CIS. Year 8. Geometry. Angles on Parallel Lines - Maths with David - Free Printable
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Step-by-step solution for: CIS. Year 8. Geometry. Angles on Parallel Lines - Maths with David
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Step-by-step solution for: CIS. Year 8. Geometry. Angles on Parallel Lines - Maths with David
I will solve each of the 16 geometry problems (A1 through C4) by analyzing the diagrams and applying geometric principles like the properties of parallel lines, triangles, and angles. I'll start with problem A1 and work my way through systematically.
- First, look closely: The diagram shows two parallel lines cut by a transversal, forming a triangle. We are given angles of 72° and 113°, and need to find x and y.
- Next, find information: I need to use the fact that the sum of angles in a triangle is 180° and properties of parallel lines.
- Then, review the findings:
- The angle adjacent to 113° on the straight line is 180° - 113° = 67°.
- In the triangle, we have angles 72°, 67°, and x. So, x = 180° - 72° - 67° = 41°.
- Angle y is vertically opposite to the 67° angle, so y = 67°.
Answer for A1: x = 41°, y = 67°
---
- First, look closely: Two parallel lines cut by two transversals. Given angles are 64° and 76°.
- Next, find information: Use alternate interior angles and corresponding angles.
- Then, review the findings:
- Angle y is alternate interior to 64°, so y = 64°.
- Angle x is corresponding to 76°, so x = 76°.
Answer for A2: x = 76°, y = 64°
---
- First, look closely: Two parallel lines cut by a transversal, with angles 83° and 71° given.
- Next, find information: Use corresponding angles and linear pairs.
- Then, review the findings:
- Angle x is vertically opposite to 83°, so x = 83°.
- Angle y is corresponding to 71°, so y = 71°.
Answer for A3: x = 83°, y = 71°
---
- First, look closely: Two parallel lines cut by a transversal, with 53° given.
- Next, find information: Use corresponding angles, vertical angles, and linear pairs.
- Then, review the findings:
- Angle x is corresponding to 53°, so x = 53°.
- Angle y is vertically opposite to x, so y = 53°.
- Angle z is supplementary to y, so z = 180° - 53° = 127°.
- Angle w is vertically opposite to z, so w = 127°.
Answer for A4: x = 53°, y = 53°, w = 127°, z = 127°
---
- First, look closely: Triangle with angles 105° and 54°, and x at the top.
- Next, find information: Sum of angles in a triangle is 180°.
- Then, review the findings:
- x = 180° - 105° - 54° = 21°.
Answer for B1: x = 21°
---
- First, look closely: Two parallel lines cut by two transversals, forming a triangle with angles 48° and 73°.
- Next, find information: Use the fact that the sum of angles in a triangle is 180°.
- Then, review the findings:
- x = 180° - 48° - 73° = 59°.
Answer for B2: x = 59°
---
- First, look closely: Two parallel lines cut by a transversal, with angles 127° and 55°.
- Next, find information: Use corresponding angles and linear pairs.
- Then, review the findings:
- The angle adjacent to 127° is 180° - 127° = 53°.
- x is vertically opposite to this 53° angle, so x = 53°.
Answer for B3: x = 53°
---
- First, look closely: Two parallel lines cut by a transversal, with angles 142° and 101°.
- Next, find information: Use linear pairs and corresponding angles.
- Then, review the findings:
- Angle x is supplementary to 142°, so x = 180° - 142° = 38°.
- Angle y is supplementary to 101°, so y = 180° - 101° = 79°.
Answer for B4: x = 38°, y = 79°
---
- First, look closely: Triangle with angles 73° and 34°, and x and y inside.
- Next, find information: Use the fact that the sum of angles in a triangle is 180° and properties of parallel lines.
- Then, review the findings:
- The third angle of the triangle is 180° - 73° - 34° = 73°.
- Since the lines are parallel, y is alternate interior to 34°, so y = 34°.
- x is part of the 73° angle, but since it's a straight line, x = 180° - 73° - 34° = 73° (same as the third angle).
Wait, let me reconsider:
- The angle at the vertex with x and y is split into x and y.
- The total angle at that vertex is 73° (since the triangle has angles 73°, 34°, and 73°).
- But x and y are not parts of the same angle; they are separate angles formed by the parallel lines.
- Actually, y is alternate interior to 34°, so y = 34°.
