How can we view AP and BP given that they are on an arc?
See solution.
Practice makes perfect
Let's use the plan's step-by-step to prove the construction is valid.
Proving △ APQ ≅ △ BPQ
The two triangles share PQ as a side so we know this side is congruent by the Reflexive Property of Congruence. Additionally, A and B on the horizontal line have been marked by drawing an arc with P as the center. Therefore, PA and PB are congruent since they are the radii of that same arc.
Similarly, Q has been marked by drawing two additional identical arcs with centers in A and B. Therefore, AQ and BQ are congruent since they are the radii of identical arcs.
As △ APQ ≅ △ BPQ, we know that ∠ APQ ≅ ∠ BPQ as these are congruent corresponding angles. Examining the diagram, we also see that ∠ APQ and ∠ BPQ forms a linear pair which means they are supplementary angles:
m∠ APQ+m∠ BPQ=180^(∘)
Since these angles are congruent, they are right angles. Thus, the construction is correct.
Proof
Two-Column Proof
Let's show this as a two-column proof as well.
Statement
Reason
1.
AP≅BP, AQ≅BQ
1.
Given
2.
PQ≅ PQ
2.
Reflexive Property of Congruence
3.
△ APQ ≅ △ BPQ
3.
SSS Congruence Theorem
4.
∠ APQ ≅ ∠ BPQ
4.
Corresponding parts of congruent triangles are congruent
