Question

what is the value of z?
triangle diagram with points g, j, i, h; angle at i is 68°, markings indicating congruent sides
z = \boxed{}°

Explanation:

Step1: Identify triangle type (isosceles)

In $\triangle HIJ$, $HJ = HI$ (marked congruent), so it's isosceles. Thus, $\angle I = \angle HJI = 68^\circ$.

Step2: Calculate $\angle JHI$

Sum of angles in a triangle: $180^\circ$. So $\angle JHI = 180 - 68 - 68 = 44^\circ$.

Step3: Use angle bisector (or congruent angles)

Since $HJ$ and $HG$ are congruent (marked), and $HJ = HI$, $\angle GHJ = \angle JHI = z$. Wait, actually, the angles at $H$ are congruent (marked), so $z = \angle JHI$? Wait, no—wait, the two triangles $\triangle GHJ$ and $\triangle IHJ$: $HJ$ is common, $HG = HI$ (marked), $GJ = IJ$? Wait, no, the markings: the sides $HG$ and $HI$ have one mark, $GJ$ and $IJ$? Wait, the diagram: $HJ$ is a segment, $HG$ and $HI$ have one tick, $GJ$ and $IJ$? Wait, no, the angle at $H$: two angles marked equal (the red arcs). Wait, actually, the key is: in $\triangle HIJ$, $HJ = HI$ (so isosceles), angle at $I$ is $68^\circ$, so base angles? Wait, no—$HJ = HI$? Wait, the marks: the side $HI$ and $HG$ have one tick, so $HG = HI$. And $GJ$ and $IJ$? Wait, the segment $HJ$ is common. Wait, maybe $\triangle GHJ \cong \triangle IHJ$ by SSS? Because $HG = HI$, $GJ = IJ$, $HJ = HJ$. So then $\angle GHJ = \angle IHJ = z$, and in $\triangle HIJ$, angle at $I$ is $68^\circ$, sides $HJ = HI$ (wait, no—if $HG = HI$ and $GJ = IJ$, then $HJ$ is common, so SSS congruence. Then $\angle HJI = \angle HJG$, but in $\triangle HIJ$, angle at $I$ is $68^\circ$, so angle at $J$: $\angle HJI = 68^\circ$, so angle at $H$: $180 - 68 - 68 = 44^\circ$. Then since $\triangle GHJ \cong \triangle IHJ$, $\angle GHJ = \angle IHJ = z$, so $z = 44^\circ$? Wait, no—wait, the angle at $H$ in $\triangle HIJ$ is $44^\circ$, and since the two angles at $H$ ( $\angle GHJ$ and $\angle JHI$) are equal (marked), so $z = 44^\circ$? Wait, no, let's re-express:

Wait, the triangle $\triangle HIJ$: sides $HJ$ and $HI$ are equal (marked with one tick? Wait, the diagram: $HI$ has one tick, $HG$ has one tick, so $HG = HI$. $GJ$ and $IJ$: do they have ticks? The diagram shows $GJ$ and $IJ$? Wait, the original diagram: $G$---$J$---$I$ on the top, $H$ at the bottom. So $GJ$ and $IJ$: if $GJ = IJ$ (marked), $HG = HI$ (marked), $HJ = HJ$ (common). So SSS congruence: $\triangle GHJ \cong \triangle IHJ$. Therefore, corresponding angles: $\angle GHJ = \angle IHJ = z$, and $\angle HJG = \angle HJI$. In $\triangle HIJ$, angle at $I$ is $68^\circ$, angle at $J$ is $\angle HJI$, and angle at $H$ is $\angle JHI = z$. Wait, no—if $\triangle HIJ$ has $HJ = HI$ (wait, maybe I misread the marks). Let's start over:

1. In $\triangle HIJ$, the sides $HJ$ and $HI$ are marked congruent (one tick), so $HJ = HI$. Therefore, $\triangle HIJ$ is isosceles with $HJ = HI$. Thus, the base angles are $\angle I$ and $\angle HJI$. Wait, no—$HJ = HI$, so the equal sides are $HJ$ and $HI$, so the angles opposite them are $\angle I$ and $\angle HJI$. So $\angle I = \angle HJI = 68^\circ$.

2. Then, the sum of angles in $\triangle HIJ$: $\angle JHI + \angle I + \angle HJI = 180^\circ$. So $\angle JHI = 180 - 68 - 68 = 44^\circ$.

3. Now, the angle $z$ is equal to $\angle JHI$ because the two angles at $H$ ( $\angle GHJ$ and $\angle JHI$) are marked congruent (the red arcs), meaning they are equal. Alternatively, since $\triangle GHJ$ and $\triangle IHJ$ are congruent (by SSS, as $HG = HI$, $GJ = IJ$, $HJ = HJ$), their corresponding angles $\angle GHJ$ and $\angle IHJ$ (which is $z$) are equal. Wait, but in $\triangle HIJ$, $\angle IHJ$ is $44^\circ$, so $z = 44^\circ$? Wait, no—wait, $\angle JHI$ is $44^\circ$, and $z$ is that angle? Wait, the diagram: the angle labeled $z$ is $\angle JHI$, which we calculated as $44^\circ$.

Answer:

$z = \boxed{44}$