Question

part a
indicate whether each statement is true or false.

the sum of the measures of ∠abg and ∠cba is always equal to 90°.	true	false
the measure of ∠cbg is always equal to the measure of ∠fbd.	true	false
the sum of the measures of ∠dbc, ∠cba, and ∠abg is always equal to 180°.	true	false
the measure of ∠gbf is always equal to the measure of ∠ebd.	true	false
the sum of the measures of ∠cba, ∠fbg, and ∠ebd is always equal to 180°.	true	false
the sum of the measures of ∠abg, ∠gbf, ∠fbe, and ∠dbc is always equal to 360°.	true	false

art b
and m∠gbe = 110°.

Explanation:

Step1: Analyze the first statement

$\angle ABG$ and $\angle CBA$ are complementary angles. By the definition of complementary angles, the sum of complementary angles is $90^{\circ}$. So, $\angle ABG+\angle CBA = 90^{\circ}$ is True.

Step2: Analyze the second statement

$\angle CBG$ and $\angle FBD$ are not necessarily equal. There is no geometric relationship (such as vertical angles, congruent triangles etc.) shown in the general case (without specific information about the figure's construction) to support their equality. So, $\angle CBG=\angle FBD$ is False.

Step3: Analyze the third statement

$\angle DBC+\angle CBA+\angle ABG=\angle DBA$. Since $\angle DBA = 180^{\circ}$ (a straight - angle). So, $\angle DBC+\angle CBA+\angle ABG = 180^{\circ}$ is True.

Step4: Analyze the fourth statement

$\angle GBF$ and $\angle EBD$ are vertical angles. By the vertical - angles theorem, vertical angles are equal. So, $\angle GBF=\angle EBD$ is True.

Step5: Analyze the fifth statement

$\angle CBA+\angle FBG+\angle EBD=\angle CBA+\angle FBG+\angle GBF$ (since $\angle EBD=\angle GBF$). And $\angle CBA+\angle FBG+\angle GBF=\angle CBA+\angle FBA$. Since $\angle CBA+\angle FBA = 180^{\circ}$ (a straight - angle). So, $\angle CBA+\angle FBG+\angle EBD = 180^{\circ}$ is True.

Step6: Analyze the sixth statement

$\angle ABG+\angle GBF+\angle FBE+\angle DBC=\angle ABG+\angle GBF+\angle FBE+\angle DBC$. But in general, these angles do not form a full - circle ($360^{\circ}$) configuration. For example, if we consider the angles around a point, these angles do not cover all the angles around a single point. So, $\angle ABG+\angle GBF+\angle FBE+\angle DBC = 360^{\circ}$ is False.

Answer:

1. True
2. False
3. True
4. True
5. True
6. False