Notice that PR is a common side for △ KPR and △ MRP. Could you show that these triangles are congruent? After that, notice that at L there is a pair of vertical angles.
Statements
Reasons
1.
∠ K ≅ ∠ M and KP⊥ PR and MR⊥ PR
1.
Given
2.
m∠ KPR =90^(∘) =m∠ MRP
2.
Definition of perpendicular segments
3.
∠ KPR ≅ ∠ MRP
3.
Definition of congruent angles
4.
PR ≅ PR
4.
Reflexive Property of Congruent Segments
5.
△ KPR ≅ △ MRP
5.
Angle-Angle-Side (AAS) Congruence Postulate
6.
KP ≅ MR
6.
Definition of congruent polygons
7.
∠ KLP ≅ ∠ MLR
7.
Vertical Angles Theorem
8.
△ KLP ≅ △ MLR
8.
Angle-Angle-Side (AAS) Congruence Postulate
9.
∠ KPL ≅ ∠ MRL
9.
Definition of congruent polygons
Practice makes perfect
In the given diagram, we can see the triangles △ KPR and △ MRP. Let's separate them, but notice that they have a side in common, PR.
Besides, since KP⊥ PR and MR⊥ PR, we have that ∠ KPR and ∠ MRP are both right angles. This means that ∠ KPR ≅ ∠ MRP. Remember that ∠ K ≅ ∠ M.
cc
∠ K ≅ ∠ M & Angle
∠ KPR ≅ ∠ MRP & Angle
PR ≅ PR & Non-included Side
Consequently, by the Angle-Angle-Side (AAS) Congruence Postulate we have that △ KPR ≅ △ MRP. This implies that KP ≅ MR. Knowing this, let's consider the two triangles below.
Notice that ∠ KLP and ∠ MLR are vertical angles, so by the Vertical Angles Theorem we obtain ∠ KLP ≅ ∠ MLR.
cc
∠ KLP ≅ ∠ MLR & Angle
∠ K ≅ ∠ M & Angle
KP ≅ MR & Non-included Side
Once again, we apply the Angle-Angle-Side (AAS) Congruence Postulate to obtain that △ KLP ≅ △ MLR. Consequently, by definition we get ∠ KPL ≅ ∠ MRL.
Two-Column Proof Table
In the following table we summarize the proof we did before.
