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Homi · Accepted Answer

This diagram shows how a quadrilateral can be divided into congruent right-angled triangles.

Step 1 — Identify properties from the main figure

Let us look at the main figure, quadrilateral ABCD. The diagonals AC and BD intersect at point O. The square symbol at O means that the diagonals are perpendicular. So, $\angle AOB = \angle AOD = \angle BOC = \angle DOC = \mathbf{90^\circ}$ . Double tick marks on BO and OD mean BO equals OD. So, $\mathbf{BO = OD}$ . Single tick marks on AB and AD mean AB equals AD. So, $\mathbf{AB = AD}$ . Single tick marks on BC and CD mean BC equals CD. So, $\mathbf{BC = CD}$ .

Diagram 1

Step 2 — Prove congruence of triangles

Let us use the information to prove the congruence of the triangles shown.

First, consider $\triangle AOB$ and $\triangle AOD$ . We know that AO is a common side to both triangles. $AO = AO$ We know that $\angle AOB$ and $\angle AOD$ are both right angles. $\angle AOB = \angle AOD = 90^\circ$ We also know that BO is equal to OD from the tick marks. $BO = OD$ This is the Side-Angle-Side (SAS) congruence criterion. So, $\triangle AOB$ is congruent to $\triangle AOD$ .

$\boxed{\triangle AOB \cong \triangle AOD}$ This means their corresponding parts are equal. For example, AB = AD, which matches the single tick marks.

Next, consider $\triangle BOC$ and $\triangle DOC$ . We know that CO is a common side to both triangles. $CO = CO$ We know that $\angle BOC$ and $\angle DOC$ are both right angles. $\angle BOC = \angle DOC = 90^\circ$ We also know that BO is equal to OD from the tick marks. $BO = OD$ This is the Side-Angle-Side (SAS) congruence criterion. So, $\triangle BOC$ is congruent to $\triangle DOC$ .

$\boxed{\triangle BOC \cong \triangle DOC}$ This means their corresponding parts are equal. For example, BC = CD, which matches the single tick marks.

Diagram 2

Answer

(i) $\triangle AOB \cong \triangle AOD$ (ii) $\triangle BOC \cong \triangle DOC$

Question 18

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