The two-column proof that proves that ∠O ≅ ∠R using the SSS and CPCTC theorems is shown in the image attached below.
<h3>What is the Side-side-side Congruence Theorem (SSS)?</h3>The side-side-side congruence theorem (SSS) states that when three sides of a triangle are congruent to the corresponding three sides of another triangle, then both triangles are considered congruent to each other.
<h3>What is the CPCTC Theorem?</h3>The CPCTC theorem states that when two triangles are proven to be congruent by any triangle congruent theorem, then all the corresponding parts of the congruent triangle are congruent. This implies that the corresponding sides and angles of both triangle are equal or congruent.
We are asked to prove that ∠O ≅ ∠R.
With the given image, we would draw segment UP. Thus, based on the reflexive property, UP ≅ UP (one pair of congruent sides).
We are given two pairs of corresponding sides already, which are: UO ≅ UR and PO ≅ PR.
This means that triangles OUP and RUP have three pairs of corresponding congruent sides. Therefore, ΔOUP ≅ ΔRUP.
Applying the CPCTC theorem, ∠O ≅ ∠R.
The two-column proof is attached in the image below.
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