quadrilateral stuv is a kite. what is m∠t?
v
u
97°
s
121°
t
m∠t = \boxed{}°

Question

Accepted Answer

# Explanation:
## Step1: Recall kite angle property
In a kite, one pair of opposite angles (between unequal sides) are equal, and the sum of interior angles of a quadrilateral is $360^\circ$. Here, $\angle S$ and $\angle U$ are not the equal pair (since $ \angle S = 121^\circ$, $ \angle U = 97^\circ$), so $\angle T$ and $\angle V$ are not the pair, wait—actually, in a kite, the angles between the unequal sides: wait, no, the correct property is that one pair of opposite angles (the ones between the two distinct pairs of adjacent equal sides) are equal? Wait, no, the sum of interior angles of a quadrilateral is $360^\circ$. Let's denote $ \angle T = x$, $ \angle V = y$. But in a kite, one pair of opposite angles are equal? Wait, no, actually, in a kite, two distinct pairs of adjacent sides are equal. So in kite $STUV$, $SV = ST$ (double marks) and $VU = UT$ (single marks). So the angles between the unequal sides: $\angle S$ and $\angle U$ are not equal, but $\angle T$ and $\angle V$? Wait, no, the correct formula: sum of interior angles of quadrilateral is $360^\circ$. So $ \angle S + \angle T + \angle U + \angle V = 360^\circ$. But in a kite, one pair of opposite angles are equal? Wait, no, actually, in a kite, the angles between the two pairs of equal sides: the angles that are between the unequal sides (the ones not between the equal sides) are equal? Wait, maybe I confused. Wait, let's check the sides: $SV = ST$ (double hash), $VU = UT$ (single hash). So the vertices: $S$ is between $SV$ and $ST$, $U$ is between $VU$ and $UT$. So the angles at $S$ and $U$ are the ones between the equal sides? No, wait, in a kite, one pair of opposite angles are equal. Wait, maybe the correct approach is: sum of angles in quadrilateral is $360^\circ$. We know $ \angle S = 121^\circ$, $ \angle U = 97^\circ$, and in a kite, $\angle T = \angle V$? Wait, no, maybe $\angle S$ and $\angle U$ are not the equal pair, but $\angle T$ and $\angle V$ are? Wait, no, let's calculate. Let's assume that in a kite, two angles are equal. Wait, maybe the problem is that in a kite, the sum of angles is $360^\circ$, and we have two angles given: $121^\circ$ and $97^\circ$, and the other two angles ( $\angle T$ and $\angle V$) are equal? Wait, no, maybe I made a mistake. Wait, let's do the math. Sum of angles: $121 + 97 + x + y = 360$. But in a kite, one pair of opposite angles are equal. Wait, maybe $\angle T = \angle U$? No, $ \angle U = 97$, but that doesn't fit. Wait, no, maybe $\angle S$ and $\angle T$ are not, but $\angle V$ and $\angle U$? No. Wait, maybe the correct property is that in a kite, the angles between the two pairs of equal sides: the angles that are between the unequal sides (the ones not between the equal sides) are equal. Wait, perhaps I should just use the sum of interior angles. Let's denote $ \angle T = x$, and since in a kite, one pair of opposite angles are equal? Wait, no, maybe the problem is that in this kite, $ \angle T = \angle U$? No, $ \angle U = 97$, but that would not sum. Wait, wait, let's recalculate. Sum of interior angles of quadrilateral: $ (4 - 2) \times 180 = 360^\circ$. So $ 121 + 97 + x + y = 360$. But in a kite, two angles are equal. Wait, maybe $ \angle T = \angle U$? No, that would be $x = 97$, then $121 + 97 + 97 + y = 360$, $121 + 194 + y = 360$, $315 + y = 360$, $y = 45$, but that doesn't make sense. Wait, maybe $ \angle T = \angle S$? No, $ \angle S = 121$, then $121 + 121 + 97 + y = 360$, $339 + y = 360$, $y = 21$, also not. Wait, maybe I got the equal angles wrong. Wait, in a kite, the angles between the unequal sides (the ones that are not between the two pairs of equal sides) are equal. So in kite $STUV$, sides $SV = ST$ (double hash) and $VU = UT$ (single hash). So the two pairs of equal sides are $SV, ST$ and $VU, UT$. So the angles at $S$ (between $SV$ and $ST$) and at $U$ (between $VU$ and $UT$) are the ones between the equal sides, and the angles at $T$ (between $ST$ and $UT$) and at $V$ (between $SV$ and $VU$) are the ones between the unequal sides, and these two angles ($\angle T$ and $\angle V$) are equal? Wait, no, that's not right. Wait, maybe the correct property is that in a kite, one pair of opposite angles are equal. Let me check a reference: In a kite, one pair of opposite angles are equal. Wait, no, actually, in a kite, the angles between the two pairs of equal sides: the angles that are between the unequal sides (the ones not between the equal sides) are equal. Wait, maybe I should just use the sum. Let's suppose that $\angle T = \angle U$? No, that's not. Wait, the problem is that in the kite, $ \angle S = 121^\circ$, $ \angle U = 97^\circ$, and we need to find $ \angle T$. Wait, maybe the other two angles ($\angle T$ and $\angle V$) are equal? Wait, no, maybe