Notice that PR is a common side for â–³ KPR and â–³ 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 â–³ KPR and â–³ 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.
