Question

use the diagram showing ( m parallel n ), as well as the relationships between interior and exterior angles of ( \triangle abc ), to answer the questions. the measure of angle ( abc ) is ( square ) ( circ ). the measure of angle ( bac ) is ( square ) ( circ ). the measure of angle ( acb ) is ( square ) ( circ ).

Explanation:

Step1: Find the measure of angle ABC

Since \(m\parallel n\), and the given angle adjacent to \(\angle ABC\) is \(60^{\circ}\), using the linear - pair property (supplementary angles), \(\angle ABC = 180^{\circ}- 60^{\circ}=120^{\circ}\).

Step2: Find the measure of angle BAC

The angle adjacent to the \(70^{\circ}\) angle (vertical angles) and using the property of parallel lines (alternate interior angles). The angle adjacent to \(70^{\circ}\) is \(180 - 70=110^{\circ}\). But for \(\angle BAC\), since \(m\parallel n\), \(\angle BAC\) and the angle adjacent to \(70^{\circ}\) (in the non - parallel line intersection) form a linear pair. Wait, another approach: \(\angle BAC\) and the angle adjacent to \(70^{\circ}\) (vertical angles) and using the parallel line property. The angle adjacent to \(70^{\circ}\) (vertical angles) is \(70^{\circ}\), and \(\angle BAC = 180-(70 + 60)=50^{\circ}\) (using the property of the sum of angles on a straight line formed by the transversal cutting the parallel lines \(m\) and \(n\)).

Step3: Find the measure of angle ACB

We know that the sum of interior angles of a triangle \(\triangle ABC\) is \(180^{\circ}\). Let \(\angle ABC = 120^{\circ}\), \(\angle BAC = 50^{\circ}\). Using the formula \(\angle ABC+\angle BAC+\angle ACB = 180^{\circ}\). Then \(\angle ACB=180-(120 + 50)=10^{\circ}\)

Answer:

The measure of angle \(ABC\) is \(120^{\circ}\).
The measure of angle \(BAC\) is \(50^{\circ}\).
The measure of angle \(ACB\) is \(10^{\circ}\).