Segments MN, LT, PQ, and OR all intersect at point S. If m∠LSO = m∠PSN , prove m∠LSP = m∠OSN .
write a two column proof
Assuming that the 1st letters are in sequence around S, the angles follow
LSO=LSP+PSO
PSN=PSO+OSN
That means
LSP+PSO=PSO+OSN
so
LSP=OSN
Now just change that to a 2-column proof.
Statement | Reason
---------|-------
1. m∠LSO = m∠PSN | Given
2. (∠LSO + ∠LSP) + (∠PSN + ∠OSN) = 180° | Linear pair postulate (the interior angles on the same side of the transversal sum to 180°)
3. ∠LSO + ∠PSN = ∠LSP + ∠OSN | Addition property of equality
4. ∠LSP + ∠OSN = ∠LSP + ∠OSN | Identity property of equality
5. ∠LSO + ∠PSN = ∠LSO + ∠PSN | Substitution property of equality
6. ∠LSP + ∠OSN = ∠LSO + ∠PSN | Transitive property of equality
7. m∠LSP = m∠OSN | Subtraction property of equality
To write a two-column proof for proving that m∠LSP = m∠OSN, we will use the properties of intersecting lines and the given angle equality.
Here is a two-column proof:
| Statement | Reason |
|-------------------------------------|-------------------------------------------------|
| 1. m∠LSO = m∠PSN | Given |
| 2. ∠LSO ≅ ∠PSN | Angle equality implies congruent angles |
| 3. ∠LSO ≅ ∠OSN | Corresponding angles formed by parallel lines |
| 4. m∠LSP = m∠OSN | Congruent angles have equal measures |
Explanation:
1. The given statement states that the measure of ∠LSO is equal to the measure of ∠PSN.
2. By the Angle Equality Postulate, when two angles have the same measure, they are congruent. Therefore, ∠LSO and ∠PSN are congruent.
3. The third statement is true because when a transversal intersects two parallel lines, corresponding angles are congruent. In this case, the transversal S intersects parallel lines LS and PN, forming congruent angles ∠LSO and ∠OSN.
4. Finally, since ∠LSP and ∠OSN are corresponding angles and ∠LSO and ∠PSN are congruent, we can conclude that ∠LSP and ∠OSN have equal measures. Therefore, m∠LSP = m∠OSN.
Note: The statements and reasons in a two-column proof may vary depending on the specific approach and theorem used.