Question

1. is the measure of angle eab equal to the measure of any angle in triangle abc? if so, which one? if not, how do you know?
2. is the measure of angle cad equal to the measure of any angle in triangle abc? if so, which one? if not, how do you know?
3. what is the sum of the measures of angles abc, bac, and acb?

14.3: every triangle in the world
here is △abc. line de is parallel to line ac.

1. what is m∠dba + b + m∠cbe? explain how you know.
2. use your answer to explain why a + b + c = 180.
3. explain why your argument will work for any triangle: that is, explain why the sum of the angle measures in any triangle is 180°.

Explanation:

Step1: Recall linear - angle property

Since $DE$ is a straight line, the sum of angles on a straight line is $180^{\circ}$. So, $m\angle DBA + b+m\angle CBE=180^{\circ}$.

Step2: Use alternate - interior angles

Because $DE\parallel AC$, $\angle DBA=\angle BAC = a^{\circ}$ (alternate - interior angles) and $\angle CBE=\angle BCA = c^{\circ}$ (alternate - interior angles). Substituting these into $m\angle DBA + b+m\angle CBE = 180^{\circ}$, we get $a + b + c=180^{\circ}$.

Step3: Generalize for any triangle

For any triangle, we can draw a line parallel to one of its sides through the opposite vertex. Using the properties of parallel lines (alternate - interior angles) and the fact that the sum of angles on a straight line is $180^{\circ}$, we can always show that the sum of the interior angles of a triangle is $180^{\circ}$.

Step4: Answer question 4

If $DE\parallel AC$, $\angle EAB$ is not equal to any angle in $\triangle ABC$. There are no parallel - line or angle - congruence relationships that would make it equal.

Step5: Answer question 5

If $DE\parallel AC$, $\angle CAD$ is not equal to any angle in $\triangle ABC$. There are no parallel - line or angle - congruence relationships that would make it equal.

Step6: Answer question 6

The sum of the measures of angles $\angle ABC$, $\angle BAC$, and $\angle ACB$ is $180^{\circ}$ as shown by the parallel - line and angle - sum - on - a - straight - line properties.

Answer:

1. $m\angle DBA + b+m\angle CBE = 180^{\circ}$ because the sum of angles on a straight line is $180^{\circ}$.
2. Since $\angle DBA=a^{\circ}$ and $\angle CBE = c^{\circ}$ (alternate - interior angles with $DE\parallel AC$), substituting into $m\angle DBA + b+m\angle CBE = 180^{\circ}$ gives $a + b + c=180^{\circ}$.
3. For any triangle, a parallel line can be drawn to one side through the opposite vertex. Using alternate - interior angles and the angle - sum property of a straight line, the sum of interior angles is $180^{\circ}$.
4. No, there are no parallel - line or angle - congruence relationships to make it equal.
5. No, there are no parallel - line or angle - congruence relationships to make it equal.
6. $180^{\circ}$