AB is a line segment and P is its mid-point. D and E are points o… - Vidyadip

Question

$AB$ is a line segment and $P$ is its mid$-$point. $D $and $E$ are points on the same side of $AB$ such that $\angle BAD = \angle ABE$ and $\angle EPA = \angle DPB.$
Show that:
$i. \triangle DAP \cong \triangle EBP$
$ii. AD = BE$

✓

Answer

Given: $AB$ is a line segment and $P$ is its mid$-$point.
$D$ and $E$ are points on the same side of $AB$
such that $ \angle BAD = \angle ABE$ and $\angle EPA = \angle DPB.$
To prove:
$i. \text{DDAP} \cong \text{DEBP}$
$ii. AD = BE$
Proof :$(ii)$
$\angle EPA = \angle DPB ...[$Given$]$
$\angle EPA + \angle EPD = \angle EPD + \angle DPB ...[$Adding $\angle EPD$ to both sides$]$
$∠APD = ∠BPE ...(1)$
In $DDAP$ and $DEBP$
$\angle DAP = \angle EBP ...[$Given$]$
$AP = BP ...[$As $P$ is the mid$-p$oint of the line $AB]$
$\angle APD = \angle BPE ...[$From $(1)]$
$\therefore DDAP \cong DEBP$ proved$ ...[\text{ASA}$ property$] ...(2)$
$(i)$ As $DDAP \cong DEBP ...[$From $(2)]$
$\therefore AD = BE ... [\text{c.p.c.t.}]$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "3 Marks Question" from Triangles→Triangles — all question types→Maths STD 9 question papers→Gujarat Board STD 9 question papers→Gujarat Board question paper generator→

Similar questions

Simplify: $\frac{1}{\sqrt{3}+\sqrt{2}}-\frac{2}{\sqrt{5}-\sqrt{3}}-\frac{3}{\sqrt{2}-\sqrt{5}}$

Write the coordinates of each of the points $P, Q, R, S, T$ and $O$ from the:

Locate $\sqrt{17}$ on the number line.

The radius of a cone is $7\ cm$ and area of curved surface is $176cm^2.$ Find the slant height.

A circus tent is cylindrical to a height of $3$ metres and conical above it. If its diameter is $105m$ and the slant height of the conical portion is $53m,$ calculate the length of the canvas 5m wide to make the required tent. $\Big(\text{Use}\ \pi=\frac{22}{7}\Big).$

Read the following bar graph and answer the following questions:
$i.$ What is the information given by the bar graph?
$ii.$ State each of the following whether true or false.
$a.$ The number of government companies in $1957$ is that of $1982$ is $1 : 9$.
$b.$ The number of government companies has decreased over the year $1957$ to $1983$.

A school provides milk to the students daily in a cylindrical glasses of diameter $7\ cm$. If the glass is filled with milk upto an height of $12\ cm$, find how many litres of milk is needed to serve $1600$ students.

Locate $\sqrt{5}$ on the number line.

Find the mean of daily wages of $40$ workers in a factory as per data given below:

Daily wages (in Rs.)$(x_i)$	$250$	$300$	$350$	$400$	$450$
Number of workes $(f_i)$	$8$	$11$	$6$	$10$	$5$

In figure, show that $AB\ ||\ EF$.