Question

in exercises 11–14, find ( mangle 1 ). then classify the triangle by its angles.

1. triangle with angles ( 78^circ ), ( 31^circ ), and ( angle 1 ) (marked in red)
2. triangle with angles ( 40^circ ), ( 30^circ ), and ( angle 1 ) (marked in red)
3. right triangle with angles ( 38^circ ), right angle, and ( angle 1 ) (marked in red)
4. triangle with two ( 60^circ ) angles and ( angle 1 ) (marked in red)

Explanation:

Exercise 11:

Step1: Recall triangle angle sum

The sum of angles in a triangle is \(180^\circ\). Let \(m\angle1 = x\).
So, \(x + 78^\circ + 31^\circ = 180^\circ\)

Step2: Solve for \(x\)

Simplify the equation: \(x + 109^\circ = 180^\circ\)
Subtract \(109^\circ\) from both sides: \(x = 180^\circ - 109^\circ = 71^\circ\)
All angles (\(78^\circ\), \(31^\circ\), \(71^\circ\)) are acute (less than \(90^\circ\)), so it's an acute triangle.

Step1: Use triangle angle sum

Let \(m\angle1 = x\). Then \(x + 40^\circ + 30^\circ = 180^\circ\)

Step2: Calculate \(x\)

Simplify: \(x + 70^\circ = 180^\circ\)
Subtract: \(x = 180^\circ - 70^\circ = 110^\circ\)
Since \(110^\circ\) is obtuse (greater than \(90^\circ\)), the triangle is obtuse.

Step1: Recall right triangle angle sum

In a right triangle, one angle is \(90^\circ\). Let \(m\angle1 = x\). So, \(x + 38^\circ + 90^\circ = 180^\circ\)

Step2: Solve for \(x\)

Simplify: \(x + 128^\circ = 180^\circ\)
Subtract: \(x = 180^\circ - 128^\circ = 52^\circ\)
Since there's a right angle (\(90^\circ\)), the triangle is right - angled.

Answer:

\(m\angle1 = 71^\circ\), Acute triangle

Exercise 12: