Question

lines, angles, and triangles drill

1. if ( a = 42 ), then what is the value of ( b )?
2. the three interior angles of a triangle are ( (x + 10)^{circ} ), ( (x - 20)^{circ} ), and ( (x + 40)^{circ} ). what is the value of ( x )?
3. in triangle ( abc ), side ( ab ) is 8 and side ( bc ) is 5. what are the minimum and maximum possible lengths of side ( ac )?

Explanation:

Question 1

Step1: Use the property of parallel lines and transversal

When two parallel lines \(l_1\) and \(l_2\) are cut by a transversal, \(a\) and \(b\) are supplementary angles (they add up to \(180^{\circ}\)).

Step2: Calculate the value of \(b\)

We know that \(a + b=180\). Given \(a = 42\), then \(b=180 - a\). Substitute \(a = 42\) into the equation: \(b=180-42\)

Step1: Use the triangle - angle sum property

The sum of the interior angles of a triangle is \(180^{\circ}\). So \((x + 10)+(x - 2)+(x + 40)=180\)

Step2: Simplify the left - hand side of the equation

Combine like terms: \((x+x+x)+(10 - 2+40)=180\), which is \(3x+48 = 180\)

Step3: Solve for \(x\)

Subtract \(48\) from both sides: \(3x=180 - 48=132\). Then divide both sides by \(3\): \(x=\frac{132}{3}\)

Step1: Use the triangle inequality theorem

For a triangle with sides \(a\), \(b\), and \(c\), \(|a - b|\lt c\lt a + b\). Let \(a = 8\) and \(b = 5\)

Step2: Find the minimum and maximum values

The minimum value of \(AC\) (let \(AC=c\)): \(|8 - 5|\lt c\), so \(c>3\). The maximum value of \(AC\): \(c\lt8 + 5\), so \(c<13\)

Answer:

\(138\)

Question 2