Question

In the given figure, $\text{AP}=\text{BP}$ and $\text{CP}=\text{DP}$. Which of the following statements is true?

• A. $\Delta \mathrm{APC}\cong \Delta \mathrm{BPD}$ by $\text{ASA}$ criterion
• B. $\Delta \mathrm{APC}\cong \Delta \mathrm{BPD}$ by $\text{SSS}$ criterion
• C. $\text{AC}||\text{BD}$
• D. None of the above

Answer 1

The correct option is C $\text{AC}||\text{BD}$

In $\Delta \text{APC}$ and $\Delta \text{BPD}$

1. $\text{AP}=\text{BP}$ (given)
2. $\text{CP}=\text{PD}$ (given)
3. $\angle \text{APC}=\angle \text{BPD}$ (vertically opposite angles)

$\Delta \mathrm{APC}\cong \Delta \mathrm{BPD}$ (by $\text{SAS}$ criterion)
So, $\angle \text{CAP}=\angle \text{DBP}\left(\text{C.P.C.T.}\right)$

Hence, $\text{AC}||\text{BD}$ (alternate angles are equal)