CONGRUENT TRIANGLES

Grade 10 enriched

By Alex Pintilie

 

Together:

 

1) Consider the triangles ABC and MNP. In each of the following, name the congruent triangles that we obtain or explain why no congruent triangles are obtained. Sketch the triangles in each case.

 

a) =50^o, =60^o, AB = 5, M=60^o, N=50^o, MN = 5

b) =50^o, AB = 5, AC= 6, M=50^o, MN = 5, MP=6

c) AB = 5, BC = 7, CA =6, MN=7, NP=6, PM = 5

d) =50^o, =60^o, AB = 5, M=60^o, P=50^o, MN = 5

e) =50^o, =60^o, C = 70^o, M=60^o, N=50^o, P = 70^o

f) =50^o, =80^o, AB = 5, M=80^o, P=50^o, MN = 5

g) =60^o, =30^o, AB = 6, M=60^o, P=30^o, MN = 3

h) =50^o, AB = 6, BC= 5, M=50^o, MN = 6, NP=5

 

2) Prove that, in an isosceles triangle, the medians corresponding to equal sides are equal.

 

3) In the following diagram, we know that AB=DC and that ABC =DCB.

a) Prove that AC = BD. (Hint: use a pair of congruent triangles.)

b) Prove that OB = OC. (Hint: use another pair of congruent triangles.)

 

 

 

 

 

 

 

 

 

 

 

The homework is on the next page.

Homework:

 

1) Consider the triangles ABC and MNP. In each of the following name the congruent triangles that we obtain or explain why no congruent triangles are obtained. Sketch the triangles in each case.

 

a) B=60^o, BC = 5, C= 60^o, M=60^o, MN = 5, P=60^o

b) AB = 10, BC = 5, CA =7, MN=10, NP=7, PM = 5

c) =70^o, =60^o, AB = 5, M=60^o, N=70^o, MN = 5

d) =80^o, =60^o, C = 40^o, M=80^o, N=60^o, P = 40^o

e) A=60^o, AB = 7, AC= 6, N=60^o, MP = 7, NP=6

f) =50^o, C=60^o, AB = 5, N=70^o, M=50^o, MN = 5

g) =90^o, =30^o, BC = 6, M=60^o, P=30^o, MN = 3

h) =50^o, =80^o, AB = 5, M=80^o, P=50^o, MN = 5

 

2) Prove that in an isosceles triangle the altitudes corresponding to the equal sides are equal.

 

3) In the following diagram PS = PT and QS = TR. Prove that PQ = PR.

 

 

 

 

 

 

4) In the given diagram, PQ = PR, PS = PT. Prove that QRS =RQT.

 

 

 

 

 

 

 

 

 

Answers: 1) a) ABC, MNP  b) ABC, NMP c) ABC, NMP d) no   e) no  f) ABC, MNP

g)  ABC, NPM  h) ABC, PMN or ABC, NMP