Question

find m∠i in rhombus fghi.
i: 9s+11°
g: 7s+37°
m∠i = °

Explanation:

Step1: Recall rhombus angle property

In a rhombus, adjacent angles are supplementary (sum to \(180^\circ\)). So, \(\angle I\) and \(\angle G\) are adjacent, thus \( (9s + 11) + (7s + 37) = 180 \).

Step2: Solve for \(s\)

Combine like terms: \(9s + 7s + 11 + 37 = 180\)
\(16s + 48 = 180\)
Subtract 48: \(16s = 180 - 48 = 132\)
Divide by 16: \(s = \frac{132}{16} = \frac{33}{4} = 8.25\)? Wait, no, wait: \(180 - 48 = 132\)? Wait, 180 - 48 is 132? Wait, 48 + 132 is 180, yes. Then \(s = 132 / 16 = 8.25\)? Wait, no, wait, 16s = 132? Wait, 9s +7s is 16s, 11 +37 is 48. So 16s = 180 - 48 = 132. Then s = 132 / 16 = 8.25? Wait, but let's check again. Wait, maybe I made a mistake. Wait, 180 - 48 is 132? 48 + 132 = 180, yes. Then 16s = 132, so s = 132 / 16 = 8.25. Wait, but let's plug back. Wait, no, wait, maybe I messed up the equation. Wait, in a rhombus, adjacent angles are supplementary. So angle I and angle G are adjacent? Wait, the rhombus is F, G, H, I. So the order is F, G, H, I, so the angles at I and G: angle I is at vertex I, between sides I-F and I-H? Wait, no, the rhombus is F, G, H, I, so the sides are FG, GH, HI, IF. So angle at G is between FG and GH, angle at I is between HI and IF. So angle I and angle G: are they adjacent? Wait, in a rhombus, consecutive angles are supplementary. So angle F and angle G are consecutive, angle G and angle H, angle H and angle I, angle I and angle F. Wait, maybe the angles given are angle I (9s +11) and angle G (7s +37). So if angle I and angle G are consecutive, then they should be supplementary. So (9s +11) + (7s +37) = 180. So 16s + 48 = 180. 16s = 180 - 48 = 132. s = 132 / 16 = 8.25. Then angle I is 9s +11 = 9*(8.25) +11 = 74.25 +11 = 85.25? Wait, that can't be right. Wait, maybe I mixed up the angles. Wait, maybe angle I and angle H are equal, and angle G and angle F are equal, and consecutive angles are supplementary. Wait, no, in a rhombus, opposite angles are equal, and consecutive angles are supplementary. So angle I and angle G: are they opposite or consecutive? Wait, the rhombus is F, G, H, I. So the vertices are in order, so angle F is opposite angle H, angle G is opposite angle I? Wait, no, in a quadrilateral, opposite angles are those not adjacent. So in rhombus FGH I, the vertices are F, G, H, I, so the sides are FG, GH, HI, IF. So the angles: angle at F (between IF and FG), angle at G (between FG and GH), angle at H (between GH and HI), angle at I (between HI and IF). So opposite angles: angle F and angle H, angle G and angle I. Wait, that's the mistake! In a rhombus, opposite angles are equal. So angle G and angle I: are they opposite? Wait, no, angle F is opposite angle H, angle G is opposite angle I? Wait, no, let's label the rhombus: F connected to G and I, G connected to F and H, H connected to G and I, I connected to H and F. So the angles: angle at F: between F-G and F-I; angle at G: between G-F and G-H; angle at H: between H-G and H-I; angle at I: between I-H and I-F. So opposite angles: angle F and angle H, angle G and angle I. Wait, no, that's not right. Wait, in a rhombus, opposite angles are equal. So angle F = angle H, angle G = angle I? Wait, no, that's not correct. Wait, no, in a parallelogram (which a rhombus is), opposite angles are equal, and consecutive angles are supplementary. So angle F (at vertex F) and angle H (at vertex H) are opposite, so equal. Angle G (at vertex G) and angle I (at vertex I) are opposite, so equal. Wait, that's the mistake! So if angle G and angle I are opposite, then they should be equal. So 9s +11 = 7s +37. Then 9s -7s = 37 -11. 2s = 26. s = 13. Ah! That makes more sense. So I had the property wrong. In a rhombus, opposite angles are equal, consecutive angles are supplementary. So angle I and angle G: are they opposite? Let's see the rhombus: F, G, H, I. So the order is F, G, H, I, so the angles at G and I: G is between F and H, I is between H and F. So angle G and angle I are opposite? Wait, no, in a quadrilateral, opposite angles are those with no common side. So angle G (at G) has sides FG and GH. Angle I (at I) has sides HI and IF. So the common side between G and I? No, so they are opposite. So angle G and angle I are opposite, so they are equal. So 9s +11 = 7s +37. Then 2s = 26, so s =13. Then angle I is 9*13 +11 = 117 +11 = 128 degrees. Wait, but then consecutive angles would be supplementary. Let's check angle G: 7*13 +37 = 91 +37 = 128? Wait, no, that can't be. Wait, no, if angle G and angle I are opposite, they should be equal, but then consecutive angles would be angle F and angle G, which should be supplementary. Wait, maybe the angles given are angle I (9s +11) and angle G (7s +37) are consecutive. So then they should be supplementary. So (9s +11) + (7s +37) = 180. 16s +48 = 180. 16s = 132. s = 8.25. Then angle I is 9*8.25 +11 = 74.25 +11 = 85.25, angle G is 7*8.25 +37 = 57.75 +37 = 94.75. Then 85.25 +94.75 = 180, which works. But then opposite angles would be equal: angle F = angle H, angle I = angle H? No, wait, no, in a rhombus, opposite angles are equal. So angle F = angle H, angle G = angle I? Wait, no, that's the confusion. Let's recall: in a parallelogram (rhombus is a parallelogram), opposite angles are equal, consecutive angles are supplementary. So if angle I and angle G are consecutive (share a side), then they are supplementary. If they are opposite, they are equal. So let's look at the diagram: the rhombus is drawn with F at the top, G at the right, H at the bottom, I at the left. So the angles: at I (left) is 9s+11, at G (right) is 7s+37. So the sides: from F to G (right side), G to H (bottom right), H to I (left side), I to F (top left). So angle at I: between I-F (top left) and I-H (bottom left). Angle at G: between G-F (top right) and G-H (bottom right). So angle I and angle G: do they share a side? No, they are on opposite sides (left and right), so they are opposite angles. Therefore, they should be equal. So 9s +11 = 7s +37. Solving: 2s = 26 → s=13. Then angle I: 9*13 +11 = 117 +11 = 128. Angle G: 7*13 +37 = 91 +37 = 128. Then consecutive angles: angle F and angle G: angle F would be supplementary to angle G, so 180 - 128 = 52. Angle H would be equal to angle F, so 52. That makes sense. So the correct equation is opposite angles are equal, so 9s +11 = 7s +37.

Step1: Set opposite angles equal (rhombus property)

In a rhombus, opposite angles are equal. So \( 9s + 11 = 7s + 37 \).

Step2: Solve for \(s\)

Subtract \(7s\) from both sides: \( 2s + 11 = 37 \).
Subtract 11: \( 2s = 26 \).
Divide by 2: \( s = 13 \).

Step3: Calculate \(m\angle I\)

Substitute \(s = 13\) into \(9s + 11\):
\( 9(13) + 11 = 117 + 11 = 128 \).

Answer:

\(128\)