Question

for exercises 24 - 27, find the measure of each angle. see example 4

1. ∠1
2. ∠2
3. ∠3
4. ∠4
5. a pennant is in the shape of an isosceles triangle. one leg of the triangle is fastened to a stick. the stick forms an 84° angle with the other leg. what is the measure of each remote interior angle in the triangle? see example 5

lesson 2 - 3 parallel lines and triangle angle sums 91

Explanation:

Step1: Use the exterior angle theorem

The exterior angle theorem states that an exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles. For the triangle with angles \(37^{\circ}\), \(116^{\circ}\), and \(\angle1\), we know that the exterior angle (let's assume it's calculated from the adjacent angles) can be found using the property that the sum of angles on a straight line is \(180^{\circ}\). First, find the third angle of the first sub - triangle. The sum of angles in a triangle is \(180^{\circ}\). But we can also use the exterior angle relation. Another approach: consider the straight - line angle. The angle adjacent to \(116^{\circ}\) is \(180 - 116=64^{\circ}\). Then, using the angle - sum property of a triangle (sum of angles in a triangle is \(180^{\circ}\)), for the triangle with angles \(37^{\circ}\), \(64^{\circ}\), and \(\angle1\), we have \(\angle1=180-(37 + 64)\).

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Step2: For \(\angle2\)

We know that the sum of angles around a point on a straight line is \(180^{\circ}\). The angles \(79^{\circ}\), \(74^{\circ}\), and \(\angle2\) form a straight line (sum to \(180^{\circ}\)). So \(\angle2=180-(79 + 74)\)

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Step3: For \(\angle3\)

Using the exterior angle theorem. The exterior angle \(122^{\circ}\) is equal to the sum of the two non - adjacent interior angles (\(48^{\circ}\) and \(\angle3\)). So \(\angle3=122 - 48\)

$$ \angle3=74^{\circ} $$

Step4: For \(\angle4\)

Using the exterior angle theorem. The sum of the non - adjacent interior angles (\(16^{\circ}\) and \(74^{\circ}\)) is equal to \(\angle4\). So \(\angle4=16 + 74\)

$$ \angle4=90^{\circ} $$

Answer:

1. \(\angle1 = 79^{\circ}\)
2. \(\angle2 = 27^{\circ}\)
3. \(\angle3 = 74^{\circ}\)
4. \(\angle4 = 90^{\circ}\)