Try These Solutions for Lines and Angles

Question 1: If two lines are cut by a transversal, and the alternate angles are equal, prove that the lines are parallel.
To prove that the lines are parallel, we can use the converse of the alternate interior angles theorem. If alternate angles are equal, then the lines are parallel. We can show this by using the properties of alternate angles and the definition of parallel lines.

Proof:
Given: Two lines are cut by a transversal, and the alternate angles are equal.
To Prove: The lines are parallel.
- Given that the alternate angles are equal, we can say that the lines are parallel.
- By the definition of alternate angles, if two lines are cut by a transversal and the alternate angles are equal, then the lines are parallel.
- Therefore, the lines are parallel.
Thus, we have proved that if two lines are cut by a transversal, and the alternate angles are equal, then the lines are parallel.

Question 2: In a triangle, if one angle is equal to the sum of the other two angles, prove that the triangle is a right-angled triangle.
To prove that the triangle is a right-angled triangle, we can use the property of a right-angled triangle where one angle is equal to the sum of the other two angles. We can show this by using the properties of angles in a triangle.

Proof:
Given: In a triangle, one angle is equal to the sum of the other two angles.
To Prove: The triangle is a right-angled triangle.
- Let the angles of the triangle be A, B, and C, where angle A is equal to the sum of angles B and C.
- According to the property of a right-angled triangle, the sum of the angles in a triangle is 180 degrees.
- If angle A is equal to the sum of angles B and C, then angle A must be 90 degrees.
- Therefore, the triangle is a right-angled triangle.
Thus, we have proved that in a triangle, if one angle is equal to the sum of the other two angles, then the triangle is a right-angled triangle.