- x is the remaining angle in the triangle at that vertex, which is 73°.
But looking again, x is the angle between the two lines, which should be 73°, and y is the alternate interior angle to 34°, so y = 34°.
Answer for C1: x = 73°, y = 34°
---
- First, look closely: Two parallel lines cut by a transversal, with angles 65° and 121°.
- Next, find information: Use corresponding angles and linear pairs.
- Then, review the findings:
- The angle adjacent to 121° is 180° - 121° = 59°.
- x is corresponding to 65°, but wait, let's see the diagram.
- Actually, x is the angle between the two lines, and we have a triangle with angles 65° and 59°.
- So x = 180° - 65° - 59° = 56°.
Answer for C2: x = 56°
---
- First, look closely: Two parallel lines cut by a transversal, with angles 133° and 143°.
- Next, find information: Use corresponding angles and linear pairs.
- Then, review the findings:
- The angle adjacent to 133° is 180° - 133° = 47°.
- The angle adjacent to 143° is 180° - 143° = 37°.
- x is the angle between these two lines, so x = 180° - 47° - 37° = 96°.
Answer for C3: x = 96°
---
- First, look closely: Two parallel lines cut by a transversal, with angles 327° and 71°.
- Next, find information: 327° is a reflex angle, so the actual angle is 360° - 327° = 33°.
- Then, review the findings:
- The angle at the vertex is 33°.
- x is the angle between the two lines, and we have a triangle with angles 33° and 71°.
- So x = 180° - 33° - 71° = 76°.
Answer for C4: x = 76°
---
- A1: x = 41°, y = 67°
- A2: x = 76°, y = 64°
- A3: x = 83°, y = 71°
- A4: x = 53°, y = 53°, w = 127°, z = 127°
- B1: x = 21°
- B2: x = 59°
- B3: x = 53°
- B4: x = 38°, y = 79°
- C1: x = 73°, y = 34°
- C2: x = 56°
- C3: x = 96°
- C4: x = 76°
These solutions are based on standard geometric principles including the sum of angles in a triangle, properties of parallel lines (corresponding, alternate interior, and vertically opposite angles), and linear pairs.
Problem A1: Find the values of x and y
- First, look closely: The diagram shows two parallel lines cut by a transversal, forming a triangle. We are given angles of 72° and 113°, and need to find x and y.
- Next, find information: I need to use the fact that the sum of angles in a triangle is 180° and properties of parallel lines.
- Then, review the findings:
- The angle adjacent to 113° on the straight line is 180° - 113° = 67°.
- In the triangle, we have angles 72°, 67°, and x. So, x = 180° - 72° - 67° = 41°.
- Angle y is vertically opposite to the 67° angle, so y = 67°.
Answer for A1: x = 41°, y = 67°
---
Problem A2: Find the values of x and y
- First, look closely: Two parallel lines cut by two transversals. Given angles are 64° and 76°.
- Next, find information: Use alternate interior angles and corresponding angles.
- Then, review the findings:
- Angle y is alternate interior to 64°, so y = 64°.
- Angle x is corresponding to 76°, so x = 76°.
Answer for A2: x = 76°, y = 64°
---
Problem A3: Find the values of x and y
- First, look closely: Two parallel lines cut by a transversal, with angles 83° and 71° given.
- Next, find information: Use corresponding angles and linear pairs.
- Then, review the findings:
- Angle x is vertically opposite to 83°, so x = 83°.
- Angle y is corresponding to 71°, so y = 71°.
Answer for A3: x = 83°, y = 71°
---
Problem A4: Find the values of x, y, w, and z
- First, look closely: Two parallel lines cut by a transversal, with 53° given.
- Next, find information: Use corresponding angles, vertical angles, and linear pairs.
- Then, review the findings:
- Angle x is corresponding to 53°, so x = 53°.
- Angle y is vertically opposite to x, so y = 53°.
- Angle z is supplementary to y, so z = 180° - 53° = 127°.
- Angle w is vertically opposite to z, so w = 127°.
Answer for A4: x = 53°, y = 53°, w = 127°, z = 127°
---
Problem B1: Find the value of x
- First, look closely: Triangle with angles 105° and 54°, and x at the top.
- Next, find information: Sum of angles in a triangle is 180°.
- Then, review the findings:
- x = 180° - 105° - 54° = 21°.
