Question

∠abc measures 38°. what is the measure of ∠dba? diagram: lines intersect at b (d---b---c, a---b---e). options: 52 (selected), 142, 90, 38

Explanation:

Step1: Identify angle relationship

∠ABC and ∠DBA are supplementary (form a linear pair), so their sum is \(180^\circ\).

Step2: Calculate ∠DBA

Let \(m\angle DBA = x\). Then \(x + 38^\circ = 180^\circ\). Solving for \(x\), we get \(x = 180^\circ - 38^\circ = 142^\circ\)? Wait, no, wait. Wait, the diagram: Wait, maybe I misread. Wait, no, the options: Wait, the selected option is 52, but that's wrong? Wait, no, maybe the lines are intersecting, but maybe ∠ABC and ∠DBA are adjacent and form a right angle? Wait, no, the diagram: Let me re-examine. The lines are intersecting at B. So ∠ABC and ∠DBA: if ∠ABC is 38°, and if they are supplementary, then 180 - 38 = 142. But the selected option is 52. Wait, maybe there's a right angle? Wait, no, the diagram shows two lines intersecting, so vertical angles or linear pairs. Wait, maybe the user made a mistake, but according to the problem, ∠ABC and ∠DBA: if they are adjacent and form a straight line, then they are supplementary. So 180 - 38 = 142. But the options have 142 as an option. Wait, the initial selected option is 52, which is wrong. Wait, maybe I misread the angle. Wait, maybe ∠ABC is 38°, and ∠DBA is complementary? No, complementary is 90. Wait, no, let's check again.

Wait, the diagram: D---B---C? No, D is on one line, A and E on another. So lines DC and AE intersect at B. So ∠ABC and ∠DBA: ∠ABC is 38°, so ∠DBA is 180 - 38 = 142°. So the correct answer should be 142. But the initial selected option is 52, which is incorrect. Wait, maybe the problem has a right angle? No, the diagram doesn't show a right angle. So let's do the calculation again.

Step1: Linear pair property

∠ABC and ∠DBA are a linear pair, so \(m\angle ABC + m\angle DBA = 180^\circ\).

Step2: Substitute and solve

Given \(m\angle ABC = 38^\circ\), so \(m\angle DBA = 180^\circ - 38^\circ = 142^\circ\).

Answer:

142 (corresponding to the option "142")