We are asked to prove Theorem 6.16. Let's focus on the two triangles on the two sides of one of the diagonals.
All four sides of a rhombus are congruent. This allows us to write several congruence relations between pairs of segments. Let's choose only two of these, keeping in mind that we would like to show that ∠ 1 is congruent to ∠ 2 and ∠ 3 is congruent to ∠ 4.
R N&≅R Q
P N&≅P Q
Since R P is common side of △ R N P and △ R Q P, we now know that these two triangles have three pairs of congruent sides. The coloring indicates corresponding vertices in the triangles. According to the Side-Side-Side (SSS) Congruence Postulate, we can conclude that these triangles are congruent.
△ R N P≅△ R Q P
This congruence implies the congruence of corresponding angles.
∠ 1&≅ ∠ 2
∠ 3&≅ ∠ 4
These two congruence relations prove that diagonal RP bisects both ∠ NPQ and ∠ NRQ. Let's focus now on the other diagonal.
As above, we can show that triangles △ Q R N and △ Q P N are congruent, so the corresponding angles are congruent.
∠ 5&≅ ∠ 6
∠ 7&≅ ∠ 8
We can summarize the process above in a paragraph proof.
Completed Proof
2
&Given:&& NPQR is a rhombus.
&Prove:&& ∠ 1≅ ∠ 2, ∠ 3≅ ∠ 4
& && ∠ 5≅ ∠ 6, ∠ 7≅ ∠ 8
Proof:
In a rhombus all sides are congruent. Using SSS, this means that △ RNP and △ RQP are congruent isosceles triangles with a common base. Corresponding angles of these congruent triangles are congruent, so diagonal RP bisects both ∠ NPQ and ∠ NRQ. This means that ∠ 1≅∠ 2 and ∠ 3≅∠ 4. Diagonal NQ also cuts the rhombus to two congruent isosceles triangles, so it bisects angles ∠ RNP and ∠ RQP. This means that ∠ 5≅∠ 6 and ∠ 7≅∠ 8.
