One of the angles formed by two intersecting lines is 50° less than the other. Find the measure of all angles. Write the solution in statement-reason format.
Answer (1)
To solve this problem, we need to find the measures of the angles formed by two intersecting lines. We'll denote one of these angles as x . According to the given information, the other angle is 50° less than x , so we can denote it as x − 50° .
When two lines intersect, they form two pairs of vertically opposite angles that are equal. Vertically opposite angles are the same, and the sum of the angles around a point is 360°.
Solution in Statement-Reason Format:
Statement : Let one angle be x .
Reason : To represent one of the angles formed by the intersecting lines.
Statement : The second angle is x − 50° .
Reason : As per the problem, one angle is 50° less than the other.
Statement : The sum of the angles at a point is 360°.
Reason : Basic property of angles around a point.
Statement : The vertically opposite angles are equal, so two pairs of x and x − 50° angles are formed.
Statement : x + ( x − 50° ) + x + ( x − 50° ) = 360° .
Reason : Sum of all angles around a point.
Statement : Simplify the equation:
4 x − 100° = 360°
Statement : Add 100° to both sides to solve for x :
4 x = 460°
Statement : Divide both sides by 4 to find x :
x = 115°
Reason : Solve for x .
Statement : The other angle is x − 50° = 115° − 50° = 65° .
Reason : Find the second angle using the relation given.
Conclusion:
The measures of the angles are 115° and 65°. The other two angles formed are also 115° and 65° since they are vertically opposite angles.
Thus, the angles formed by the intersecting lines are 115°, 65°, 115°, and 65°.
