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Draw several diagrams to help you decide whether each statement is true or false. If it is false, show a counterexample. It is true, draw and label a diagram you could use in a proof. List, in terms of the diagram, what is given and what is to be proved. Do not write a proof.

If two triangles have equal perimeters, then they have congruent sides.

Short Answer

Expert verified

If a triangle has two congruent sides, then the angles opposite those sides are congruent istrue.

Step by step solution

01

Step 1. Draw the diagram.

Draw a diagram of two triangles with equal perimeters.

In above figure, there is one equilateralΔABC and a right-angled triangle ΔPQR.

02

Step 2. Verify if the perimeters are equal.

To verify if: Perimeter of ΔABC= Perimeter of ΔPQR

The Perimeter of equilateral triangle ΔABC=3a, wherea=4

Perimeter of ΔABC=34=12

Perimeter of right triangleΔPQR=a+b+c, where a=3,b=4andc=5

Perimeter of ΔPQR=3+4+5=12

03

Step 3. Prove if AB≅PQ, BC≅QR and AC≅PR is true or not. 

So according to above definitions

Perimeter of ΔABC= Perimeter of ΔPQR

But sides BCQRand ACPR.

Therefore, if a triangle has equal perimeters, then they have congruent sides is false.

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