1. **State the problem:** We need to find the size of angle $t$ in a triangle formed by two parallel lines intersected by a transversal, where two angles are given: $123^\circ$ and $109^\circ$, and the triangle is isosceles with two equal sides. 2. **Identify key facts and formulas:** - The sum of angles in any triangle is $180^\circ$. - Alternate interior angles formed by a transversal with parallel lines are equal. - In an isosceles triangle, the two angles opposite the equal sides are equal. 3. **Analyze the given angles:** - The $123^\circ$ angle is on the left side, outside the triangle. - The $109^\circ$ angle is on the top right corner, outside the triangle. - The pink angle $t$ is inside the triangle. 4. **Find the third angle of the triangle:** Since the triangle is isosceles with two equal sides, the two base angles are equal. Let these equal angles be $t$ and $t$. 5. **Use the fact that the sum of angles on a straight line is $180^\circ$:** - The angle adjacent to $123^\circ$ inside the triangle is $180^\circ - 123^\circ = 57^\circ$. - The angle adjacent to $109^\circ$ inside the triangle is $180^\circ - 109^\circ = 71^\circ$. 6. **Since the triangle is isosceles, the two equal angles are $t$ and $t$, and the third angle is either $57^\circ$ or $71^\circ$. We check which matches the isosceles property.** 7. **Sum of angles in triangle:** $$ t + t + 57 = 180 $$ $$ 2t = 180 - 57 $$ $$ 2t = 123 $$ $$ t = \frac{123}{2} = 61.5^\circ $$ 8. **Check with the other angle:** $$ t + t + 71 = 180 $$ $$ 2t = 109 $$ $$ t = 54.5^\circ $$ Since the triangle is isosceles, the two equal angles must be the same. The only consistent value is $t = 61.5^\circ$ when the third angle is $57^\circ$. **Final answer:** $$ t = 61.5^\circ $$