Question

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

Explanation:

Step1: Recall property of parallelogram

In a parallelogram, opposite - angles are equal. So, $\angle T=\angle V$ and $\angle S = \angle U$, and the sum of adjacent angles is $180^{\circ}$. Also, we can set up an equation using the fact that opposite - angles are equal: $4x + 9=6x-29$.

Step2: Solve the equation for $x$

Subtract $4x$ from both sides: $9 = 6x-4x - 29$. Then $9=2x - 29$. Add 29 to both sides: $2x=9 + 29=38$. Divide both sides by 2: $x = 19$.

Step3: Find the measure of an angle

Substitute $x = 19$ into the expression for $\angle U$ or $\angle S$. For $\angle U=4x + 9$, then $\angle U=4\times19+9=76 + 9=85^{\circ}$. For $\angle S=6x-29$, then $\angle S=6\times19-29=114 - 29 = 85^{\circ}$.
The adjacent angles to $\angle U$ and $\angle S$ are $\angle T$ and $\angle V$. Since the sum of adjacent angles in a parallelogram is $180^{\circ}$, if $\angle U=\angle S = 85^{\circ}$, then $\angle T=\angle V=180 - 85=95^{\circ}$. This is wrong. Let's use the fact that we want to find which angles are $91^{\circ}$. In a parallelogram, opposite angles are equal. If one angle is $91^{\circ}$, its opposite angle is also $91^{\circ}$. In parallelogram $T U V S$, $\angle T$ and $\angle V$ are opposite angles, and $\angle S$ and $\angle U$ are opposite angles.

Answer:

angles T and V