EXERCISE

$\mathrm{\Delta }\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{}\mathrm{}$ and $\mathrm{\Delta }\mathrm{D}\mathrm{B}\mathrm{C}$ are two isosceles triangles on the same base $\mathrm{B}\mathrm{C}$ and vertices $\mathrm{A}$ and $\mathrm{D}$ are on the same side of $\mathrm{B}\mathrm{C}$ as shown in figure. If $\mathrm{A}\mathrm{D}$ is extended to intersect $\mathrm{B}\mathrm{C}$ at $\mathrm{P},$ show that $∆\mathrm{ABP}\cong ∆\mathrm{ACP}$.

 

$\mathrm{\Delta }\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{}\mathrm{}$ and $\mathrm{\Delta }\mathrm{D}\mathrm{B}\mathrm{C}$ are two isosceles triangles on the same base $\mathrm{B}\mathrm{C}$ and vertices $\mathrm{A}$ and $\mathrm{D}$ are on the same side of $\mathrm{B}\mathrm{C}$ as shown in figure. If $\mathrm{A}\mathrm{D}$ is extended to intersect $\mathrm{B}\mathrm{C}$ at $\mathrm{P},$ show that $\mathrm{A}\mathrm{P}$ bisects $\mathrm{\angle }\mathrm{A}$ as well as $\mathrm{\angle }\mathrm{D}.$

$\mathrm{\Delta }\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{}\mathrm{}$ and $\mathrm{\Delta }\mathrm{D}\mathrm{B}\mathrm{C}$ are two isosceles triangles on the same base $\mathrm{B}\mathrm{C}$ and vertices $\mathrm{A}$ and $\mathrm{D}$ are on the same side of $\mathrm{B}\mathrm{C}$ as shown in figure. If $\mathrm{A}\mathrm{D}$ is extended to intersect $\mathrm{B}\mathrm{C}$ at $\mathrm{P},$ show that $\mathrm{A}\mathrm{P}$ is the perpendicular bisector of $\mathrm{B}\mathrm{C}.$

$\mathrm{A}\mathrm{D}$ is an altitude of an isosceles triangle $\mathrm{A}\mathrm{B}\mathrm{C}$ in which $\mathrm{A}\mathrm{B}=\mathrm{A}\mathrm{C}.$ Show that $\mathrm{A}\mathrm{D}$ bisects $\mathrm{\angle }\mathrm{A}$.

Two sides $\mathrm{A}\mathrm{B}$ and $\mathrm{B}\mathrm{C}$ and median $\mathrm{A}\mathrm{M}$ of one triangle $\mathrm{A}\mathrm{B}\mathrm{C}$ are respectively equal to sides $\mathrm{P}\mathrm{Q}$ and $\mathrm{Q}\mathrm{R}$ and median $\mathrm{PN}$ of $∆\mathrm{P}\mathrm{Q}\mathrm{R}$ as shown in figure. Show that $∆\mathrm{ABM}\cong ∆\mathrm{PQN}$.

 
 

Two sides $\mathrm{A}\mathrm{B}$ and $\mathrm{B}\mathrm{C}$ and median $\mathrm{A}\mathrm{M}$ of one triangle $\mathrm{A}\mathrm{B}\mathrm{C}$ are respectively equal to sides $\mathrm{P}\mathrm{Q}$ and $\mathrm{Q}\mathrm{R}$ and median $\mathrm{PN}$ of $∆\mathrm{P}\mathrm{Q}\mathrm{R}$ as shown in figure. Show that $∆\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{}\cong ∆\mathrm{P}\mathrm{Q}\mathrm{R}$

$\mathrm{B}\mathrm{E}$ and $\mathrm{C}\mathrm{F}$ are two equal altitudes of a triangle $\mathrm{A}\mathrm{B}\mathrm{C}.$ Using $\mathrm{R}\mathrm{H}\mathrm{S}$ congruence rule, prove that the triangle $\mathrm{A}\mathrm{B}\mathrm{C}$ is isosceles.

$\mathrm{A}\mathrm{B}\mathrm{C}$ is an isosceles triangle with $\mathrm{A}\mathrm{B}=\mathrm{A}\mathrm{C}.$ Draw $\mathrm{A}\mathrm{P}\perp \mathrm{BC}$ to show that $\mathrm{\angle }\mathrm{B}=\mathrm{\angle }\mathrm{C}$.