Question

kuta software - infinite geometry
angles in a triangle
find the measure of each angle indicated.
1)
2)
3)
4)
5)
6)
7)
8)

Explanation:

Step1: Apply the triangle - angle sum theorem

The sum of the interior angles of a triangle is \(180^{\circ}\).

For problem 1:

Let the unknown angle be \(x\). Then \(x + 65^{\circ}+57^{\circ}=180^{\circ}\).
\(x=180^{\circ}-(65^{\circ} + 57^{\circ})=180^{\circ}-122^{\circ}=58^{\circ}\).

For problem 2:

In a right - triangle (one angle is \(90^{\circ}\)), let the unknown angle be \(y\). Then \(y+40^{\circ}+90^{\circ}=180^{\circ}\).
\(y = 180^{\circ}-(40^{\circ}+90^{\circ})=50^{\circ}\).

For problem 3:

Let the unknown angle be \(z\). Then \(z+20^{\circ}+130^{\circ}=180^{\circ}\).
\(z=180^{\circ}-(20^{\circ}+130^{\circ})=30^{\circ}\).

For problem 4:

Let the unknown angle be \(a\). Then \(a + 85^{\circ}+50^{\circ}=180^{\circ}\).
\(a=180^{\circ}-(85^{\circ}+50^{\circ})=45^{\circ}\).

Step2: Use the exterior - angle theorem and angle - sum relationships

The exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles. Also, linear pairs of angles sum to \(180^{\circ}\).

For problem 5:

First, find the non - adjacent interior angle to the \(137^{\circ}\) angle. Let's call it \(b\). The angle adjacent to \(137^{\circ}\) is \(180 - 137=43^{\circ}\). Then, using the angle - sum of a triangle (\(43^{\circ}+102^{\circ}+c = 180^{\circ}\), \(c = 180-(43 + 102)=35^{\circ}\)). The unknown exterior angle (let's call it \(d\)) is \(180 - 35=145^{\circ}\).

For problem 6:

The angle adjacent to \(35^{\circ}\) (linear pair) is \(180 - 35 = 145^{\circ}\). Let the unknown angle be \(e\). Using the angle - sum of a triangle: \(e+100^{\circ}+145^{\circ}=180^{\circ}\) (wait, no, better way: the exterior angle. The exterior angle (let's use the property). The angle inside the triangle adjacent to the \(100^{\circ}\) - related part: The angle formed by the intersection (vertical angles: the angle inside the triangle is \(35^{\circ}\)). Then, using the exterior - angle theorem (exterior angle of a triangle is equal to the sum of two non - adjacent interior angles). Let the unknown angle be \(f\). \(f+35^{\circ}=100^{\circ}\), \(f = 65^{\circ}\).

For problem 7:

Using the exterior - angle theorem. Let the unknown angle be \(g\). \(g=30^{\circ}+20^{\circ}=50^{\circ}\).

For problem 8:

The angle adjacent to \(155^{\circ}\) (linear pair) is \(180 - 155=25^{\circ}\). Let the unknown angle be \(h\). Using the exterior - angle theorem (assuming the figure's structure, if we consider the triangle part: \(h=25^{\circ}+60^{\circ}=85^{\circ}\)).

Answer:

1. \(58^{\circ}\)
2. \(50^{\circ}\)
3. \(30^{\circ}\)
4. \(45^{\circ}\)
5. \(145^{\circ}\)
6. \(65^{\circ}\)
7. \(50^{\circ}\)
8. \(85^{\circ}\)