Complete the two-column proof.
DiagramTwo horizontal lines that appear to be parallel are cut by a transversal. Where the transversal intersects the top horizontal line, the angle in the top right corner is obtuse and labeled with a 2. Where the transversal intersects the bottom horizontal line, the angle in the top left corner is acute and labeled with a 1, the angle in the bottom left corner is obtuse and labeled with a 4, and the angle in the bottom right corner is acute and labeled with a 3.
Drawing not to scale.
Given: Angle 2 is congruent to angle 4, the measure of angle 2 is 110 degrees
Prove: m angle D equals 70 degrees
Statement Proof
Angle 2 is congruent to angle 4, the measure of angle 2 is 110 degrees Given
measure of angle 2 equal to measure of angle 4 Definition of congruent angles
m angle 4 equals 110 degrees a.
m angle 3 and m angle 4are a linear pair Definition of a linear pair (shown in diagram)
m angle 3 and m angle 4are supplementary Linear Pair Postulate
m angle 3 plus m angle 4 equals 180 degrees b.
m angle 3 plus 110 degrees equals 180 degrees Substitution Property
m angle D equals 70 degrees c.
m angle 3 plus 110 degrees equals 180 degrees Subtraction Property of Equality
m angle 3 equals 70 degrees Subtraction Property of Equality
m angle D equals 70 degrees Angle Addition Postulate
Statement Proof
Angle 2 is congruent to angle 4, the measure of angle 2 is 110 degrees Given
Measure of angle 2 is equal to measure of angle 4 Definition of congruent angles
Measure of angle 4 is equal to 110 degrees a.
Angles 3 and 4 are a linear pair Definition of a linear pair (shown in diagram)
Angles 3 and 4 are supplementary Linear Pair Postulate
Measure of angle 3 plus measure of angle 4 equals 180 degrees b.
Measure of angle 3 plus 110 degrees equals 180 degrees Substitution Property
Measure of angle 3 equals 70 degrees c.
Angle D is congruent to angle 3 Supplementary angles are congruent
Measure of angle D is equal to measure of angle 3 Definition of congruent angles
Measure of angle D equals 70 degrees d.
To complete the two-column proof, we need to provide a logical step-by-step explanation for each statement. Here's how we can do that:
Statement | Proof
--- | ---
Angle 2 is congruent to angle 4, the measure of angle 2 is 110 degrees | Given
The measure of angle 2 is equal to the measure of angle 4 | Definition of congruent angles
The measure of angle 4 is 110 degrees | Use substitution - we know angle 2 is 110 degrees and angle 2 is congruent to angle 4, so angle 4 must also be 110 degrees.
Angle 3 and angle 4 are a linear pair | Definition of a linear pair - a linear pair consists of adjacent angles formed when two lines intersect.
Angle 3 and angle 4 are supplementary | Linear Pair Postulate - if two angles form a linear pair, then they are supplementary, meaning their measures add up to 180 degrees.
The measure of angle 3 plus the measure of angle 4 equals 180 degrees | Definition of supplementary angles
The measure of angle 3 plus 110 degrees equals 180 degrees | Substitution Property - since angle 4 is 110 degrees (from statement 3), we can substitute it in the equation.
The measure of angle D is 70 degrees | Subtraction - subtract 110 degrees from both sides of the equation to solve for the measure of angle 3.
m angle D equals 70 degrees | Definition of angle D - since angle 3 and angle D are alternate interior angles, and they are congruent, the measure of angle D must also be 70 degrees.
By following these logical steps, we have completed the two-column proof and have shown that m angle D is equal to 70 degrees.