Can the following equalities be used to establish that ? (proved above) (vertically opposite angles)

Question

Homi · Accepted Answer

To prove that two triangles are congruent, we need to satisfy specific conditions like SSS (Side-Side-Side), SAS (Side-Angle-Side), ASA (Angle-Side-Angle), AAS (Angle-Angle-Side), or RHS (Right angle-Hypotenuse-Side).

Step 1 — List the given equalities

We are given a rectangle ABCD with diagonals AC and BD intersecting at point O. We are asked if the following equalities can be used to establish that $\Delta AOD \cong \Delta COB$ :

$AO = CO$ (This means side $AO$ of $\Delta AOD$ is equal to side $CO$ of $\Delta COB$ .)
$\angle AOB = \angle COD$ (These are vertically opposite angles, so they are equal.)
$AD = CB$ (This means side $AD$ of $\Delta AOD$ is equal to side $CB$ of $\Delta COB$ . Opposite sides of a rectangle are equal.)

Step 2 — Analyze the given information for $\Delta AOD$ and $\Delta COB$

Let us examine the parts we have for $\Delta AOD$ and $\Delta COB$ based on the given equalities.

We have a pair of equal sides: $AO = CO$ .
We have another pair of equal sides: $AD = CB$ .
The angle $\angle AOB$ is not an angle within $\Delta AOD$ . Similarly, $\angle COD$ is not an angle within $\Delta COB$ . Therefore, these angles cannot be directly used for the congruence of $\Delta AOD$ and $\Delta COB$ .

However, the first diagram shows that $\angle AOD$ and $\angle COB$ are vertically opposite angles. These angles are part of $\Delta AOD$ and $\Delta COB$ respectively, and they are equal. So, if we consider these relevant angles for the congruence of $\Delta AOD$ and $\Delta COB$ :

We have side $AO$ .
We have side $AD$ .
We have angle $\angle AOD$ .

For $\Delta AOD$ , the angle $\angle AOD$ is opposite to side $AD$ . It is not the angle included between sides $AO$ and $AD$ . The included angle between $AO$ and $AD$ would be $\angle DAO$ . This situation, where we have two sides and a non-included angle, is called the Side-Side-Angle (SSA) condition.

Step 3 — Conclude on congruence

The SSA condition (Side-Side-Angle, where the angle is not the included angle) is generally not a valid rule for proving triangle congruence. This means that even if two triangles have two pairs of equal sides and one pair of equal non-included angles, they are not necessarily congruent.

Therefore, the given equalities, even when considering the relevant vertically opposite angles, are not sufficient to establish that $\Delta AOD \cong \Delta COB$ .

Answer

No.

Diagram 1

Question 4

Answer

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