If line AB is perpendicular to line EF, and line CD is perpendicular to EF, then the measure of angle 1=the measure of angle 2. Give an arguement for the statement and a reason for each step.
There is also a diagram drawn on my homework which is a vertical line with two horizontal lines going across. Angle 1 is the interior angle on the right and below the first horizontal line. Angle 2 is the bottom right exterior angle. AB is the first horizontal line. CD is the bottom horizontal line. Point E is on the top of the vertical line and Point F is on the bottom, and together they are line EF
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sounds like it involves the fact that vertical angles are equal.
To show that the measure of angle 1 is equal to the measure of angle 2, we can use the following argument:
Step 1: Given that line AB is perpendicular to line EF, and line CD is perpendicular to EF.
Reason: Perpendicular lines form right angles.
Step 2: By definition, angle 1 is an interior angle formed by line AB and line EF.
Reason: Angle 1 is formed on the inside of the intersection of AB and EF.
Step 3: By definition, angle 2 is an exterior angle formed by line CD and line EF.
Reason: Angle 2 is formed on the outside of the intersection of CD and EF.
Step 4: Since both AB and CD are perpendicular to EF, they are parallel to each other.
Reason: Perpendicular lines are always parallel to each other.
Step 5: When a transversal (EF) intersects two parallel lines (AB and CD), the corresponding angles are congruent.
Reason: Corresponding angles formed by a transversal and two parallel lines are congruent.
Step 6: Angle 1 and angle 2 are corresponding angles.
Reason: Angle 1 and angle 2 are formed in corresponding positions when line EF intersects parallel lines AB and CD.
Step 7: Therefore, the measure of angle 1 is equal to the measure of angle 2.
Reason: Corresponding angles are congruent, so angle 1 and angle 2 have equal measures.
To prove that the measure of angle 1 equals the measure of angle 2, we can use the fact that line AB is perpendicular to line EF and line CD is perpendicular to line EF.
Argument:
Step 1: Given that line AB is perpendicular to line EF and line CD is perpendicular to line EF.
Reason: The information is given in the statement.
Step 2: When a line is perpendicular to another line, it forms right angles.
Reason: Perpendicular lines always intersect at right angles.
Step 3: Angle 1 is an interior angle formed by line AB and line EF.
Reason: Angle 1 is the angle formed inside the triangle formed by line AB and line EF.
Step 4: Angle 2 is an exterior angle formed by line CD and line EF.
Reason: Angle 2 is the angle formed outside the triangle formed by line CD and line EF.
Step 5: The sum of the measures of an exterior angle and its corresponding interior angle is always 180 degrees.
Reason: This is a property of exterior angles of a triangle.
Step 6: Therefore, the measure of angle 1 + the measure of angle 2 = 180 degrees.
Reason: By applying the property stated in step 5.
Step 7: Substituting the measures of angle 1 and angle 2 in the equation, we have: the measure of angle 1 + the measure of angle 2 = 180 degrees.
Reason: Algebraic substitution.
Step 8: Since the measure of angle 1 + the measure of angle 2 = 180 degrees, it implies that the measure of angle 1 is equal to the measure of angle 2.
Reason: Transitive property of equality.
Conclusion:
Therefore, we have reasoned and demonstrated that the measure of angle 1 is equal to the measure of angle 2.