Question

egents practice:
date
period
1 in an equilateral triangle, what is the difference between the sum of the exterior angles
and the sum of the interior angles?

1. ( 180^{circ} )
2. ( 120^{circ} )
3. ( 90^{circ} )
4. ( 60^{circ} )

2 the angles of triangle ( a b c ) are in the ratio of ( 8: 3: 4 ). what is the measure of the smallest
angle?

1. ( 12^{circ} )
2. ( 24^{circ} )
3. ( 36^{circ} )
4. ( 72^{circ} )

3 what is the measure of the largest angle in the accompanying triangle?

1. 41
2. 46.5
3. 56
4. 83

Explanation:

Question 1

Step1: Recall sum of exterior and interior angles

The sum of exterior angles of any triangle is \(360^{\circ}\). The sum of interior angles of a triangle is \((n - 2)\times180^{\circ}\), for \(n=3\) (triangle), it is \(180^{\circ}\).

Step2: Calculate the difference

Difference \(=360^{\circ}-180^{\circ}=180^{\circ}\)

Step1: Find the sum of ratio parts

The ratio of angles is \(8:3:4\), sum of ratio parts \(=8 + 3+4=15\)

Step2: Use the angle - sum property of a triangle

Since the sum of interior angles of a triangle is \(180^{\circ}\). Let the angles be \(8x\), \(3x\), and \(4x\). Then \(8x+3x + 4x=180^{\circ}\), \(15x=180^{\circ}\), \(x = 12^{\circ}\)

Step3: Find the smallest angle

The smallest angle is \(3x\). Substitute \(x = 12^{\circ}\), \(3x=3\times12^{\circ}=36^{\circ}\)

Step1: Use the angle - sum property of a triangle

\((2x + 1)+(x + 15)+x=180\)

Step2: Simplify the equation

\(2x+1+x + 15+x=180\), \(4x+16 = 180\), \(4x=180 - 16=164\), \(x = 41\)

Step3: Find the angles

The angles are:

• \(x=41^{\circ}\)
• \(x + 15=41+15 = 56^{\circ}\)
• \(2x+1=2\times41+1=83^{\circ}\)

Answer:

1. \(180^{\circ}\)

Question 2