Question

in \\( \triangle a b c \\), side \\( \overline{b c} \\) is extended through \\( c \\) to \\( d \\). if \\( m \angle a=30^{circ} \\) and \\( m \angle a c d=110^{circ} \\), what is the longest side of \\( \triangle a b c \\)?

1. \\( \overline{a c} \\)
2. \\( \overline{b c} \\)
3. \\( \overline{a b} \\)
4. \\( \overline{c d} \\)

Explanation:

Step1: Find the measure of ∠ACB

Since ∠ACD and ∠ACB are supplementary (they form a linear pair), we know that \(m\angle ACB=180^{\circ}-m\angle ACD\).
Given \(m\angle ACD = 110^{\circ}\), then \(m\angle ACB=180 - 110=70^{\circ}\).

Step2: Find the measure of ∠B

Using the angle - sum property of a triangle (\(m\angle A+m\angle B+m\angle ACB = 180^{\circ}\)), we substitute \(m\angle A = 30^{\circ}\) and \(m\angle ACB=70^{\circ}\).
So, \(30^{\circ}+m\angle B + 70^{\circ}=180^{\circ}\).
Simplifying the left - hand side gives \(m\angle B+100^{\circ}=180^{\circ}\).
Subtracting \(100^{\circ}\) from both sides, we get \(m\angle B=180 - 100=80^{\circ}\).

Step3: Relate side lengths to angle measures

In a triangle, the side opposite the largest angle is the longest side.
We have \(m\angle A = 30^{\circ}\), \(m\angle ACB=70^{\circ}\), and \(m\angle B = 80^{\circ}\).
The side opposite ∠B is \(\overline{AC}\), the side opposite ∠ACB is \(\overline{AB}\), and the side opposite ∠A is \(\overline{BC}\).

Answer:

1. \(\overline{AB}\)