Question

1. in the diagram below, line ab is parallel to line cd, that is, (l_{ab}parallel l_{cd}). the measure of (angle abc) is (21^{circ}), and the measure of (angle edc) is (42^{circ}). find the measure of (angle ced). explain why you are correct by presenting an informal argument that uses the angle - sum of a triangle.
2. in the diagram below, line ab is parallel to line cd, that is, (l_{ab}parallel l_{cd}). the measure of (angle abe) is (38^{circ}), and the measure of (angle edc) is (16^{circ}). find the measure of (angle bed). explain why you are correct by presenting an informal argument that uses the angle - sum of a triangle. (hint: find the measure of (angle ced) first, and then use that measure to find the measure of (angle bed).)

Explanation:

Step1: Use alternate - interior angles

Since \(AB\parallel CD\), \(\angle ABE\) and \(\angle BEC\) are alternate - interior angles. So \(\angle BEC=\angle ABE\).

Step2: Consider triangle \(ECD\)

In \(\triangle ECD\), we know that the sum of the interior angles of a triangle is \(180^{\circ}\). Let's first find \(\angle CED\) in terms of the given angle \(\angle EDC\).
In \(\triangle ECD\), we know that \(\angle CED = 180^{\circ}-\angle ECD-\angle EDC\). Since \(AB\parallel CD\), we can use angle - relationship properties. But we can also use the fact that we want to find the required angle in terms of the given ones.
Let's consider the first problem:
We know that \(\angle ABE = 21^{\circ}\) and \(\angle EDC=42^{\circ}\). Since \(AB\parallel CD\), \(\angle ABE\) and \(\angle BEC\) are equal (alternate - interior angles).
In \(\triangle ECD\), we know that the sum of the interior angles of a triangle \(\triangle ECD\) is \(180^{\circ}\). Let \(\angle CED=x\).
We know that \(\angle ECD\) and \(\angle ABE\) are related by the parallel lines. Since \(AB\parallel CD\), \(\angle ABE\) and \(\angle BEC\) are equal.
We use the angle - sum property of a triangle in \(\triangle ECD\).
\(\angle CED=180^{\circ}-\angle ECD - \angle EDC\). But since \(AB\parallel CD\), \(\angle ABE\) and \(\angle BEC\) are alternate - interior angles, so \(\angle BEC = 21^{\circ}\).
In \(\triangle ECD\), we know that \(\angle CED=180^{\circ}- 21^{\circ}-42^{\circ}=117^{\circ}\)

Let's consider the second problem:
Since \(AB\parallel CD\), \(\angle ABE\) and \(\angle BEC\) are alternate - interior angles, so \(\angle BEC = 38^{\circ}\)
In \(\triangle ECD\), we know that the sum of the interior angles of a triangle is \(180^{\circ}\). Given \(\angle EDC = 16^{\circ}\)
Let \(\angle CED=x\), then \(x=180^{\circ}-\angle ECD-\angle EDC\). Since \(\angle ECD=\angle ABE = 38^{\circ}\) (alternate - interior angles)
\(\angle CED=180^{\circ}-38^{\circ}-16^{\circ}=126^{\circ}\)
\(\angle BED = 180^{\circ}-\angle CED\) (linear - pair of angles)
\(\angle BED=180^{\circ}-126^{\circ}=54^{\circ}\)

For the first problem:

Answer:

\(117^{\circ}\)

For the second problem: