Angles - Test Questions I

Question 1

Question:   In two right triangles, an acute angle and a side of one are equal to the the angle and corresponding side of the other. Prove that the triangles are congruent.

Answer:    Given:
In triangles ABC and DEF,

BC = EF

To prove:

Proof:

In triangles ABC and DEF.

BC = EF ( given)

Question 2

Question:   AB is a line segment AX and BY are two equal line segments drawn on opposite sides of line AB such that AX||BY. If line segment AB and XY intersect each other at the point P, then prove that

ii) a line segments AB and XY bisect each other at P.

Answer:    Given:
AB is a line segment
AX||BY
AB and XY intersect at P
To prove:

ii) AP=PB and PY=PX
Proof:

Compare triangles APX and BPY,
(given)




AP = PB

( corresponding parts of congruent triangles)
PY = PX

Question 3

Question:  

Answer:    Given:
AC=BC


To prove:

Proof:
In triangles DBC and EAC,
BC=AC (given)
(given)
From the figure,



From the figure,

From (i) and (ii),

Question 4

Question:   In the adjoining figure, AB ^ BD, DE ^ BD, BC = CD and

Answer:    Given:

BC=CD

To prove:

Proof:

Compare triangles ABC and ECD,

BC=CD ( given)
(given)
(given)

Question 5

Question:   In the adjoining figure, AB||CD. BC and AD are transversals intersecting at O such that OA=OD. Show that

Answer:    Given:
AB||CD.
AD and BC intersect at O.
OA = OD.
To prove:

Proof:

In triangles AOB and COD,
OA=OD (given)
( alternate angles)
( vertically opposite angles)

Question 6

Question:   In D PQR, P=40^o, Q=65^o and R=75^o. Name the greatest side and the least side.

Answer:   
(theorem on inequalities)

(theorem on inequalities)

Question 7

Question:   In D XYZ, XY=6.5cm, YZ=5.5cm and XZ=7.5cm. Name the greatest and smallest angle.

Answer:   
(theorem on inequalities)

(theorem on inequalities)

Question 8

Question:   In the adjoining figure PQR is an equilateral triangle. N is any point on PQ. Show that (i) NR>PN (ii) NR>NQ.

Answer:    Given:
In D PQR, PQ=PR=QR
To prove:
(i) NR>PN
(ii) NR>NQ
Proof:
In D PQR,

PQ=PR=QR ( given)
( Equilateral triangle is equiangular)
In D PNR,

( is only a part of PRQ)
( given)
NR>PN (theorem on inequalities)
In DNQR,


NR>NQ (theorem on inequalities)

Question 9

Question:   In a quadrilateral ABCD, AB=AD and BC>CD. Prove that . (Hint: Join BD)

Answer:    Given:

To prove:

Construction:
Join BD.
Proof:
In DBC,
BC>CD ( given)
...(i) (theorem on inequalities)
In D ABD, AB = AD ( given)
...(ii)
By adding (i) and (ii), we get


(adding equal thing to both sides
does not alter inequation)

Question 10

Question:   In the adjoining figure D ABC, D is the mid point of BC and AD>BD or DC. Prove that .

Answer:    Given:
In D ABC, D is the mid-point of BC. AD>BD.
To prove:

Proof:
AD>BD ( given)
.....(i) (theorem on inequalities)
AD>DC ( given)
.....(ii) (theorem on inequalities)
By adding (i) and (ii), we get