Let's start by considering the given information and the desired outcome.
Given & ∠ QRS and ∠ PSR are supplementary.
Prove & ∠ QRL ≅ ∠ PSR
We can choose any proof format, so let's use a flow-chart proof. Here, arrows show the logical connections between the statements. Reasons are written below the statements. First, let's mark ∠ QRS, ∠ PSR, and ∠ QRL on the given diagram.
Notice that ∠ QRS and ∠ QRL form a linear pair. We can write the first statement.
Statement1
∠ QRS and ∠ QRL are a linear pair
Given on the diagram
Therefore, by the Linear Pair Postulate, we know that the measures of ∠ QRS and ∠ QRL add up to 180^(∘).
Statement2
m∠ QRS+m∠ QRL=180^(∘)
Linear Pair Postulate
We are also given that ∠ QRS and ∠ PSR are supplementary angles.
Statement3
∠ QRS and ∠ PSR are supplementary
Given
By the definition of supplementary angles, the sum of measures of these two angles is equal to 180^(∘) as well.
Statement4
m∠ QRS+m∠ PSR=180^(∘)
Definition of supplementary angles
By the Transitive Property of Equality, we can equate the left-hand sides of the equations from Statements 2 and 4.
Statement5
m∠ QRS+m∠ QRL=m∠ QRS+m∠ PSR
Transitive Property of Equality
By the Subtraction Property of Equality, we can eliminate m∠ QRS from both sides of the equation.
Statement6
m∠ QRL=m∠ PSR
Subtraction Property of Equality
Finally by the definition of congruent angles, we obtain the desired statement.
Statement7
∠ QRL≅∠ PSR
Definition of congruent angles
Completed Proof
Let's consider once again the given information and the desired outcome.
&Given ∠ QRSand∠ PSRare supplementary.
&Prove ∠ QRL≅∠ PSR
Combining the statements, we can write our final proof.
