Question

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

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

Step-by-step solution

Step 1. Draw the diagram.

Draw a diagram of two triangles with equal perimeters.

In above figure, there is one equilateral$\Delta ABC$ and a right-angled triangle $\Delta PQR$.

Step 2. Verify if the perimeters are equal.

To verify if: $\text{Perimeter\hspace{0.33em}of\hspace{0.33em}}\Delta ABC=\text{\hspace{0.33em}Perimeter\hspace{0.33em}of\hspace{0.33em}}\Delta PQR$

The Perimeter of equilateral triangle $\Delta ABC=3a$, where$a=4$

$\begin{array}{c}\text{Perimeter\hspace{0.33em}of\hspace{0.33em}}\Delta ABC=3\left(4\right)\\ =12\end{array}$

$\text{Perimeter\hspace{0.33em}of\hspace{0.33em}right\hspace{0.33em}triangle}\Delta PQR=a+b+c,\text{\hspace{0.33em}where\hspace{0.33em}}a=3,b=4$and$c=5$

$\begin{array}{c}\text{Perimeter\hspace{0.33em}of\hspace{0.33em}}\Delta PQR=3+4+5\\ =12\end{array}$

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

So according to above definitions

$\text{Perimeter\hspace{0.33em}of\hspace{0.33em}}\Delta ABC=\text{\hspace{0.33em}Perimeter\hspace{0.33em}of\hspace{0.33em}}\Delta PQR$

But sides $BC\ne QR$and $AC\ne PR$.

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