Question

figure tuvs is a parallelogram. which angles equal 91°? (4x + 9)° (6x - 29)° angles t and v angles s and u angles u and v angles s and t

Explanation:

Step1: Recall parallelogram angle property

In a parallelogram, opposite angles are equal. So, angle \( U \) and angle \( V \) (wait, no, looking at the figure, angle \( U \) is \( (4x + 9)^\circ \) and angle \( V \) is \( (6x - 29)^\circ \)? Wait, no, in a parallelogram, consecutive angles are supplementary, but opposite angles are equal. Wait, actually, in a parallelogram, opposite angles are equal. Wait, the figure is TUV S, so sides TU and SV are opposite, TV and SU are opposite? Wait, no, the angles at U and V: wait, no, in a parallelogram, opposite angles are equal. Wait, the angles given are \( (4x + 9)^\circ \) at U and \( (6x - 29)^\circ \) at V? Wait, no, maybe U and V are adjacent? Wait, no, maybe U and S are opposite? Wait, no, let's correct. In a parallelogram, opposite angles are equal. So angle U and angle S? No, wait, the figure: T connected to U and S, U connected to T and V, V connected to U and S, S connected to V and T. So TUV S is a parallelogram, so TU is parallel to SV, and TV is parallel to SU. So angle U and angle S are consecutive? Wait, no, angle at U: \( (4x + 9)^\circ \), angle at V: \( (6x - 29)^\circ \). Wait, in a parallelogram, opposite angles are equal. Wait, maybe angle U and angle V are not opposite. Wait, maybe the two angles given are opposite angles? Wait, no, in a parallelogram, opposite angles are equal, so \( 4x + 9 = 6x - 29 \). Let's solve for x.

Step2: Solve for x

Set \( 4x + 9 = 6x - 29 \)
Subtract \( 4x \) from both sides: \( 9 = 2x - 29 \)
Add 29 to both sides: \( 38 = 2x \)
Divide by 2: \( x = 19 \)

Step3: Calculate angle measure

Now plug x = 19 into \( 4x + 9 \): \( 4(19) + 9 = 76 + 9 = 85 \)? Wait, that can't be. Wait, maybe I made a mistake. Wait, maybe the angles are consecutive angles? No, consecutive angles in a parallelogram are supplementary. Wait, maybe the two angles are adjacent? No, wait, maybe the angles at U and V are adjacent? Wait, no, maybe the angles given are opposite angles. Wait, no, maybe I misread the figure. Wait, the figure is a parallelogram, so angle T and angle V are opposite, angle U and angle S are opposite? Wait, no, let's re-express. Let's assume that angle U and angle S are opposite? No, the angles given are at U: \( (4x + 9)^\circ \) and at V: \( (6x - 29)^\circ \). Wait, maybe the two angles are adjacent and supplementary? Wait, no, in a parallelogram, consecutive angles are supplementary. Wait, maybe I messed up the property. Wait, in a parallelogram, opposite angles are equal, so if angle U and angle S are opposite, but the angles given are at U and V. Wait, maybe the figure is a parallelogram, so TU is parallel to SV, and TV is parallel to SU. So angle at U: between TU and UV, angle at V: between UV and VS. So TU is parallel to VS, so angle U and angle V are same-side interior angles, hence supplementary. Wait, no, same-side interior angles are supplementary. Wait, but the problem says "which angles equal 91°". Wait, maybe my initial assumption is wrong. Wait, let's solve \( 4x + 9 = 6x - 29 \):

\( 4x + 9 = 6x - 29 \)

\( 9 + 29 = 6x - 4x \)

\( 38 = 2x \)

\( x = 19 \)

Then angle U: \( 4(19) + 9 = 85 \), angle V: \( 6(19) - 29 = 114 - 29 = 85 \). Wait, that's not 91. Wait, maybe the angles are consecutive and supplementary? Wait, no, maybe I made a mistake in the angle labels. Wait, maybe angle T and angle U are consecutive, and angle T and angle V are opposite? Wait, the options are angles T and V, S and U, U and V, S and T.

Wait, let's re-express. Let's suppose that in parallelogram TUV S, angle T and angle V are opposite, angle U and angle S are opposite. So angle T = angle V, angle U = angle S.

Wait, maybe the two angles given are angle U and angle S? No, the angles given are \( (4x + 9)^\circ \) at U and \( (6x - 29)^\circ \) at V. Wait, maybe the problem is that the two angles are adjacent and equal? No, in a parallelogram, adjacent angles are supplementary. Wait, maybe the figure is a rectangle? No, it's a parallelogram. Wait, maybe I made a mistake in the equation. Wait, maybe the two angles are opposite, so \( 4x + 9 = 6x - 29 \), but that gives 85. But the question is which angles equal 91. Wait, maybe I messed up the angle labels. Wait, maybe angle U and angle V are not the ones given. Wait, maybe the angles at T and V are equal, or S and U. Wait, let's try another approach.

Wait, maybe the two angles given are consecutive angles, so they are supplementary? No, consecutive angles in a parallelogram are supplementary, so \( (4x + 9) + (6x - 29) = 180 \)

\( 10x - 20 = 180 \)

\( 10x = 200 \)

\( x = 20 \)

Then angle U: \( 4(20) + 9 = 89 \), angle V: \( 6(20) - 29 = 120 - 29 = 91 \). Ah! There we go. So I made a mistake earlier, thinking they were opposite, but they are consecutive angles? No, wait, consecutive angles are supplementary, so their sum is 180. So solving \( (4x + 9) + (6x - 29) = 180 \) gives x = 20. Then angle V is \( 6(20) - 29 = 91^\circ \). Now, in a parallelogram, opposite angles are equal. So angle V is equal to angle T, and angle U is equal to angle S. Wait, angle V is 91°, so angle T is also 91°. Let's check the options:

Options:

- angles T and V: if angle T = angle V, then yes, because they are opposite angles.

Wait, let's confirm. If x = 20, angle U: 4*20 +9=89, angle V:6*20 -29=91. Then angle T is opposite angle V, so angle T = angle V =91°, angle S is opposite angle U, so angle S=angle U=89°. So angles T and V are equal to 91°.

Step4: Verify the options

So angles T and V are equal (opposite angles in parallelogram), and each is 91°. Let's check the options:

- angles T and V: correct, because they are opposite angles, so equal, and we found angle V is 91°, so angle T is also 91°.

- angles S and U: angle S = angle U =89°, not 91°.

- angles U and V: 89° and 91°, not equal.

- angles S and T: 89° and 91°, not equal.

So the correct option is angles T and V.

Answer:

angles T and V