Question : ABCD is a quadrilateral in which BD and AC are diagonals. Then, which of the following is true:

Option 1: AB + BC + CD + DA < (AC + BD)

Option 2: AB + BC + CD + DA > (AC + BD)

Option 3: AB + BC + CD + DA = (AC + BD)

Option 4: AB + BC + CD + DA > 2(AC + BD)

Question : In $\triangle \mathrm{ABC}$, $\angle \mathrm{ABC} = 90^{\circ}$, $\mathrm{BP}$ is drawn perpendicular to $\mathrm{AC}$. If $\angle \mathrm{BAP} = 50^{\circ},$ what is the value of $\angle \mathrm{PBC}?$

Option 1: $30^{\circ}$

Option 2: $45^{\circ}$

Option 3: $50^{\circ}$

Option 4: $60^{\circ}$

Question : If ABC is an equilateral triangle and D is a point in BC such that AD is perpendicular to BC, then:

Option 1: AB : BD = 1 : 1

Option 2: AB : BD = 1 : 2

Option 3: AB : BD = 2 : 1

Option 4: AB : BD = 3 : 2

Question : $\triangle ABC$ and $\triangle PQR$ are two triangles. AB = PQ = 6 cm, BC = QR =10 cm, and AC = PR = 8 cm. If $\angle ABC = x$, then what is the value of $\angle PRQ$?

Option 1: $(180 ^{\circ}–x)$

Option 2: $x$

Option 3: $(90 ^{\circ}–x)$

Option 4: $(90 ^{\circ}+x)$

Question : $D$ is a point on the side $BC$ of a triangle $ABC$ such that $AD\perp BC$. $E$ is a point on $AD$ for which $AE:ED=5:1$. If $\angle BAD=30^{\circ}$ and $\tan \left ( \angle ACB \right )=6\tan \left ( \angle DBE \right )$, then $\angle ACB =$

Option 1: $30^{\circ}$

Option 2: $45^{\circ}$

Option 3: $60^{\circ}$

Option 4: $15^{\circ}$