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

From the protractor, ∠DBA is 40° (since the ray D is at 40° from A on the protractor).

Step2: Determine the measure of ∠EBA

∠EBA is 90° (since E is along the 90° mark from A, as it's a right angle from the horizontal line AC).

Step3: Calculate ∠DBE

To find ∠DBE, we subtract ∠DBA from ∠EBA. So, \( m\angle DBE = m\angle EBA - m\angle DBA = 90^\circ - 40^\circ = 50^\circ \)? Wait, no, wait. Wait, actually, looking at the protractor, the position of D: let's re-examine. Wait, the straight line is AC, with B at the center. The protractor has A on the left, C on the right. The ray D: looking at the protractor, the angle from A to D: the left scale (since A is on the left, the inner scale maybe? Wait, the protractor shows that D is at 40° from A (the left side), and E is at 90° from A (since it's the vertical line? Wait, no, E is at 90°? Wait, the protractor has 0° at A (left) and 180° at C (right). Wait, the ray E: looking at the protractor, the angle for E: the inner scale (red) shows that E is at 90°? Wait, no, the numbers: the left side (A) is 0°, then as we go up, the numbers increase. Wait, D is at 40° from A (so ∠ABD = 40°), and E is at 90° from A? Wait, no, the ray E: looking at the protractor, the angle between A and E: the inner scale (red) has E at 90°? Wait, no, the diagram: A---B---C is a straight line (180°). D is a ray from B, making some angle with BA, E is another ray, F is another. Wait, the protractor: the red numbers, from A (left) to C (right), 0° to 180°. So D is at 40° from A (so ∠ABD = 40°), E is at 90° from A? Wait, no, the ray E: looking at the protractor, the angle between BA and BE: if BA is 0°, then BE is at 90°? Wait, no, the protractor shows E at 90°? Wait, the user already said m∠DBC = 140°, which is correct because ∠DBC is supplementary to ∠DBA (since A---B---C is straight, 180°), so ∠DBA = 180° - 140° = 40°, which matches. Now, ∠DBE: the angle between D and E. Let's see, the angle between BA (0°) and BE: let's check the protractor. The ray E: looking at the protractor, the inner scale (red) has E at 90°? Wait, no, the numbers: from A (left) to E: the red numbers go from 0° (A) up to 90° at E? Wait, no, the diagram shows E at the 90° mark? Wait, no, the protractor has 0° at A, 180° at C. The ray E: the angle between BA and BE: if BA is 0°, then BE is at 90°? Wait, no, the user's diagram: the protractor has A on the left, C on the right, B in the middle. D is at 40° from A (so ∠ABD = 40°), E is at 90° from A? Wait, no, the angle between BD and BE: let's calculate. ∠ABE: if BA is 0°, and BE is at 90° (since it's the vertical line), then ∠ABE = 90°. Then ∠DBE = ∠ABE - ∠ABD = 90° - 40° = 50°? Wait, but maybe I got the direction wrong. Wait, alternatively, the angle between BD and BE: let's look at the protractor. The ray D is at 40° from A (left), and E is at 90° from A? Wait, no, the protractor's numbers: the outer scale (maybe) or inner. Wait, the user's diagram: D is at 40° (the left side, 40°), E is at 90° (the top, 90°), F is at 30° from C? Wait, no, F is at 30° from C? Wait, C is at 180°, so F is at 180° - 30° = 150° from A? Wait, no, the protractor shows F at 30° from C (right side), so from A, it's 180° - 30° = 150°? Wait, no, the numbers on the protractor: the red numbers, from A (left) to C (right), 0° to 180°. So D is at 40° (so ∠ABD = 40°), E is at 90° (so ∠ABE = 90°), F is at 150° (so ∠ABF = 150°). Then ∠DBE = ∠ABE - ∠ABD = 90° - 40° = 50°? Wait, but let's check ∠DBC: ∠DBC is 180° - ∠ABD = 180° - 40° = 140°, which matches the given answer. Now, ∠DBE: between D and E. So BD is at 40° from BA, BE is at 90° from BA, so the angle between them is 90° - 40° = 50°. Wait, but maybe I made a mistake. Wait, another way: the protractor shows D at 40° (from A), E at 90° (from A), so the angle between D and E is 90° - 40° = 50°. So m∠DBE = 50°.

Wait, but let's confirm. Let's see: ∠ABD = 40°, ∠ABE = 90°, so ∠DBE = ∠ABE - ∠ABD = 90 - 40 = 50. Yes, that makes sense.

Step4: Calculate ∠EBF

Now, ∠ABF: from the protractor, F is at 30° from C (right side), so ∠FBC = 30°, so ∠ABF = 180° - 30° = 150°? Wait, no, ∠ABF is the angle from A to F. Since A---B---C is 180°, and ∠FBC is 30°, then ∠ABF = 180° - 30° = 150°? Wait, no, that can't be. Wait, the protractor: F is at 30° from C, so from B, the angle between BC and BF is 30°, so ∠FBC = 30°, so ∠ABF = 180° - 30° = 150°? But ∠ABE is 90°, so ∠EBF = ∠ABF - ∠ABE = 150° - 90° = 60°? Wait, no, maybe the protractor shows F at 30° from the right (C), so the angle between BE and BF: BE is at 90° from A, BF is at 180° - 30° = 150° from A, so the angle between BE (90°) and BF (150°) is 150° - 90° = 60°, so ∠EBF = 60°.

Step5: Calculate ∠DBF

∠DBF is the angle between D and F. D is at 40° from A, F is at 150° from A, so ∠DBF = 150° - 40° = 110°? Wait, or alternatively, ∠DBF = ∠DBE + ∠EBF = 50° + 60° = 110°, which matches.

But the question first asks for m∠DBE. Let's focus on that. From the protractor, ∠ABD = 40°, ∠ABE = 90°, so ∠DBE = 90° - 40° = 50°.

Answer:

50