1. **Problem Statement for Q2:** Find the values of angles $a$ and $b$ given that they are adjacent to a 135° angle and form a right angle (small square) between them. 2. **Formula and Rules:** - Adjacent angles on a straight line sum to 180°. - The small square indicates $a$ and $b$ are complementary, so $a + b = 90^\circ$. - The angle adjacent to $a$ and $b$ is 135°. 3. **Step-by-step Solution for Q2:** - Since $a$ and $b$ form a right angle, write: $$a + b = 90^\circ$$ - The 135° angle and the angle formed by $a$ and $b$ are supplementary (on a straight line), so: $$135^\circ + (a + b) = 180^\circ$$ - Substitute $a + b = 90^\circ$: $$135^\circ + 90^\circ = 225^\circ$$ which is incorrect, so re-examine. - Actually, the 135° angle is between the vertical and one of the lines forming $a$ and $b$. The right angle between $a$ and $b$ means: $$a + b = 90^\circ$$ - The straight angle at the vertex is 180°, so the other angle adjacent to 135° is: $$180^\circ - 135^\circ = 45^\circ$$ - Since $a$ and $b$ are complementary and adjacent to the 135° angle, one of them is 45° and the other is 45° to sum to 90°. - Therefore: $$a = 45^\circ, \quad b = 45^\circ$$ --- 4. **Problem Statement for Q4:** Find the unknown angles in the quadrilateral with vertices $r, s, t,$ and one unlabeled vertex, given angles 40°, 65°, and 75° inside two triangles formed by intersecting lines. 5. **Formula and Rules:** - Sum of angles in a triangle is 180°. - Sum of angles in a quadrilateral is 360°. 6. **Step-by-step Solution for Q4:** - Consider the two triangles formed by the intersection inside the quadrilateral. - For the triangle with angles 40°, 65°, and unknown angle $x$: $$40^\circ + 65^\circ + x = 180^\circ$$ $$x = 180^\circ - 105^\circ = 75^\circ$$ - For the other triangle with angles 75°, $y$, and $z$ (unknowns), use the quadrilateral angle sum: $$40^\circ + 65^\circ + 75^\circ + y = 360^\circ$$ $$y = 360^\circ - 180^\circ = 180^\circ$$ which is impossible for a single angle, so re-examine. - Instead, use the fact that the two triangles share the intersection and the angles around the intersection sum to 360°. - The angles around the intersection are 40°, 65°, 75°, and $t$: $$40^\circ + 65^\circ + 75^\circ + t = 360^\circ$$ $$t = 360^\circ - 180^\circ = 180^\circ$$ again impossible. - Since the problem is ambiguous without a diagram, the best we can do is find the missing angle in the triangle with 40° and 65°: $$x = 75^\circ$$ **Final answers:** - Q2: $a = 45^\circ$, $b = 45^\circ$ - Q4: Missing angle in triangle with 40° and 65° is $75^\circ$.