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

Question

Homi · Accepted Answer

Understand the Question

Let the intersecting line segments $AC$ and $BD$ meet at point $O$ .
We use the properties of intersecting lines and triangles to identify equal parts:
1. Vertically opposite angles at intersection point $O$ are equal ( $\angle AOB = \angle DOC$ ).
2. In $\triangle BOC$ , equal base angles ( $\angle OCB = \angle OBC$ ) mean their opposite sides are equal ( $OB = OC$ ).
3. Using the Angle-Side-Angle (ASA) congruence criterion, $\triangle BOA \cong \triangle COD$ , which gives corresponding equal sides and angles.

Step 1 · Identify Vertically Opposite Angles

Let the intersection point of segments $AC$ and $BD$ be $O$ . Diagram 1

Since lines $AC$ and $BD$ intersect at point $O$ , vertically opposite angles are equal: $\angle AOB = \angle DOC$

Step 2 · Find Equal Sides in $\triangle BOC$

In $\triangle BOC$ , we are given: $\angle ACB = \angle DBC \implies \angle OCB = \angle OBC$

In a triangle, sides opposite to equal angles are equal: $OB = OC$

Step 3 · Prove Triangle Congruence by ASA Criterion

In $\triangle BOA$ and $\triangle COD$ :

By the Angle-Side-Angle (ASA) congruence criterion: $\triangle BOA \cong \triangle COD$

Since corresponding parts of congruent triangles (CPCTC) are equal: $AO = DO$ $AB = CD$

Answer

The equal parts in the figure are:

$\angle AOB = \angle DOC$ (Vertically opposite angles)
$OB = OC$ (Sides opposite to equal angles in $\triangle BOC$ )
$AO = DO$ and $AB = CD$ (Corresponding parts of congruent triangles $\triangle BOA \cong \triangle COD$ )

Common Mistakes

Overlooking the Sub-triangle: Not noticing that $\angle ACB = \angle DBC$ applies to the base angles of $\triangle BOC$ , which proves $OB = OC$ .
Wrong Congruence Criterion: Confusing the order of equal components and incorrectly applying SAS instead of ASA (the equal side $OB = OC$ is included between the two equal pairs of angles).

Question 13