Identify the equal parts in the following figure, given that and .

Question

Homi · Accepted Answer

We will use the given equal angles to find congruent triangles and their matching parts.

Step 1 — Name the intersection point Let us call the point where lines AC and BD cross each other O.

Diagram 1

Step 2 — Identify vertically opposite angles When two straight lines cross, they form pairs of angles opposite to each other. These are called vertically opposite angles. The angles $\angle AOB$ and $\angle DOC$ are vertically opposite angles. Vertically opposite angles are always equal. So, we know that $\angle AOB = \angle DOC$ .

Step 3 — Find equal segments from given angles We are given that $\angle ACB = \angle DBC$ . Let us look at the small triangle $\triangle BOC$ . In $\triangle BOC$ , the angle at C is $\angle OCB$ . This is the same as $\angle ACB$ . The angle at B is $\angle OBC$ . This is the same as $\angle DBC$ . Since $\angle ACB = \angle DBC$ , it means $\angle OCB = \angle OBC$ . When two angles in a triangle are equal, the sides opposite to these angles are also equal. The side opposite to $\angle OCB$ is OB. The side opposite to $\angle OBC$ is OC. So, we know that OB = OC.

Step 4 — Prove triangle congruence Now let us look at the small triangles $\triangle BOA$ and $\triangle COD$ . We have three pieces of information to compare them:

We are given that $\angle ABD = \angle DCA$ . This means that $\angle OBA = \angle OCD$ . (This is an Angle)
We just found that OB = OC in Step 3. (This is a Side)
We found that $\angle BOA = \angle DOC$ in Step 2. (This is an Angle)

Since we have an Angle-Side-Angle (ASA) match ( $\angle OBA$ , side OB, $\angle BOA$ for $\triangle BOA$ and $\angle OCD$ , side OC, $\angle DOC$ for $\triangle COD$ ), the two triangles are congruent. So, $\triangle BOA \cong \triangle COD$ by the ASA condition. When triangles are congruent, all their corresponding parts are equal. From this congruence, the side AO in $\triangle BOA$ corresponds to the side DO in $\triangle COD$ . So, we know that AO = DO.

Answer

The equal parts in the figure are:

(i) $\angle AOB = \angle DOC$ (Vertically opposite angles) (ii) AO = DO (iii) CO = BO (iv) $\triangle COD \cong \triangle BOA$ by the Angle-Side-Angle (ASA) condition.

Question 13

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