I made a mistake. Wait, let's calculate: $360 - 121 - 97 = 142$. Then, if $\angle T = \angle V$, then each would be $142 / 2 = 71$? No, that doesn't fit. Wait, wait, maybe the equal angles are $\angle S$ and $\angle U$? No, 121 and 97 are not equal. Wait, maybe the kite has two distinct pairs of adjacent equal sides, so $SV = ST$ and $VU = UT$, so the angles at $S$ and $U$ are the vertex angles, and the angles at $T$ and $V$ are the base angles? Wait, no, in a kite, the vertex angles are between the unequal sides, and the base angles are between the equal sides. Wait, maybe the correct formula is that in a kite, the sum of the two distinct angles (the ones not equal) is $360 - 2x$, where $x$ is the equal angle. Wait, I'm confused. Let's do the math again. Sum of interior angles: $360^\circ$. Given $ \angle S = 121^\circ$, $ \angle U = 97^\circ$, so $ \angle T + \angle V = 360 - 121 - 97 = 142^\circ$. But in a kite, one pair of opposite angles are equal. Wait, maybe $ \angle T = \angle U$? No, $ \angle U = 97$, so $ \angle T = 97$, then $ \angle V = 142 - 97 = 45$, but that doesn't make sense. Wait, maybe $ \angle T = \angle S$? No, 121, then $ \angle V = 142 - 121 = 21$, also no. Wait, maybe the problem is that the kite has $SV = ST$ and $VU = UT$, so the angles at $T$ and $V$ are equal? Wait, no, that would be $ \angle T = \angle V$, so $2x = 142$, $x = 71$. But that doesn't seem right. Wait, maybe I made a mistake in the property. Wait, let's check the diagram again. The diagram shows $SV$ and $ST$ with double marks (equal), $VU$ and $UT$ with single marks (equal). So the vertices: $S$ is connected to $V$ and $T$ (equal sides), $U$ is connected to $V$ and $T$ (equal sides). So the angles at $S$ (between $SV$ and $ST$) and at $U$ (between $VU$ and $UT$) are the angles between the equal sides? No, $SV$ and $ST$ are equal, so $ \angle S$ is between them, and $VU$ and $UT$ are equal, so $ \angle U$ is between them. Then the angles at $T$ (between $ST$ and $UT$) and at $V$ (between $SV$ and $VU$) are between the unequal sides. In a kite, the angles between the unequal sides are equal. So $ \angle T = \angle V$. Then sum of angles: $ \angle S + \angle U + \angle T + \angle V = 360$. So $121 + 97 + x + x = 360$. So $218 + 2x = 360$. Then $2x = 360 - 218 = 142$. Then $x = 71$. Wait, but that would be $ \angle T = 71$? But that doesn't seem right. Wait, maybe I mixed up the equal angles. Wait, maybe the equal angles are $ \angle S$ and $ \angle U$? No, they are 121 and 97. Wait, maybe the problem is that in a kite, the sum of the two angles between the equal sides is $360 - 2x$, where $x$ is the angle between the unequal sides. Wait, I think I made a mistake. Let's check the sum again. $360 - 121 - 97 = 142$. If $ \angle T = \angle U$, then $ \angle T = 97$, but then $ \angle V = 142 - 97 = 45$, which is not equal to any. If $ \angle T = \angle S$, $ \angle T = 121$, then $ \angle V = 142 - 121 = 21$, also not. Wait, maybe the correct property is that in a kite, one pair of opposite angles are equal, and the other two are not. Wait, maybe the diagram is different. Wait, the diagram shows $S$ with 121, $U$ with 97, and we need to find $T$. Wait, maybe the kite has $SV = VU$ and $ST = TU$? No, the marks are double on $SV$ and $ST$, single on $VU$ and $UT$. So $SV = ST$ (double), $VU = UT$ (single). So the two pairs of equal adjacent sides are $SV, ST$ and $VU, UT$. So the angles at $S$ (between $SV$ and $ST$) and at $U$ (between $VU$ and $UT$) are the angles between the equal sides, and the angles at $T$ (between $ST$ and $UT$) and at $V$ (between $SV$ and $VU$) are between the unequal sides. In a kite, the angles between the unequal sides are equal. So $ \angle T = \angle V$. Then sum: $121 + 97 + 2x = 360$. So $218 + 2x = 360$, $2x = 142$, $x = 71$. So $ \angle T = 71^\circ$? Wait, but let's verify. If $ \angle T = 71$, $ \angle V = 71$, then sum is $121 + 97 + 71 + 71 = 360$. Yes, that works. So the measure of $ \angle T$ is 71 degrees.
## Step1: Sum of quadrilateral angles
The sum of interior angles of a quadrilateral is $360^\circ$. So $ \angle S + \angle T + \angle U + \angle V = 360^\circ$.
## Step2: Identify equal angles in kite
In kite $STUV$, $SV = ST$ (double marks) and $VU = UT$ (single marks). The angles between the unequal sides ($\angle T$ and $\angle V$) are equal, so $ \angle T = \angle V$. Let $ \angle T = x$, so $ \angle V = x$.
## Step3: Substitute known angles
We know $ \angle S = 121^\circ$ and $ \angle U = 97^\circ$. Substitute into the sum:
$121 + x + 97 + x = 360$.
Simplify: $218 + 2x = 360$.
## Step4: Solve for $x$
Subtract 218 from both sides: $2x = 360 - 218 = 142$.
Divide by 2: $x = \frac{142}{2} = 71$.

# Answer:
$71$

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Step1: Recall kite angle property

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