Question

find the value of t in rhombus hijk.
h
2t-24°
k
i
t
j
t =
°
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work it out
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Explanation:

Step1: Recall rhombus angle property

In a rhombus, adjacent angles are supplementary (sum to \(180^\circ\))? Wait, no, actually in a rhombus, opposite angles are equal, and adjacent angles are supplementary. Wait, looking at the diagram, angle \(H\) is \(2t - 24^\circ\) and angle \(I\) is \(t\). Wait, maybe they are adjacent? Wait, no, in a rhombus, consecutive angles (adjacent) are supplementary. Wait, but maybe angle \(H\) and angle \(I\) are adjacent? Wait, no, let's check the labels: \(H\), \(I\), \(J\), \(K\). So the vertices are \(H\), \(I\), \(J\), \(K\) in order. So angle at \(H\) and angle at \(I\) are adjacent? Wait, no, \(H\) to \(I\) to \(J\) to \(K\) to \(H\). So angle at \(H\) (between \(K\) and \(I\)) and angle at \(I\) (between \(H\) and \(J\)) are adjacent. Wait, but in a rhombus, adjacent angles are supplementary? Wait, no, actually in a parallelogram (and rhombus is a parallelogram), consecutive angles are supplementary. Wait, but also, in a rhombus, opposite angles are equal. Wait, maybe angle \(H\) and angle \(J\) are equal, angle \(I\) and angle \(K\) are equal. But in the diagram, we have angle \(H = 2t - 24^\circ\) and angle \(I = t\). Wait, maybe angle \(H\) and angle \(I\) are supplementary? Wait, no, maybe I made a mistake. Wait, no, in a rhombus, adjacent angles are supplementary. Wait, but let's re - examine. Wait, the problem: in rhombus \(HIJK\), angle at \(H\) is \(2t - 24^\circ\), angle at \(I\) is \(t\). Wait, maybe they are adjacent? Wait, no, maybe they are opposite? Wait, no, \(H\) and \(I\) are adjacent vertices. Wait, no, \(H\) is connected to \(K\) and \(I\), \(I\) is connected to \(H\) and \(J\). So angle at \(H\) (between \(K\) and \(I\)) and angle at \(I\) (between \(H\) and \(J\)) are adjacent. Wait, but in a parallelogram, consecutive angles are supplementary. Wait, but maybe the diagram shows that angle \(H\) and angle \(I\) are supplementary? Wait, no, maybe I got it wrong. Wait, no, let's think again. Wait, in a rhombus, opposite angles are equal. So if angle \(H\) and angle \(J\) are equal, angle \(I\) and angle \(K\) are equal. But the problem gives angle \(H = 2t - 24\) and angle \(I = t\). Wait, maybe angle \(H\) and angle \(I\) are supplementary? Wait, no, maybe they are equal? Wait, that can't be. Wait, maybe the diagram is such that angle \(H\) and angle \(I\) are adjacent and supplementary? Wait, no, let's check the sum. Wait, maybe the problem is that in a rhombus, adjacent angles are supplementary. Wait, no, in a parallelogram, consecutive angles are supplementary. So if \(HIJK\) is a rhombus (hence a parallelogram), then angle \(H\) + angle \(I=180^\circ\)? Wait, no, that would be if they are consecutive. Wait, but let's check the equation. Wait, maybe the angles at \(H\) and \(I\) are supplementary. So \( (2t - 24)+t = 180\)? Wait, no, that would be if they are supplementary. Wait, but let's solve that: \(3t-24 = 180\), \(3t=204\), \(t = 68\). But wait, maybe they are equal? Wait, no, if they are opposite angles, then \(2t - 24=t\), which would give \(t = 24\), but that doesn't make sense. Wait, maybe I misread the diagram. Wait, the diagram: angle at \(H\) is \(2t - 24^\circ\), angle at \(I\) is \(t\). Let's assume that angle \(H\) and angle \(I\) are adjacent and supplementary. Wait, no, in a rhombus, adjacent angles are supplementary. So let's set up the equation: \(2t - 24 + t=180\)? Wait, no, that would be if they are supplementary. Wait, but let's check again. Wait, maybe the angles are equal? Wait, no, in a rhombus, opposite angles are equal. So if angle \(H\) and angle \(J\) are equal, angle \(I\) and angle \(K\) are equal. But the problem gives angle \(H\) and angle \(I\). Wait, maybe the diagram is labeled such that angle \(H\) and angle \(I\) are opposite? No, \(H\) and \(I\) are adjacent vertices. Wait, maybe the problem has a typo, but more likely, I made a mistake. Wait, wait, in a rhombus, all sides are equal, and opposite angles are equal. Wait, maybe angle \(H\) and angle \(I\) are supplementary? Wait, no, in a parallelogram, consecutive angles are supplementary. So if \(HIJK\) is a parallelogram (rhombus is a parallelogram), then angle \(H\) + angle \(I = 180^\circ\) (if they are consecutive). Wait, let's try that. So \(2t-24 + t=180\). Then \(3t=204\), \(t = 68\). But wait, let's check the other way. Wait, maybe angle \(H\) and angle \(I\) are equal? Then \(2t - 24=t\), so \(t = 24\). But then angle \(H=2*24 - 24 = 24^\circ\), angle \(I = 24^\circ\), and then the other angles would be \(156^\circ\) each. But is that possible? Wait, no, in a rhombus, adjacent angles are supplementary. So if angle \(H\) is \(24^\circ\), adjacent angle should be \(156^\circ\), not \(24^\circ\). So my first assumption was wrong. Wait, maybe angle \(H\) and angle \(I\) are supplementary. Wait, let's re - examine the diagram. The rhombus is labeled \(H\), \(I\), \(J\), \(K\). So the order is \(H - I - J - K - H\). So angle at \(H\) is between \(K\) and \(I\), angle at \(I\) is between \(H\) and \(J\). So they are adjacent angles, so they should be supplementary. So \( (2t - 24)+t=180\)? Wait, no, that would be if they are supplementary. Wait, but let's solve \(2t-24 + t=180\): \(3t=204\), \(t = 68\). But then angle \(H=2*68 - 24=136 - 24 = 112^\circ\), angle \(I = 68^\circ\), and \(112 + 68=180\), which works for supplementary angles. So that must be it. So the correct equation is \(2t - 24 + t=180\)? Wait, no, wait, adjacent angles in a parallelogram are supplementary. So yes, so \(2t - 24 + t=180\)? Wait, no, \(2t - 24\) and \(t\) are adjacent angles, so their sum is \(180^\circ\). So:

Step2: Set up the equation

\(2t-24 + t=180\)
Combine like terms: \(3t-24 = 180\)

Step3: Solve for t

Add 24 to both sides: \(3t=180 + 24=204\)
Divide both sides by 3: \(t=\frac{204}{3}=68\)

Wait, but let's check again. Wait, maybe the angles are equal? No, because adjacent angles in a parallelogram are supplementary. So if angle \(H\) is \(2t - 24\) and angle \(I\) is \(t\), and they are adjacent, then \(2t - 24+t = 180\), which gives \(t = 68\). Let's verify: angle \(H=2*68 - 24=136 - 24 = 112^\circ\), angle \(I = 68^\circ\), and \(112+68 = 180^\circ\), which satisfies the supplementary condition for adjacent angles in a parallelogram (rhombus). So that works.

Answer:

\(t = 68\)