This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
here are the solutions to the problems. Question 7.1 Step 1: Identify the relationship between the angles. Angles CEF and EFB are consecutive interior angles (also known as co-interior angles) because they are between the parallel lines AB and CD and on the same side of the transversal line EF. When two parallel lines are intersected by a transversal, consecutive interior angles are supplementary, meaning their sum is 180^. Step 2: Set up the equation and solve for a. CEF + EFB = 180^ (5a + 40^) + (a + 100^) = 180^ 6a + 140^ = 180^ 6a = 180^ - 140^ 6a = 40^ a = (40)/(6) a = (20)/(3) The value of a is (20)/(3). Question 7.2 Given TS PR. We need to find the values of a, b, and c. From the diagram: c = SQR a = QSR b = SRQ The angle 154^ is PQR. The angle 48^ is USQ. Step 1: Find the value of c. Angles PQR and SQR are angles on a straight line PQR. Therefore, their sum is 180^. PQR + SQR = 180^ 154^ + c = 180^ c = 180^ - 154^ c = 26^ Step 2: Use parallel lines to find relationships between a and b. Since TS PR and SQ is a transversal, TSQ and SQR are alternate interior angles. TSQ = SQR TSQ = c TSQ = 26^ Now, consider the transversal SR. Since TS PR, TSR and SRQ are alternate interior angles. TSR = SRQ From the diagram, TSR is composed of TSQ and QSR. TSR = TSQ + QSR TSR = 26^ + a So, we have the equation: b = 26^ + a (1) Step 3: Use the sum of angles in SQR. The sum of angles in a triangle is 180^. QSR + SRQ + SQR = 180^ a + b + c = 180^ Substitute the value of c = 26^: a + b + 26^ = 180^ a + b = 180^ - 26^ a + b = 154^ (2) Step 4: Solve the system of equations for a and b. Substitute equation (1) into equation (2): a + (26^ + a) = 154^ 2a + 26^ = 154^ 2a = 154^ - 26^ 2a = 128^ a = (128^)/(2) a = 64^ Now substitute the value of a back into equation (1) to find b: b = 26^ + 64^ b = 90^ (Note: The angle 48^ is extraneous information not needed to solve for a, b, c based on the standard geometric theorems and parallel line properties.) The values are: a = 64^ b = 90^ c = 26^ Drop the next question.