Triangles

Theorem

Angle Sum Property of a Triangle

The sum of the angles of a triangle is 1800.

Triangles

In the picture above, PQR is a triangle with angles 1, 2 and 3

Then according to the theorem

Angle 1+Angle 2 +Angle 3 =1800

 

Theorem

If a side of a triangle is produced then the exterior angle so formed is equal to the sum of two interior opposite angles.

 

triangle side

In the picture above XYZ is a triangle whose side YZ is extended to R. 1, 2, and 3 are the interior angles of a triangle. Angle 1 and Angle 2 are the interior angles opposite to the exterior angle 4

 

Then according to the theorem

Angle 4 = Angle 1+ Angle 2

 

Let us do some questions based on these theorems.

Ques. - In the figure if QT is perpendicular PR, Angle TQR = 400  and Angle SPR =300, find x and y.

perpendicular

Solution : In triangle QTR

Angle TQR +Angle QRT +Angle QTR =1800

400 + y + 900 =1800

  y =1800-1300

     = 500

Angle QSP = Angle SPR +Angle SRP

 

Reason: Exterior angle = sum of interior opposite angles

  x = 30+y

  x = 300 +500

  x =800

Ques. - In the figure below XY II MN , Angle YXZ  =350  and angle ZMN =530, find angle MZN.

http://2.bp.blogspot.com/-IXiwO073l8I/VboBLBdS4jI/AAAAAAAAAFQ/Ld34m4JwtKs/s400/4.png

Solution: We know that XY is parallel to MN.

Angle MNZ = Angle ZXY  ( alternate interior angles)

                      = 350

Now in triangle MZN

Angle ZMN +Angle MNZ +Angle MZN =1800

530 +350 +Angle MZN =1800

Angle MZN = 1800-880

Angle =920

 

Ques - In the figure given below  If PQ and RS intersect at T, such that angle PRT =400, angle RPT=950 and angle TSQ=750, find SQT.

http://2.bp.blogspot.com/-bl9-n7iAsrA/VboB4JyAqoI/AAAAAAAAAFY/mT-dgi77BTU/s400/5.png

Solution:

In triangle PRT

400+950+Angle RTP =1800

Angle RTP =1800-1350

Angle RTP =450

Angle STQ =Angle RTP ( vertically Opposite angle )

                =450

Again in triangle TQS

Angle STQ + Angle SQT + Angle TSQ = 1800 ( Angle sum property)

450+ Angle SQT +750 =1800

Angle SQT =1800-1200

Angle SQT = 600

 

Q. In the figure if PQ is perpendicular to PS, PQ II SR Angle SQR =280 and Angle QRT =650, then find the values of x and y .

http://4.bp.blogspot.com/-XZyyOq_DUek/VboCiMQ3IMI/AAAAAAAAAFg/Pv4hf4Szgj8/s400/6.png

Solution:

Since PQ II SR

Angle QRT = Angle PQR (alternate interior angles ) 

65= x +280

X= 650-280

    = 370

In triangle PQS

Angle PSQ +Angle PQS + QPS = 1800 ( angle sum property)

Y +x +900 =1800

Y+370+900=1800

Y =1800-1270

    = 530