1. **Problem 1: Rectangle ABCD with diagonal AC and triangle ADE inside** - Given: ABCD is a rectangle, so all angles are 90°. - Angle BAC = 38°, angle AED = 56°, find angle x inside triangle ADE. 2. **Step 1: Understand the rectangle and triangle ADE** - Since ABCD is a rectangle, diagonal AC creates two right triangles. - Triangle ADE is formed with points A, D, and E. 3. **Step 2: Use angle sum in triangle ADE** - Sum of angles in triangle ADE is 180°. - Given angles: angle AED = 56°, angle ADE = 38° (since angle BAC = 38° and corresponds to angle ADE in rectangle). 4. **Step 3: Calculate angle x** $$x = 180° - 56° - 38° = 86°$$ --- 5. **Problem 2: Equilateral triangle XYZ with angle at W = 39° and angle x at X** - XYZ is equilateral, so all angles are 60°. - Angle at W = 39°, find angle x at X. 6. **Step 1: Use properties of equilateral triangle** - Each angle in XYZ is 60°. 7. **Step 2: Calculate angle x** - Since angle at W = 39° is external or adjacent, angle x inside triangle is: $$x = 60° - 39° = 21°$$ --- 8. **Problem 3: Isosceles triangle PQR with PQ = PR, angles at S = 37°, Q = 62°, find angle x at P** 9. **Step 1: Use isosceles triangle properties** - PQ = PR means angles at Q and R are equal. - Given angle Q = 62°, so angle R = 62°. 10. **Step 2: Calculate angle x at P** - Sum of angles in triangle PQR is 180°. $$x = 180° - 62° - 62° = 56°$$