Answer for B1: x = 21°
---
Problem B2: Find the value of x
- First, look closely: Two parallel lines cut by two transversals, forming a triangle with angles 48° and 73°.
- Next, find information: Use the fact that the sum of angles in a triangle is 180°.
- Then, review the findings:
- x = 180° - 48° - 73° = 59°.
Answer for B2: x = 59°
---
Problem B3: Find the value of x
- First, look closely: Two parallel lines cut by a transversal, with angles 127° and 55°.
- Next, find information: Use corresponding angles and linear pairs.
- Then, review the findings:
- The angle adjacent to 127° is 180° - 127° = 53°.
- x is vertically opposite to this 53° angle, so x = 53°.
Answer for B3: x = 53°
---
Problem B4: Find the values of x and y
- First, look closely: Two parallel lines cut by a transversal, with angles 142° and 101°.
- Next, find information: Use linear pairs and corresponding angles.
- Then, review the findings:
- Angle x is supplementary to 142°, so x = 180° - 142° = 38°.
- Angle y is supplementary to 101°, so y = 180° - 101° = 79°.
Answer for B4: x = 38°, y = 79°
---
Problem C1: Find the values of x and y
- First, look closely: Triangle with angles 73° and 34°, and x and y inside.
- Next, find information: Use the fact that the sum of angles in a triangle is 180° and properties of parallel lines.
- Then, review the findings:
- The third angle of the triangle is 180° - 73° - 34° = 73°.
- Since the lines are parallel, y is alternate interior to 34°, so y = 34°.
- x is part of the 73° angle, but since it's a straight line, x = 180° - 73° - 34° = 73° (same as the third angle).
Wait, let me reconsider:
- The angle at the vertex with x and y is split into x and y.
- The total angle at that vertex is 73° (since the triangle has angles 73°, 34°, and 73°).
- But x and y are not parts of the same angle; they are separate angles formed by the parallel lines.
- Actually, y is alternate interior to 34°, so y = 34°.
- x is the remaining angle in the triangle at that vertex, which is 73°.
But looking again, x is the angle between the two lines, which should be 73°, and y is the alternate interior angle to 34°, so y = 34°.
Answer for C1: x = 73°, y = 34°
---
Problem C2: Find the value of x
- First, look closely: Two parallel lines cut by a transversal, with angles 65° and 121°.
- Next, find information: Use corresponding angles and linear pairs.
- Then, review the findings:
- The angle adjacent to 121° is 180° - 121° = 59°.
- x is corresponding to 65°, but wait, let's see the diagram.
- Actually, x is the angle between the two lines, and we have a triangle with angles 65° and 59°.
- So x = 180° - 65° - 59° = 56°.
Answer for C2: x = 56°
---
Problem C3: Find the value of x
- First, look closely: Two parallel lines cut by a transversal, with angles 133° and 143°.
- Next, find information: Use corresponding angles and linear pairs.
- Then, review the findings:
- The angle adjacent to 133° is 180° - 133° = 47°.
- The angle adjacent to 143° is 180° - 143° = 37°.
- x is the angle between these two lines, so x = 180° - 47° - 37° = 96°.
Answer for C3: x = 96°
---
Problem C4: Find the value of x
- First, look closely: Two parallel lines cut by a transversal, with angles 327° and 71°.
- Next, find information: 327° is a reflex angle, so the actual angle is 360° - 327° = 33°.
- Then, review the findings:
- The angle at the vertex is 33°.
- x is the angle between the two lines, and we have a triangle with angles 33° and 71°.
- So x = 180° - 33° - 71° = 76°.
Answer for C4: x = 76°
---
Final Answers:
- A1: x = 41°, y = 67°
- A2: x = 76°, y = 64°
- A3: x = 83°, y = 71°
- A4: x = 53°, y = 53°, w = 127°, z = 127°
- B1: x = 21°
- B2: x = 59°
- B3: x = 53°
- B4: x = 38°, y = 79°
- C1: x = 73°, y = 34°
- C2: x = 56°
- C3: x = 96°
- C4: x = 76°
These solutions are based on standard geometric principles including the sum of angles in a triangle, properties of parallel lines (corresponding, alternate interior, and vertically opposite angles), and linear pairs.
Parent Tip: Review the logic above to help your child master the concept of angles and parallel lines worksheet.
