Question

1. use the diagram to find the measure of \\( \angle ade \\) : \\( m\angle ade= \\)

Explanation:

Step1: Find the measure of ∠ABC

In △ABC, by the triangle - angle - sum theorem (the sum of interior angles of a triangle is \(180^{\circ}\)).
We know that \(\angle C = 50^{\circ}\) and \(\angle BAC=65^{\circ}\). So \(\angle ABC=180^{\circ}-\angle C - \angle BAC\).
\(\angle ABC = 180^{\circ}-50^{\circ}-65^{\circ}=65^{\circ}\).

Step2: Use the property of isosceles triangle

Since \(AB = AD\) (marked in the diagram), \(\triangle ABD\) is isosceles. Let \(\angle ADB=\angle ABD\).
\(\angle ABD = 180^{\circ}-\angle ABC\) (linear - pair of angles). So \(\angle ABD = 180^{\circ}- 115^{\circ}\).

Step3: Calculate \(\angle ADE\)

\(\angle ADE = 180^{\circ}-\angle ADB\).
Since \(\angle ABD = 115^{\circ}\) (from step 2) and \(\angle ADB=\angle ABD\) (isosceles triangle property), \(\angle ADB = 65^{\circ}\). Then \(\angle ADE=180^{\circ}-115^{\circ}=65^{\circ}\).

Answer:

\(65^{\circ}\)