{\circ}\) and \((x + 34)^{\circ}\). So we can set up the equation:
\(2x=39+(x + 34)\)

Step 2: Solve for \(x\)

Simplify the right - hand side of the equation: \(2x=x + 73\)
Subtract \(x\) from both sides: \(2x-x=x + 73-x\), which gives \(x = 73\)

Step 3: Find the measure of the exterior angle

Now that we know \(x = 73\), the exterior angle \(2x=2\times73 = 146^{\circ}\)

Problem 2

Step 1: Apply the exterior angle theorem

The exterior angle is \(2x^{\circ}\), and the two non - adjacent interior angles are \(71^{\circ}\) and \((x - 6)^{\circ}\). So the equation is:
\(2x=71+(x - 6)\)

Step 2: Solve for \(x\)

Simplify the right - hand side: \(2x=x + 65\)
Subtract \(x\) from both sides: \(2x-x=x + 65-x\), so \(x = 65\)

Step 3: Find the measure of the exterior angle

The exterior angle \(2x = 2\times65=130^{\circ}\)

Problem 3

Step 1: Apply the exterior angle theorem

The exterior angle is \((4x + 18)^{\circ}\), and the two non - adjacent interior angles are \(x^{\circ}\) and \((2x + 48)^{\circ}\). So we have:
\(4x+18=x+(2x + 48)\)

Step 2: Solve for \(x\)

Simplify the right - hand side: \(4x+18=3x + 48\)
Subtract \(3x\) from both sides: \(4x-3x+18=3x-3x + 48\), which gives \(x+18 = 48\)
Subtract 18 from both sides: \(x+18 - 18=48 - 18\), so \(x = 30\)

Step 3: Find the measure of the exterior angle

Substitute \(x = 30\) into \(4x + 18\): \(4\times30+18=120 + 18=138^{\circ}\)

Problem 4

Step 1: Use the angle - sum property of a triangle

The sum of the interior angles of a triangle is \(180^{\circ}\). The interior angles are \(x^{\circ}\), \((x + 85)^{\circ}\), and \((x - 7)^{\circ}\). So:
\(x+(x + 85)+(x - 7)=180\)

Step 2: Solve for \(x\)

Simplify the left - hand side: \(3x+78 = 180\)
Subtract 78 from both sides: \(3x+78 - 78=180 - 78\), so \(3x = 102\)
Divide both sides by 3: \(x=\frac{102}{3}=34\)
The exterior angle (since it forms a linear pair with the angle adjacent to it) is supplementary to the adjacent interior angle. But we can also note that if we consider the exterior angle related to the triangle, but in this case, since the sum of interior angles gives us \(x = 34\), and if we assume the exterior angle is related to the straight line, but from the angle - sum, we found \(x = 34\)

Answer:

s:

1. The measure of the exterior angle is \(\boldsymbol{146^{\circ}}\)
2. The measure of the exterior angle is \(\boldsymbol{130^{\circ}}\)
3. The measure of the exterior angle is \(\boldsymbol{138^{\circ}}\)
4. The value of \(x\) (related to the angle) is \(\boldsymbol{34}\) (if we consider the angle sum result)