Question

the diagram shows ae intersecting cf and dg at point b. statement true false the sum of the measures of ∠abg, ∠gbf, and ∠fbe is always equal to 180° the measure of ∠abg is always equal to the measure of ∠fbe the sum of the measures of ∠gbf, ∠fbe, and ∠dbc is always equal to 180° the sum of the measures of ∠abg, ∠fbe and ∠dbc is always equal to 180°

Explanation:

Step1: Analyze the first statement

$\angle ABG+\angle GBF+\angle FBE$ forms a straight - line (since $A - B - E$ is a straight line). By the definition of a straight - line angle, the sum of angles on a straight line is $180^{\circ}$. So, $\angle ABG+\angle GBF+\angle FBE = 180^{\circ}$.

Step2: Analyze the second statement

$\angle ABG$ and $\angle FBE$ are not vertical angles. There is no geometric property (such as vertical angles, alternate interior angles etc.) that would make them always equal.

Step3: Analyze the third statement

$\angle GBF+\angle FBE+\angle DBC=\angle GBE+\angle DBC$. There is no reason for this sum to be $180^{\circ}$. For example, if we rotate the lines $CF$ and $DG$ around point $B$, the sum of these angles changes.

Step4: Analyze the fourth statement

$\angle ADG+\angle FBE+\angle DBC=\angle ADB+\angle FBE+\angle DBC$. Since $\angle ADB+\angle DBC = 180^{\circ}$ (they form a linear pair along line $AE$ and line $DG$ intersection), and $\angle FBE
eq0^{\circ}$, $\angle ADG+\angle FBE+\angle DBC
eq180^{\circ}$.

Answer:

1. True
2. False
3. False
4. False