The diagram shows a rhombus ABCD. ABQ, PBC and DRQ are straight lines and AQ = BP.

(a)
State the triangle that is similar to triangle BRQ.

Answer Triangle ...
(b)
Prove that triangle DQA is congruent to triangle APB.

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Worked solution

14(b)

1. Identify the two corresponding equal sides.

Since $ABCD$ is a rhombus,
$AD = AB.$
Also, it is given that
$AQ = BP.$

2. Show the included angles are equal and conclude congruence.

Since $AD \parallel BC$ and $P, B, C$ are collinear, $AD \parallel BP$ .

Since $A, B, Q$ are collinear, $AQ$ lies on the same straight line as $AB$ .

Therefore,
$\angle DAQ = \angle ABP.$

Hence, with $AD = AB$ , $AQ = BP$ and the included angle equal,
$\triangle DQA \cong \triangle APB \quad \text{(SAS).}$

Answer

In the rhombus $ABCD$ , $AD = AB$ .

Given that $AQ = BP$ .

Since $AD \parallel BC$ and $P, B, C$ are collinear, $AD \parallel BP$ .
Also, since $A, B, Q$ are collinear, $AQ$ and $AB$ are on the same straight line.

Hence,
$\angle DAQ = \angle ABP$ .

Therefore, by SAS congruence,
$\triangle DQA \cong \triangle APB.$

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The diagram shows a rhombus ABCD. ABQ, PBC and DRQ are straight lines and AQ = BP.

Worked solution

14(b)

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