Question

given: △gpk is equilateral; l is the mid - point of (overline{gp}). prove: △glk≅△plk 1. △gpk is equilateral. 1. given 2. definition of an equilateral triangle 3. l is the mid - point of (overline{gp}). 4. (overline{gl}congoverline{lp})

Explanation:

Step1: State given information

Given $\triangle GPK$ is equilateral and $L$ is mid - point of $\overline{GP}$.

Step2: Apply equilateral triangle property

Since $\triangle GPK$ is equilateral, by definition, all its sides are equal, so $GP = PK = GK$.

Step3: Recall mid - point definition

As $L$ is mid - point of $\overline{GP}$, by definition of mid - point, $\overline{GL}\cong\overline{LP}$.

Step4: Identify common side

The side $\overline{LK}$ is common to both $\triangle GLK$ and $\triangle PLK$, so $\overline{LK}=\overline{LK}$ by reflexive property.

Step5: Use SSS congruence

We have $\overline{GL}\cong\overline{LP}$, $\overline{LK}=\overline{LK}$ and $\overline{GK}=\overline{PK}$ (from equilateral triangle property), so by SSS congruence criterion, $\triangle GLK\cong\triangle PLK$.

Answer:

1. $\triangle GPK$ is equilateral. - Given
2. $GP = PK=GK$ - Definition of an equilateral triangle
3. $L$ is the mid - point of $\overline{GP}$. - Given
4. $\overline{GL}\cong\overline{LP}$ - Definition of a mid - point
5. $\overline{LK}=\overline{LK}$ - Reflexive property of congruence
6. $\triangle GLK\cong\triangle PLK$ - Side - Side - Side (SSS) congruence criterion