Question

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nd the missing angle using your understanding of the exterior angle theorem.
exterior angle theorem

Explanation:

Step1: Recall the Exterior Angle Theorem

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

Step2: Solve for the first triangle (Problem 1)

For the triangle with angles \(25^{\circ}\) and \(80^{\circ}\), let the exterior angle be \(x\).
By the Exterior Angle Theorem, \(x = 25^{\circ}+80^{\circ}\)
\(x = 105^{\circ}\)

Step3: Solve for the second triangle (Problem 2)

For the triangle with angles \(70^{\circ}\) and \(50^{\circ}\), let the exterior angle be \(y\).
By the Exterior Angle Theorem, \(y=70^{\circ}+50^{\circ}\)
\(y = 120^{\circ}\)

Step4: Solve for the third triangle (Problem 3)

Let the unknown angle be \(z\). The exterior angle is \(120^{\circ}\) and one non - adjacent interior angle is \(50^{\circ}\).
By the Exterior Angle Theorem, \(120^{\circ}=z + 50^{\circ}\)
\(z=120^{\circ}-50^{\circ}\)
\(z = 70^{\circ}\)

Step5: Solve for the fourth triangle (Problem 5)

First, find the interior angle adjacent to \(110^{\circ}\). Let it be \(a\), \(a = 180^{\circ}-110^{\circ}=70^{\circ}\). Let the exterior angle be \(b\).
By the Exterior Angle Theorem, \(b=80^{\circ}+70^{\circ}\)
\(b = 150^{\circ}\)

Step6: Solve for the fifth triangle (Problem 5 - another way)

Let the unknown angle be \(m\). The exterior angle is \(140^{\circ}\) and one non - adjacent interior angle is \(45^{\circ}\).
By the Exterior Angle Theorem, \(140^{\circ}=m + 45^{\circ}\)
\(m=140^{\circ}-45^{\circ}\)
\(m = 95^{\circ}\)

Step7: Solve for the sixth triangle (Problem 6)

Let the unknown angle be \(n\). The exterior angle (related to \(95^{\circ}\)): First, find the interior angle adjacent to \(95^{\circ}\), let it be \(k\), \(k = 180^{\circ}-95^{\circ}=85^{\circ}\).
By the Exterior Angle Theorem, \(n=35^{\circ}+85^{\circ}\)
\(n = 60^{\circ}\)

Answer:

1. \(105^{\circ}\)
2. \(120^{\circ}\)
3. \(70^{\circ}\)
4. \(95^{\circ}\)
5. \(60^{\circ}\)