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 equals 70 degrees Subtraction Property of Equality
m angle D equals 70 degrees Definition of congruent angles (angle D is congruent to angle 3)
Let's complete the two-column proof:
Statement Proof
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1. Angle 2 is congruent to angle 4, the measure of angle 2 is 110 degrees Given
2. Measure of angle 2 equal to measure of angle 4 Definition of congruent angles
3. Measure of angle 4 equals 110 degrees Algebraic substitution (Using statement 1)
4. Angle 3 and angle 4 are a linear pair Definition of a linear pair (shown in diagram)
5. Angle 3 and angle 4 are supplementary Linear Pair Postulate
6. Measure of angle 3 plus measure of angle 4 equals 180 degrees Definition of supplementary angles (Using statement 5)
7. Measure of angle 3 plus 110 degrees equals 180 degrees Algebraic substitution (Using statement 3)
8. Measure of angle 3 equals 70 degrees Simplification (Subtracting 110 from both sides)
9. Angle D is congruent to angle 3 Definition of congruent angles
10. Measure of angle D equals measure of angle 3 Definition of congruent angles
11. Measure of angle D equals 70 degrees Algebraic substitution (Using statement 8)