Question

Prove OPR=QRP using a paragraph proof. (Write out the words for symbols. For example, write "Z" as "angle" and "x" as

"congruent to.")

Answer 1

`\triangle OPR \cong \triangle QRP` as per AAS congruence.

Explanation:

We are given two triangles
`\triangle OPR \ and\ \triangle QRP`.

Side OP is parallel to Side QR.

OP || QR

`\angle O = \angle Q`

To prove:

The two triangles are congruent i.e.
`\triangle OPR \cong \triangle QRP`

Solution:

Let us have a look at the triangles:

`\triangle OPR \ and\ \triangle QRP`

1. Given that
`\angle O = \angle Q`.

2. Side OP || RQ so,
`\angle OPR = \angle QRP` because they are the alternate angles of parallel sides. (Alternate angels are equal when a line cuts two parallel sides).

3. Side PR is common to both the sides i.e. PR = RP

Hence, by AAS congruence i.e. two angles and a side which is not between the two angles are same.

`\therefore \triangle OPR \cong \triangle QRP` as per AAS congruence.