Question

in exercises 11 - 12, find the value of the variables.
11.
12.

1. your friend claims that the measure of an exterior angle of a triangle can never be acute because it is the sum of the two nonadjacent angles of the triangle. is your friend correct? explain your reasoning. (it may help to draw a picture.)
2. the figure shows the measures of various angles of a roof and its supports. find the measure of ∠1. the angle between an eave and a horizontal support beam.

Explanation:

Step1: Recall the exterior - angle property of a triangle

The measure of an exterior angle of a triangle is equal to the sum of the measures of the two non - adjacent interior angles.

Step2: For exercise 11

In the triangle, the exterior angle is \(4x - 18\), and the two non - adjacent interior angles are \(2x+6\) and \(36\). So, we set up the equation \(4x - 18=(2x + 6)+36\).
First, simplify the right - hand side: \((2x + 6)+36=2x+42\).
Then, solve the equation \(4x - 18=2x + 42\).
Subtract \(2x\) from both sides: \(4x-2x-18=2x-2x + 42\), which gives \(2x-18 = 42\).
Add 18 to both sides: \(2x-18 + 18=42+18\), so \(2x=60\).
Divide both sides by 2: \(x = 30\).

Step3: For exercise 12

In a right - triangle, one angle is \(90^{\circ}\), another is \(24^{\circ}\), and the exterior angle is \(2x+9\).
Using the exterior - angle property, \(2x + 9=24 + 90\).
Simplify the right - hand side: \(24+90 = 114\).
So, \(2x+9=114\).
Subtract 9 from both sides: \(2x+9 - 9=114 - 9\), which gives \(2x = 105\).
Divide both sides by 2: \(x=\frac{105}{2}=52.5\).

Step4: Analyze the claim in exercise 13

The claim that an exterior angle of a triangle can never be acute is correct.
The measure of an exterior angle of a triangle \(E\) is \(E=a + b\), where \(a\) and \(b\) are non - adjacent interior angles. Since the interior angles of a triangle are non - negative, and at least two of them are non - zero, the sum \(a + b\geq90^{\circ}\) (in a non - degenerate triangle). So, an exterior angle is either right (in a right - triangle for the exterior angle adjacent to the right angle) or obtuse or straight (in a degenerate case), but never acute.

Step5: For exercise 14

The sum of the angles in the given figure: Let the unknown angle be \(y\).
We know that the sum of angles around a point is \(360^{\circ}\). If we consider the angles related to the roof and support beam, assume the angles in the figure form a complete rotation around a point. Given one angle is \(110^{\circ}\), and we assume the other two angles are equal (symmetry or based on the problem context, if not, more information is needed). Let the angle we want to find be \(\angle1\).
If we consider the linear - pair or angle - sum relationships, if the \(110^{\circ}\) angle and \(\angle1\) are part of a linear pair, then \(\angle1 = 180-110=70^{\circ}\).

Answer:

1. \(x = 30\)
2. \(x = 52.5\)
3. The claim is correct.
4. \(\angle1 = 70^{\circ}\)