Question

g.co.c.10: isosceles triangle theorem 2
1 in the diagram below of △gjk, h is a point on (overline{gj}), (overline{hj}congoverline{jk}), (mangle g = 28), and (mangle gjk=70). determine whether △ghk is an isosceles triangle and justify your answer.

Explanation:

Step1: Find $\angle GKJ$ in $\triangle GJK$

In $\triangle GJK$, by the angle - sum property of a triangle ($\angle G+\angle GJK+\angle GKJ = 180^{\circ}$), we have $\angle GKJ=180^{\circ}-\angle G - \angle GJK$. Substituting $\angle G = 28^{\circ}$ and $\angle GJK = 70^{\circ}$, we get $\angle GKJ=180 - 28-70=82^{\circ}$.

Step2: Find angles in $\triangle HJK$

Since $\overline{HJ}\cong\overline{JK}$, in $\triangle HJK$, $\angle JHK=\angle JKH$. Let $\angle JHK = \angle JKH=x$. In $\triangle HJK$, $\angle JHK+\angle JKH+\angle HJK = 180^{\circ}$. And $\angle HJK = 180 - \angle GJK=180 - 70 = 110^{\circ}$. So $x+x + 110^{\circ}=180^{\circ}$, $2x=70^{\circ}$, $x = 35^{\circ}$.

Step3: Find $\angle GHK$

$\angle GHK = 180^{\circ}-\angle JHK$. So $\angle GHK=180 - 35=145^{\circ}$.

Step4: Find $\angle GKH$

$\angle GKH=\angle GKJ-\angle JKH$. So $\angle GKH=82 - 35 = 47^{\circ}$.

Step5: Find $\angle HGK$

$\angle HGK=\angle G = 28^{\circ}$.

Step6: Check for isosceles - triangle

In $\triangle GHK$, we check if any two angles are equal. $\angle HGK = 28^{\circ}$, $\angle GKH=47^{\circ}$, $\angle GHK = 145^{\circ}$. Since no two angles are equal, $\triangle GHK$ is not an isosceles triangle.

Answer:

$\triangle GHK$ is not an isosceles triangle.