Isosceles Triangle Property

Q. In $\Delta ABC$, if BC = AB and $\angle B={80}^{\circ }$ then $\angle A$ is equal to

1. ${80}^{\circ }$
2. ${40}^{\circ }$
3. ${50}^{\circ }$
4. ${100}^{\circ }$

Q. In $△ABC$, if AB =AC and $\angle B={50}^{\circ }$, then $\angle C$ is equal to

1. ${40}^{\circ }$
2. ${50}^{\circ }$
3. ${80}^{\circ }$
4. ${130}^{\circ }$

Q. In $△PQR$, if $\angle R=\angle P$, $QR=4$ cm and $PR=5$ cm, then PQ = ____.

1. 4 cm
2. 8 cm
3. 6 cm
4. 5 cm

Q.

If in the given figure, ABCD is a rectangle and DCE is an equilateral triangle, then which of the following statements is/are correct?

1. $\Delta DAE\cong \Delta CBE$
2. $\angle DAE={15}^{\circ }$
3. $\Delta AEB$ is isosceles.
4. $\Delta DAE$ is isosceles.

Q.

$\mathrm{ABC}$ is a triangle in which $\angle \mathrm{B}=2\angle \mathrm{C}$. $\mathrm{D}$ is a point on $\mathrm{BC}$ such that $\mathrm{AD}$ bisects $\angle \mathrm{BAC}$ and $\mathrm{AB}=\mathrm{CD}$. Find $\angle \mathrm{BAC}$?

Q.

In the given figure, BD = AD and DC = AC. Find the value of x.

1. 32°

2. 34°

3. 48°

Q.

Question 1 (i)
In an isosceles triangle ABC, with AB = AC, the bisectors of $\angle Band\angle Cintersect$
each other at O. Join A to O. Show that:
OB = OC

Q. Question 11
ABC and ADC are two right triangles with common hypotenuse AC. Prove that $\angle CAD=\angle CBD$.

Q. $\Delta ABC$ is an isosceles triangle with AC = BC. Find the value of 'x - y'.

1. -7
2. -4
3. 4
4. 7

Q. Show that in an isosceles triangle angles opposite to equal sides are equal

Q. State true or false.
In an equilateral triangle, a perpendicular drawn from one of the vertices to the opposite side bisects the side.

1. True
2. False

Q. In an isosceles triangle ABC, with AB = AC, the bisectors of $\angle B$ and $\angle C$ intersect each other at O. Join A to O. Which of the following is true?

1. OB = OC
2. OA = OC
3. AB = OB
4. AC = OC

Q. In triangleABC, side AB is produced to D so that BD = BC. If angle $B={60}^{\circ }$ and angle $A={70}^{\circ }$ prove that
1.AD>CD
2.AD>AC.

Q.

In Fig. AB > AC. Show that AB > AD. [4 MARKS]

Q. Question 4
P is a point on the bisector of $\angle ABC$. If the line through P, parallel to BA meet BC at Q, prove that BPQ is an isosceles triangle.

Q. ABC is a right-angled triangle in which $\angle A={90}^{\circ }$ and AB = AC. Then, $\angle B$ equals

1. $\angle C$
2. ${45}^{\circ }$
3. ${90}^{\circ }$
4. $\angle A$

Q. In a right angle triangle $\Delta ABC$, a perpendicular BD is drawn onto the largest side from the opposite vertex. Which of the following does NOT give the ratio of the areas of $\Delta ABD$and $\Delta BCD$ ?

1. ${\left(\frac{AB}{BC}\right)}^2$
2. ${\left(\frac{AB}{AC}\right)}^2$
3. ${\left(\frac{BD}{DC}\right)}^2$
4. ${\left(\frac{AD}{BD}\right)}^2$

Q. P is a point on the bisector of $\angle ABC$. If a line through P is drawn parallel to BA to meet BC at Q, then BPQ is a/an ______ triangle.

1. isosceles
2. equilateral
3. right angled
4. scalene

Q. $ABC$ is a triangle right angled at $C$. A line through the mid-point $M$ of hypotenuse $AB$ and parallel to $BC$ intersects AC at D. Then, mark the correct answer option.

1. $CM=CD$
2. $CM=MA=\frac{1}{2}AC$
3. $Noneofthese$
4. $CM=MA=\frac{1}{2}AB$

Q.

In $△$ABC if $\angle$B = $\angle$C = ${45}^{\circ }$, which of the following is the longest side?

1. AC

2. AB

3. None of these

4. BC

Q. Bisectors of the angles B and C of an isosceles triangle with AB = AC intersect each other at O. BO is produced to a point M. Then which of the following options is correct?

1. $\angle MOC=\angle BAC$
2. $\angle MOC=\angle ABC$
3. $\angle MOC=\angle MBC$
4. $\angle MOC=\angle OCB$

Q. In the given figure, $AB=AC$ and $\angle GBC=\angle HCB={90}^{\circ }$.
Which of the following statements is true?

1. BG = CB and AG = AH
2. AG = AB and BG = CH
3. $\angle BAC={120}^{\circ }$
4. BG = CH and AG = AH

Q. Construct an angle of ${45}^{\circ }$ from a horizontal line and justify the construction.

Q.

Find $x$ in the following figure:

1. ${54}^{\circ }$
2. ${42}^{\circ }$
3. ${63}^{\circ }$
4. ${73}^{\circ }$

Q. In an isosceles triangle $ABC$, with $AB=AC$, the bisectors of $\angle B$ and $\angle C$ intersect each other $O$. Join $A$ to $O$. Show that:
$OB=OC$

Q. In an isosceles triangle ABC, AB = AC and BA is produced to D, such that AB = AD. What is the value of $\angle BCD$?

1. ${70}^{\circ }$
2. ${45}^{\circ }$
3. ${60}^{\circ }$
4. ${90}^{\circ }$

Q.

In the given figure, AB = AC, $\angle A={48}^{\circ }$ and $\angle ACD={18}^{\circ }$. Then which of the given options is correct?

1. BC = BD
2. BC = CD
3. BD = DC
4. BC = 2 BD

Q. If the quadrilateral formed by joining the mid-points of the adjacent sides of quadrilateral $ABCD$ is a rectangle, show that the diagonals $AC$ and $BD$ intersect at right angle. State true or false.

1. True
2. False

Q.

In $△NVY$, if $NV=VY$ and $VY\ne YN$, then which of the following is true?

1. $\angle NVY$ = $\angle VYN$

2. $\angle NVY$ = $\angle YNV$

3. $\angle NVY$+ $2\angle VYN$ = ${180}^{\circ }$

4. $\angle VYN$ + $2\angle NVY$ = ${180}^{\circ }$

Q. In the figure, if $\frac{NT}{AB}=\frac{9}{5}$, and MB= 10 cm, then MN will be

1. $18cm$
2. $20cm$
3. $30cm$
4. $90cm$