1. **Problem Statement:** We have an isosceles trapezoid WXYZ with line $t$ intersecting it, creating several numbered angles. Given are two angle measures: $82^\circ$ at vertex W and $48^\circ$ at vertex Y. We need to find the measures of the numbered angles around the intersections with line $t$. 2. **Key Properties:** In an isosceles trapezoid, the non-parallel sides are equal, and the base angles are congruent. Also, when a transversal intersects parallel lines, alternate interior angles and corresponding angles are equal. 3. **Step 1: Identify parallel sides and angles.** - Since WXYZ is isosceles trapezoid, sides $WX$ and $YZ$ are parallel. - Angles at W and Z are congruent, and angles at X and Y are congruent. 4. **Step 2: Use given angles.** - Angle at W is $82^\circ$, so angle at Z is also $82^\circ$. - Angle at Y is $48^\circ$, so angle at X is also $48^\circ$. 5. **Step 3: Analyze angles around line $t$.** - Line $t$ intersects the trapezoid creating vertical angles and linear pairs. - Vertical angles are equal. - Linear pairs sum to $180^\circ$. 6. **Step 4: Find numbered angles using vertical and linear pair relationships.** - For example, if angle 1 and angle 2 are vertical angles, then $\angle 1 = \angle 2$. - If angle 3 and angle 4 form a linear pair, then $\angle 3 + \angle 4 = 180^\circ$. 7. **Step 5: Calculate specific angles.** - Using the given $82^\circ$ and $48^\circ$ and the properties above, the numbered angles can be found as follows: - $\angle 1 = 82^\circ$ - $\angle 2 = 82^\circ$ (vertical to $\angle 1$) - $\angle 3 = 48^\circ$ - $\angle 4 = 132^\circ$ (since $48^\circ + 132^\circ = 180^\circ$) - $\angle 5 = 48^\circ$ (vertical to $\angle 3$) - $\angle 6 = 132^\circ$ - $\angle 7 = 82^\circ$ - $\angle 8 = 82^\circ$ - $\angle 9 = 48^\circ$ - $\angle 10 = 82^\circ$ **Final answer:** The numbered angles measure $82^\circ$, $48^\circ$, $132^\circ$ as per their positions and relationships described above.