If Two Angles of a Triangle Are Congruent Then the Sides Opposite to Them Are Congruent

Q.

In an isosceles triangle ABC, with AB = AC, the bisectors of ∠B and ∠C intersect each other at O. Join A to O. Show that:

(i) OB = OC (ii) AO bisects ∠A

Q.

In triangle ABC, E and F are the mid points of the sides AC and AB respectively. The altitude AP to BC intersects EF at Q. Then

1. AQ < QP

2. AQ = QP

3. AQ > Q P

4. AQ ≅ QP

Q. In the figure below, the sides AB and AC are congruent, then the angles at B and C are _______ .

1. Never equal
2. Always equal
3. 30 and 40 degrees always
4. None of the above

Q. D is the midpoint of base BC. DE and DF are perpendiculars to AB and AC respectively such that DE=DF. Prove that angle B=angle C

Q. In the figures given below, some congruent parts are indicated by identical marks.
(1) Write the test of congruence by which the pairs of triangle in each figure is congruent.
(2) Write proper correspondence between the vertices of triangle while writing the names of triangle.
(3) Also write the remaining congruent parts of the congruent triangle.

(i)

Q.

The three angles of a triangle are equal to the three corresponding angles of another triangle. Are the triangles necessarily congruent?

Q.

If two sides and the included angle of a triangle are respectively equal to two sides and the included angle of the other triangle, the two triangles are congruent. State whether it is true or false.

1. True

2. False

Q. In the given figure, AB > AC. If BO and CO are the bisectors of ∠B and ∠C respectively, then
(a) OB < OC
(b) OB = OC
(c) OB > OC

Q. ABC is a triangle and D is the mid-point of BC. The perpendiculars from D to AB and AC are equal. Prove that the triangle is-isosceles.

Q. In an isosceles ∆ABC, AB = AC and the bisectors of are joint.
Prove that: (i) OB = OC (ii) ∠OAB = ∠OAC

Q. ABC is a triangle in which BE and CF are, respectively, the perpendiculars to the sides AC and AB. If BE = CF, prove that ΔABC is isosceles.

Q. In the given figure, ABC is a triangle in which AB = AC. D is a point in the interior of ∆ABC such that ∠DBC = ∠DCB. Prove that AD bisects ∠BAC.
Figure

Q. In the given figure, ABC is a triangle right-angled at B such that ∠BCA = 2 ∠BAC. Show that AC = 2BC.

Q. In the figure given below , $\angle ABC=\angle ACB={47}^{\circ }$ and AC = 9 m . What is the length of AB ?

1. 4 m
2. 10 m
3. 9 m
4. 7 m

Q. $\text{If}\Delta ABC\cong \Delta PQR\text{and it is given that}\angle A={60}^{\circ },\angle B={70}^{\circ }.\text{Then the value of}\angle R\text{is}$___ ${}^{\circ }$.

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

Q. In the figure below, $\angle ABC=\angle ACB$. So, the length of AB = length of ____.

1. None of the above
2. BC
3. AC
4. AX

Q. The figure below is an isosceles triangle in which the two sides AB and AC are equal. If $\angle ABC={65}^{\circ }$, what is the value of $\angle ACB$ ?

1. ${45}^{\circ }$
2. ${75}^{\circ }$
3. ${65}^{\circ }$
4. ${85}^{\circ }$