Question: 3) Find the measure of the given angle. M P N L R 3 Find $m\angle LPN$ if $\angle 3$ is a right angle and $\angle MPN \cong \angle RPN$. DRAW TO HELP FOR #9 AND 10 9) If $RS = 12$, $RT = 5$, $ST = 7$, and points $R$, $S$, and $T$ are collinear, what is the relationship between points $R$, $S$, and $T$? ○ Point $R$ is between point $S$ and point $T$. ○ Point $S$ is between point $R$ and point $T$. ○ Point $T$ is between point $R$ and point $S$. ○ There is not enough information to determine the relationship.

1. **State the problem:** We need to find the measure of $m\angle LPN$ given that $\angle 3$ is a right angle and $\angle MPN \cong \angle RPN$. 2. **Analyze the given information:** - $\angle 3$ is a right angle, so $m\angle 3 = 90^\circ$. - $\angle MPN \cong \angle RPN$ means these two angles have equal measures. 3. **Identify the angles around point $P$:** - The angles around point $P$ formed by rays $PM$, $PN$, $PR$, and $PL$ sum to $360^\circ$. - Since $\angle 3$ is between $PL$ and $PR$ and is $90^\circ$, the remaining angles around $P$ are $\angle MPN$, $\angle RPN$, and $\angle LPN$. 4. **Use the congruence of angles:** - $\angle MPN \cong \angle RPN$ means $m\angle MPN = m\angle RPN = x$ (say). 5. **Express the sum of angles around $P$:** $$ m\angle MPN + m\angle RPN + m\angle LPN + m\angle 3 = 360^\circ $$ Substitute known values: $$ x + x + m\angle LPN + 90^\circ = 360^\circ $$ $$ 2x + m\angle LPN = 270^\circ $$ 6. **Find $m\angle LPN$ in terms of $x$:** $$ m\angle LPN = 270^\circ - 2x $$ 7. **Use the fact that $\angle LPN$ is adjacent to $\angle MPN$ and $\angle RPN$:** - Since $\angle MPN$ and $\angle RPN$ are adjacent and equal, and $\angle LPN$ is the angle between $PN$ and $PL$, the problem implies $m\angle LPN$ is the sum of $\angle MPN$ and $\angle RPN$. 8. **Therefore, $m\angle LPN = m\angle MPN + m\angle RPN = x + x = 2x$.** 9. **From step 6 and 8, set equal:** $$ m\angle LPN = 270^\circ - 2x = 2x $$ 10. **Solve for $x$:** $$ 270^\circ = 4x $$ $$ x = \frac{270^\circ}{4} = 67.5^\circ $$ 11. **Find $m\angle LPN$:** $$ m\angle LPN = 2x = 2 \times 67.5^\circ = 135^\circ $$ **Final answer:** $$ \boxed{135^\circ} $$