Question

in the figure below, \\(\overline{ad}\\) and \\(\overline{ec}\\) intersect at point g, \\(m \angle agb = 35^\circ\\), and \\(m \angle age = 69^\circ\\).

note: picture not drawn to scale

what is \\(m \angle bgc\\)?

\\(\circ\\) a. \\(111^\circ\\)
\\(\circ\\) b. \\(76^\circ\\)
\\(\circ\\) c. \\(145^\circ\\)
\\(\circ\\) d. \\(104^\circ\\)

Explanation:

Step1: Find ∠BGE

First, we know that ∠AGB and ∠BGE are adjacent angles on a straight line? Wait, no, actually, ∠AGE and ∠AGB and ∠BGE? Wait, no, let's look at the angles around point G. Wait, AD is a straight line? Wait, AD is a line, so ∠AGE and ∠AGB and ∠BGC? Wait, maybe we can use the fact that angles on a straight line sum to 180°, or vertical angles. Wait, first, let's find ∠BGE. Wait, ∠AGE is 69°, ∠AGB is 35°, so ∠BGE would be... Wait, no, maybe we need to find the angle on the straight line. Wait, AD is a straight line, so ∠AGC? No, wait, EC and AD intersect at G. So, ∠AGE and ∠CGD are vertical angles? Wait, no, let's think again.

Wait, the sum of angles around a point is 360°, but maybe we can use linear pairs. Wait, first, let's find ∠BGD? No, wait, let's find the angle adjacent to ∠BGC. Wait, ∠AGB + ∠BGC + ∠CGD = 180°? No, AD is a straight line, so ∠AGD is 180°. Wait, ∠AGE is 69°, so ∠DGE is 180° - 69° = 111°? No, that's not right. Wait, maybe we can find ∠BGC by using the fact that ∠AGB + ∠AGE + ∠EGC? No, wait, let's look at the angles at point G.

Wait, ∠AGE is 69°, ∠AGB is 35°, so ∠BGE = ∠AGE - ∠AGB? Wait, no, if A, G, D are colinear, then ∠AGD is 180°. Wait, maybe ∠BGC is equal to 180° - ∠AGB - ∠AGE? No, that would be 180 - 35 - 69 = 76? But that's option B. Wait, no, maybe not. Wait, let's check again.

Wait, EC is a line, so ∠AGE and ∠CGD are vertical angles? No, ∠AGE and ∠CGD: if EC is a straight line, then ∠AGE + ∠AGC = 180°? No, EC is a straight line, so ∠EGC is 180°. Wait, maybe the correct approach is:

Since AD is a straight line, the sum of angles on one side of AD is 180°. Wait, ∠AGE is 69°, so ∠DGE is 180° - 69° = 111°? No, that's not helpful. Wait, maybe we can find ∠BGC by using the fact that ∠BGC = 180° - ∠AGB - ∠AGE? Wait, 180 - 35 - 69 = 76, which is option B. But wait, let's check the diagram again.

Wait, the points are A---G---D (straight line), and E---G---C (straight line). So, ∠AGB is 35°, ∠AGE is 69°. So, on line EC, ∠AGE + ∠EGC = 180°? No, EC is E---G---C, so ∠EGC is 180°, but ∠AGE is 69°, so ∠AGC is 180° - 69° = 111°? Then, ∠AGB is 35°, so ∠BGC = ∠AGC - ∠AGB = 111° - 35° = 76°? Wait, that makes sense. So ∠AGC is 111° (since ∠AGE is 69°, and EC is straight, so ∠AGC = 180° - 69° = 111°). Then, ∠AGB is 35°, so ∠BGC = ∠AGC - ∠AGB = 111° - 35° = 76°. So the answer is 76°, which is option B.

Wait, but let's verify again. If EC is a straight line, then ∠AGE + ∠AGC = 180° (since they are adjacent angles on a straight line). So ∠AGC = 180° - ∠AGE = 180° - 69° = 111°. Then, ∠AGB is 35°, so ∠BGC = ∠AGC - ∠AGB = 111° - 35° = 76°. Yes, that works. So the measure of ∠BGC is 76 degrees.

Step1: Calculate ∠AGC

Since EC is a straight line, ∠AGE and ∠AGC are supplementary (they form a linear pair). So,

$$ \angle AGC = 180^\circ - \angle AGE $$

Substituting \( \angle AGE = 69^\circ \):

$$ \angle AGC = 180^\circ - 69^\circ = 111^\circ $$

Step2: Calculate ∠BGC

Now, ∠AGB and ∠BGC are adjacent angles that form ∠AGC. So,

$$ \angle BGC = \angle AGC - \angle AGB $$

Substituting \( \angle AGC = 111^\circ \) and \( \angle AGB = 35^\circ \):

$$ \angle BGC = 111^\circ - 35^\circ = 76^\circ $$

Answer:

B. 76°