Question

What is the first error Alycia made in her proof.

Answer 1

Alycia used a criterion that does not guarantee congruence.

Explanation:

A proof is a formal description of how we can know that something is true based on the information from a mathematical scenario.

Each step of the proof is a transformation, a construction, or a statement about what we can conclude from earlier steps.

For a two-column proof we give a reason to justify each statement. Common reasons include geometric theorems, algebraic properties, and information given in diagrams.

Let's check the reason for each step and whether Alycia established the conditions for it.

Step 1

The diagram does support the claim that OP=ST=4

Step 1 is correct

Step 2

The diagram does support the claim that OP is parallel to ST because they both have an arrow mark.

Step 2 is correct.

Step 3

<O is indeed congruent to ∠S because they are alternate interior angles between parallel lines.

Step 3 is correct.

Step 4

In this step, Alycia claims that the triangles are congruent using a "side-angle" criterion. She did establish the conditions for claiming that the figures has one pair of congruent corresponding sides and one pair of congruent corresponding angles.

~From Step 1, we know OP is congruent to ST

~From step 2, we know O is congruent to S

(Segments and angles are congruent if and only if they have equal measures.)

However, side-angle congruence is not a valid reason for claiming that two triangles are congruent, so Alycia's reason is inappropriate.

Alycia used a criterion that does not guarantee congruence.

Answer 2

Answer: Alycia only established some of the necessary conditions for a congruence criterion.

Explanation: Two triangles are congruent if their corresponding sides or/and corresponding angles are the same.

There are 5 ways to find out if the triangles are congruent.

The one Alycia used seems to be the one called Side-Angle-Side, which states that if two sides and its included angle is equal to the corresponding sides and angle of the other triangle, they are congruent.

For triangles ΔOPQ and ΔSTR to be congruent, using that theorem:

1) ∠O = ∠S

2) OP = TS

And

3) RS = OQ

So, Alycia's error is that she didn't prove RS = OQ, which would definitively conclude that both triangles are congruent.