Question

Given: Δabc prove: m∠zab = m∠acb m∠cba. Triangle abc is shown with its exterior angles. Line ab extends through point x. Line ac extends through point y. Line ca extends through point x. We start with triangle abc and see that angle zab is an exterior angle created by the extension of side ac. Angles zab and cab are a linear pair by definition. We know that m∠zab + m∠cab = 180° by the angle sum property. We also know m∠cab + m∠acb + m∠cba = 180° because the sum of angles in a triangle is 180°. Using substitution, we have m∠zab + m∠cab = m∠cab + m∠acb + m∠cba. Therefore, we conclude m∠zab = m∠acb + m∠cba using the angle sum property.

Answer 1

The problem demonstrates the exterior angle theorem, which states that an exterior angle of a triangle is equal to the sum of the two opposite interior angles. By showing that the sum of the interior angles and the linear pairs each equal 180 degrees, we confirm the initial assertion.

Step-by-step explanation:

The student is asked to prove specific angle relationships within a triangle and its exterior angles. By using properties of linear pairs and the sum of angles in a triangle, we can demonstrate the relationships. If an exterior angle of a triangle is formed by extending one of its sides, then this exterior angle is equal to the sum of the two opposite interior angles. This is based on the fact that a linear pair of angles adds up to 180 degrees, and the sum of all angles in a triangle is also 180 degrees.

A linear pair is formed when two angles are adjacent and their non-common sides form a straight line. According to the angle addition postulate, if a line is drawn through the interior of an angle creating two adjacent angles, then the measure of the original angle is the sum of the measures of the two new angles. Thus, m∠ZAB + m∠CAB = 180°, since they form a linear pair. Additionally, since the sum of angles in a triangle is 180°, we have m∠CAB + m∠ACB + m∠CBA = 180°. By substitution, we can conclude that m∠ZAB is equal to m∠ACB + m∠CBA, because both expressions are equivalent to 180° minus m∠CAB.

The exterior angle theorem is essential for solving this problem because it states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles.