Question

Prove: m∠zab = m∠acb m∠cba triangle a c b is shown with its exterior angles. line a b extends through point x. line a c extends through point y. line c a 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 . we also know m∠cab m∠acb m∠cba = 180° because . 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 .

A) Vertical Angles Theorem
B) Linear Pair Definition
C) Triangle Exterior Angle Theorem
D) Substitution Property

Answer 1

The measure of exterior angle ∠ZAB of triangle ABC is proven to be equal to the sum of the measures of the non-adjacent interior angles ∠ACB and ∠CBA using the Triangle Exterior Angle Theorem and substitution.

Step-by-step explanation:

The solution to the student's problem involves applying the Triangle Exterior Angle Theorem, which states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent internal angles.

In this case, angle ∠ZAB is an exterior angle and, by the theorem, m∠ZAB = m∠ACB + m∠CBA. This effectively proves that the measure of the exterior angle is equal to the sum of the measures of the interior opposite angles.

Additionally, angles ZAB and CAB form a linear pair which sum to 180° because of properties of a straight line. Since m∠CAB + m∠ACB + m∠CBA = 180° (angle sum property of a triangle), and combining this with the previous statement, we can conclude that m∠ZAB = m∠ACB + m∠CBA, hence proving the initial statement using substitution.