Relationship between Unequal Sides of Triangle and the Angles Opposite to It.

Q.

ABC is a right angled triangle in which ∠A = 90º and AB = AC. Find ∠B and ∠C.

Q.

PQR is a triangle in which PQ = PR and S is any point on the side PQ. Through S, a line is drawn parallel to QR and intersecting PR at T. Prove that PS = PT.

Q.

ΔABC is an isosceles triangle in which AB = AC. Side BA is produced to D such that AD = AB (see the given figure). Show that ∠BCD is a right angle.

Q. Prove that sum of all interior angles of a triangle equals to ${180}^{\circ }$

Q.

$In\Delta ABC,\angle A={50}^{\circ }and\angle B={60}^{\circ }.\text{Determine the longest and shortest sides of the triangle.}$

Q.

If the bisector of the exterior vertical angle of a triangle be parallel to the base. Show that the triangle is isosceles.

Q.

In the figure, AB = AC and DB = DC, find the ratio $\angle ABD:\angle ACD.$

Q.

One of the exterior angles of a triangle is ${120}^{°}$ and the interior opposite angles are in the ratio $3:5$ Find the angles of the triangle.

Q.

$In\Delta ABC,if\angle C>\angle B,then\left(a\right)BC>AC\left(b\right)AB>AC\left(c\right)AB<AC\left(d\right)BC<AC$

Q. Question 9
In $\Delta PQR$, if $\angle R>\angle Q$, then
(A) QR > PR
(B) PQ > PR
(C) PQ < PR
(D) QR < PR

Q.

$\text{In the given figure,}\angle B<\angle Aand\angle C<\angle D.\text{Show that}AD<BC.$

Q.

In an isosceles triangle, if the vertex angle is twice the sum of the base angles, calculate the angles of the triangle.

Q.

In the given figure, $AB>AC.$ Then, which of the following is true?

(a) AB < AD

(b) AB = AD

(c) AB > AD

(d) Cannot be determined

Q.

Which of the following statements are true (T) and which are false (F):

(i) Sides opposite to equal angles of a triangle may be unequal.

(ii) Angles opposite to equal sides of a triangle are equal.

(iii) The measure of each angle of an equilateral triangle is ${60}^{\circ }$

(iv) If the altitude from one vertex of a triangle bisects the opposite side, then the triangle may be isosceles.

(v) The bisectors of two equal angles of a triangle are equal.

(vi) If the bisector of the vertical angle of a triangle bisects the base, then the triangle may be isosceles.

(vii) The two altitudes corresponding to two equal sides of the triangle need not be equal

(viii) If any two sides of a right triangle are respectively equal to two sides of another right triangle, then the two triangles are congruent.

(ix) Two right triangles are congruent if hypotenuse and a side of one triangle are respectively equal to the hypotenuse and a side of the other triangle.

Q.

If a line is drawn parallel to the base of an isosceles triangle to intersect its equal sides. Prove that the quadrilateral, so formed is cyclic.

Q.

$\text{D is any point on the side AC of}\Delta ABC\text{with AB = AC. Show that}CD<BD.$

Q. Question 8
In $\Delta PQR,\angle P={70}^{\circ }$ and $\angle R={30}^{\circ }$ which side of this triangle is the longest? Give reason for your answer.

Q. Question 5
In the given figure, PR > PQ and PS bisects $\angle$QPR. Prove that $\angle$PSR > $\angle$PSQ
.

Q. How many sides and angles does a triangle have?

Q.

If the base of an isosceles triangle is produced on both sides, prove that the exterior angles so formed are equal to each other.

Q.

For the triangle below, which option correctly gives an arrangement of its sides in ascending order of their lengths?

1. $AB,BC,AC$

2. $AB,CA,BC$

3. $AC,AB,BC$

4. $CA,BC,AB$

Q.

$\text{In the given figure, D is a point on side BC of a}\Delta ABC\text{and E is a point such that CD = DE.}\text{Prove that AB + AC}>BE.$

Q.

Prove that in a triangle, other than an equilateral triangle, angle opposite the longest side is greater than $\frac{2}{3}$ of a right angle.

Q.

$\text{In the given figure, prove that}\text{(i) CD + DA + AB}>BC\text{(ii) CD + DA + AB + BC}>2AC.$

Q. Question 6
If D is a point on the side BC of a $\Delta ABC$ such that AD bisects $\angle BAC$. Then,
(A) BD = CD
(B) BA > BD
(C) BD > BA
(D) CD > CA

Q.

If two sides of a triangle are unequal then larger side has the smaller angle opposite to it. True or False

1. True

2. False

Q.

In $\Delta ABC,\angle B={35}^{\circ },\angle C={65}^{\circ }\text{and the bisector of}\angle BAC$ meets BC in P. Arrange AP, BP and CP in descending order.

Q.

Question 5
In the given figure, PR > PQ and PS bisects $\angle$QPR. Prove that $\angle$PSR > $\angle$PSQ
.

Q.

Fill in the blanks in the following so that each of the following statements is true.

(i) Sides opposite to equal angles of a triangle are ___

(ii) Angle opposite to equal sides of a triangle are ___

(iii) In an equilateral triangle all angles are ___

(iv) In a $\Delta ABCif\angle A=\angle C,thenAB=$___

(v) If altitudes CE and BF of a triangle ABC are equal, then AB = ___

(vi) In an isosceles, triangle ABC with AB = AC, if BD and CE are its altitudes then BD is ___ CE.

(vii) In right triangles ABC and DEF, if hypotenuse AB = EF and side AC = DE, then $\Delta ABC\cong \Delta$___

Q.

$\text{In the given figure,}AB>AC.\text{If BO and CO are the bisectors of}\angle Band\angle C\text{respectively then}\left(a\right)OB=OC\left(b\right)OB>OC\left(c\right)OB<OC$