Question

in the diagram shown, what are the measures of ∠dbc, ∠dbe, ∠ebf, and ∠dbf? m∠dbc = 140° m∠dbe = □°

Explanation:

Step1: Determine the measure of ∠DBA

Since ∠DBC is 140° and ∠ABC is a straight angle (180°), we can find ∠DBA by subtracting ∠DBC from 180°. So, \( m\angle DBA = 180^\circ - 140^\circ = 40^\circ \). Wait, no, actually, looking at the protractor, the angle between BD and BE: the protractor shows that BE is at 100° from BA (since BA is 180° and BE is at 100° mark from the left? Wait, maybe better to look at the protractor scale. The straight line is AC, with B at the center. The protractor has markings: from A (left) to C (right) is 180°. BD is at 40° from A? Wait, no, the angle ∠DBC is 140°, so the angle between BD and BC is 140°, so the angle between BD and BA (which is supplementary to ∠DBC) is \( 180^\circ - 140^\circ = 40^\circ \). Then BE is at 100° from A? Wait, no, the protractor: the upper scale, from E, which is at 100°? Wait, maybe the angle between BD and BE: BD is at 40° from A (since ∠DBA is 40°), and BE is at 90° + 10°? Wait, no, let's re-examine. The straight line AC, B is the vertex. ∠DBC is 140°, so the angle between BD and BC is 140°, so the angle between BD and BA (adjacent to ∠DBC) is \( 180 - 140 = 40^\circ \). Now, BE: looking at the protractor, E is at 100° from A? Wait, no, the protractor's upper scale: from A (left) to C (right), the upper scale goes from 0° (at C) to 180° (at A). Wait, maybe the lower scale is from 0° at A to 180° at C. Wait, the diagram: BD is at 40° from A (lower scale), BE is at 100° from A (lower scale)? No, wait, ∠DBC is 140°, so BD is 40° above the left side (BA), and BC is the right side (180°). So BE is at 100° from A? Wait, no, the angle between BD and BE: BD is at 40° from BA (∠DBA = 40°), and BE is at 90° - 10°? Wait, maybe the protractor shows that BE is at 100° from the left (A), and BD is at 40° from the left (A). So the angle between BD and BE is \( 100^\circ - 40^\circ = 60^\circ \)? Wait, no, let's check again. Wait, ∠DBC is 140°, so the angle between BD and BC (right side) is 140°, so BD is 140° from BC, so from BA (left side), BD is 180 - 140 = 40°, so BD is 40° above BA. Now, BE: looking at the protractor, E is at 100° from BA? Wait, the protractor has markings: the upper scale, from E, which is at 100°? Wait, maybe the angle between BE and BC: let's see, BC is 0° (right side), so BE is at 40° from BC? No, the protractor's upper scale: from C (right) to A (left), the upper scale is 0° at C, 180° at A. So BD is at 140° from C (since ∠DBC is 140°), and BE is at 40° from C? Wait, no, the diagram: E is between D and F? Wait, the protractor: the red semicircle, with markings. Let's count the degrees. From D to E: the angle. Wait, maybe a better approach: ∠DBC is 140°, so the angle between BD and BC is 140°. The straight line AC, so ∠ABC = 180°. Then ∠DBA = 180° - 140° = 40°. Now, BE: looking at the protractor, BE is at 100° from BA? Wait, no, the protractor's lower scale (from A to C) is 0° at A, 180° at C. So BD is at 40° (since ∠DBA is 40°), and BE is at 100° (since from A, moving towards C, BE is at 100°). So the angle between BD (40° from A) and BE (100° from A) is \( 100^\circ - 40^\circ = 60^\circ \). Wait, but let's check with the straight angle. Alternatively, ∠DBE: since ∠DBC is 140°, and ∠EBC: let's see, F is at 40° from C (since from C, moving towards A, F is at 40°), so ∠FBC is 40°, and ∠EBF: let's see, E is at 100° from A, so from C (0°), E is at 80°? Wait, no, 180° - 100° = 80°, so ∠EBC is 80°? Wait, this is confusing. Wait, the given ∠DBC is 140°, so BD to BC is 140°. BE is at 100° from A (left), so from A (180°) to BE is 100°, so BE is at 180° - 100° = 80° from C (right). So ∠EBC is 80°, then ∠DBE = ∠DBC - ∠EBC = 140° - 80° = 60°? Wait, no, ∠DBC is BD to BC, ∠EBC is BE to BC, so BD to BE is ∠DBC - ∠EBC = 140° - 80° = 60°? Wait, but maybe the protractor shows that BD is at 40° from A (so ∠DBA = 40°), and BE is at 100° from A (∠EBA = 100°), so ∠DBE = ∠EBA - ∠DBA = 100° - 40° = 60°. Yes, that makes sense. So \( m\angle DBE = 60^\circ \).

Step2: Verify with the protractor markings

Looking at the protractor, BD is at 40° from A (left), BE is at 100° from A (left), so the difference is 60°, so \( m\angle DBE = 60^\circ \).

Answer:

\( 60 \)