1. Solve for the unknown values with reasons: **a)** Given angles 40°, 5x, and 65° produced by two parallel lines and a transversal. 1. Angles on a straight line sum to 180°. 2. So, $$40 + 5x + 65 = 180$$. 3. Simplify: $$105 + 5x = 180$$. 4. Subtract 105 from both sides: $$5x = 75$$. 5. Divide both sides by 5: $$x = 15$$. **b)** Angle at B is 35°, and x is the unknown angle adjacent. 1. Adjacent angles on a straight line sum to 180°. 2. So, $$x + 35 = 180$$. 3. Solve for $$x$$: $$x = 145$$. **c)** Angles given: $$3x - 15°$$ and $$45°$$. 1. If these are corresponding or vertically opposite angles, they are equal: $$3x - 15 = 45$$. 2. Add 15 to both sides: $$3x = 60$$. 3. Divide by 3: $$x = 20$$. **d)** Angles in quadrilateral sum to 360°. 1. Given angles: $$z$$, 94°, $$x$$, $$y$$, and 38°. 2. Sum equation: $$z + 94 + x + y + 38 = 360$$. 3. Simplify: $$z + x + y + 132 = 360$$. 4. So, $$z + x + y = 228$$. 5. Additional info needed to solve individually. **e)** At point O, angles around sum to 360°. 1. Given angles $$2x$$, $$y$$, $$x$$, and 30°. 2. Equation: $$2x + y + x + 30 = 360$$. 3. Simplify: $$3x + y = 330$$. 4. More info needed to solve both. **f)** Horizontal line A-B with angles $$x + 15°$$ and 45° at different points. 1. If angles are alternate interior or supplementary, then $$x + 15 = 45$$. 2. Solve: $$x = 30$$. **g)** Given $$SU = ST$$, segments labeled $$z$$, $$4y$$, and $$x$$ with angle 50°. 1. Triangle properties or segment equalities can be used. 2. Without a diagram, can't solve for all variables explicitly. **h)** Angles $$2x$$ and 160° given. 1. If supplementary, $$2x + 160 = 180$$. 2. Solve: $$2x = 20$$ and $$x = 10$$. 2. Express in terms of x: **a)** Angle $$DÊC$$ is vertical or supplementary angle related to angle $$x$$. 1. If vertical opposite, $$DÊC = x$$. 2. If supplementary, $$DÊC = 180 - x$$. **b)** Angle $$G L K$$ is related to $$x$$ by triangle or linear pair. 1. Using triangle sum or line properties, $$G L K = 180 - x$$ (assuming adjacent angles). Final answers: a) $$x=15$$ b) $$x=145$$ c) $$x=20$$ d) $$z + x + y = 228$$ (needs more info) e) $$3x + y = 330$$ (needs more info) f) $$x=30$$ h) $$x=10$$ 2a) $$DÊC = x$$ or $$180 - x$$ depending on position 2b) $$G L K = 180 - x$$