Question

kuta software - infinite geometry
angles in a triangle
find the measure of each angle indicated.

1. 65° 57°?
2. 40°?
3. 20° 130°?
4. 85° 50°?
5. 137° 102°?
6. 35° 100°?
7. 30° 20° 130?
8. 155° 60°?

Explanation:

Step1: Recall angle - sum property of a triangle

The sum of the interior angles of a triangle is 180°. For exterior - angle problems, an exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles.

Step2: Solve problem 1

Let the unknown angle be \(x\). Using the angle - sum property of a triangle, \(x+65^{\circ}+57^{\circ}=180^{\circ}\). Then \(x = 180^{\circ}-(65^{\circ}+57^{\circ})=180^{\circ}-122^{\circ}=58^{\circ}\).

Step3: Solve problem 2

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

Step4: Solve problem 3

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

Step5: Solve problem 4

Let the unknown angle be \(a\). Using the angle - sum property, \(a + 85^{\circ}+50^{\circ}=180^{\circ}\). So \(a=180^{\circ}-(85^{\circ}+50^{\circ}) = 45^{\circ}\).

Step6: Solve problem 5

The exterior angle is equal to the sum of the two non - adjacent interior angles. Let the exterior angle be \(b\). The two non - adjacent interior angles are \(137^{\circ}-102^{\circ}=35^{\circ}\) and \(102^{\circ}\). So \(b=35^{\circ}+102^{\circ}=137^{\circ}\).

Step7: Solve problem 6

The exterior angle is equal to the sum of the two non - adjacent interior angles. Let the unknown angle be \(c\). The two non - adjacent interior angles are \(100^{\circ}-35^{\circ}=65^{\circ}\) and \(35^{\circ}\). So \(c = 65^{\circ}\).

Step8: Solve problem 7

The exterior angle is equal to the sum of the two non - adjacent interior angles. Let the unknown angle be \(d\). The two non - adjacent interior angles are \(30^{\circ}\) and \(20^{\circ}\). So \(d=30^{\circ}+20^{\circ}=50^{\circ}\).

Step9: Solve problem 8

First, find the interior angle adjacent to the 155° exterior angle. It is \(180^{\circ}-155^{\circ}=25^{\circ}\). Let the unknown angle be \(e\). Using the angle - sum property of a triangle, \(e+25^{\circ}+60^{\circ}=180^{\circ}\). So \(e=180^{\circ}-(25^{\circ}+60^{\circ}) = 95^{\circ}\).

Answer:

1. \(58^{\circ}\)
2. \(50^{\circ}\)
3. \(30^{\circ}\)
4. \(45^{\circ}\)
5. \(137^{\circ}\)
6. \(65^{\circ}\)
7. \(50^{\circ}\)
8. \(95^{\circ}\)