Question

2 numeric 1 point in parallelogram mnpt, m∠m = (6x + 10)° and m∠n = (5x + 10.5)°. how many degrees is ∠t? answer previous next

Explanation:

Step1: Recall parallelogram angle property

In a parallelogram, consecutive angles are supplementary (sum to \(180^\circ\)). So, \(m\angle M + m\angle N = 180^\circ\).
Substitute the given expressions: \((6x + 10) + (5x + 10.5) = 180\).

Step2: Solve for \(x\)

Combine like terms: \(11x + 20.5 = 180\).
Subtract \(20.5\) from both sides: \(11x = 180 - 20.5 = 159.5\).
Divide by \(11\): \(x = \frac{159.5}{11} = 14.5\).

Step3: Find \(m\angle M\)

Substitute \(x = 14.5\) into \(m\angle M = 6x + 10\):
\(m\angle M = 6(14.5) + 10 = 87 + 10 = 97^\circ\).

Step4: Find \(m\angle T\)

In a parallelogram, opposite angles are equal. So, \(\angle M\) and \(\angle T\) are opposite? Wait, no—wait, in parallelogram \(MNPT\), let's confirm vertices. If \(MNPT\) is a parallelogram, then \(M\) and \(P\) are opposite, \(N\) and \(T\)? Wait, no, consecutive angles: \(M\) and \(N\) are consecutive, \(N\) and \(T\) are consecutive, \(T\) and \(P\), \(P\) and \(M\). Wait, actually, opposite angles: \(\angle M\) and \(\angle T\)? Wait, no, maybe I mixed up. Wait, in a parallelogram, opposite angles are equal. So if \(M\) and \(T\) are opposite? Wait, no, let's re-express the parallelogram: \(MNPT\), so sides \(MN \parallel PT\) and \(MP \parallel NT\). So angles: \(\angle M\) and \(\angle N\) are consecutive (supplementary), \(\angle N\) and \(\angle T\) are consecutive (supplementary), so \(\angle M\) and \(\angle T\) are equal? Wait, no, \(\angle M\) and \(\angle T\): wait, no, \(\angle M\) and \(\angle P\) are opposite, \(\angle N\) and \(\angle T\) are opposite? Wait, maybe I made a mistake. Wait, let's correct: in parallelogram \(MNPT\), the vertices are in order, so \(M\) connected to \(N\) and \(P\), \(N\) connected to \(M\) and \(T\), \(P\) connected to \(M\) and \(T\), \(T\) connected to \(N\) and \(P\). So consecutive angles: \(M\) and \(N\) (supplementary), \(N\) and \(T\) (supplementary), \(T\) and \(P\) (supplementary), \(P\) and \(M\) (supplementary). Therefore, \(\angle M\) and \(\angle T\): wait, \(\angle M\) and \(\angle T\) are not opposite. Wait, no, \(\angle M\) and \(\angle T\): let's see, \(\angle M\) is at vertex \(M\), \(\angle T\) at vertex \(T\). Since \(MN \parallel PT\) and \(NT\) is a transversal, \(\angle N\) and \(\angle T\) are same-side interior? No, wait, in a parallelogram, opposite angles are equal. So \(\angle M = \angle T\) only if \(M\) and \(T\) are opposite. Wait, maybe the parallelogram is labeled as \(M\), \(N\), \(P\), \(T\) in order, so \(M\) opposite \(P\), \(N\) opposite \(T\). Wait, that must be it. So \(\angle N\) and \(\angle T\) are opposite, so they are equal? Wait, no, the problem says "how many degrees is \(\angle T\)". Wait, maybe I messed up the first step. Wait, let's re-express: in parallelogram \(MNPT\), \(m\angle M = (6x + 10)^\circ\), \(m\angle N = (5x + 10.5)^\circ\). Since \(M\) and \(N\) are consecutive angles (supplementary), so \(m\angle M + m\angle N = 180^\circ\), which we solved \(x = 14.5\). Then \(m\angle M = 97^\circ\), \(m\angle N = 5(14.5) + 10.5 = 72.5 + 10.5 = 83^\circ\). Now, in a parallelogram, opposite angles are equal. So \(\angle M\) and \(\angle P\) are equal, \(\angle N\) and \(\angle T\) are equal? Wait, no, \(\angle M\) and \(\angle T\): wait, maybe the vertices are \(M\), \(N\), \(T\), \(P\)? No, the problem says \(MNPT\). So \(M - N - P - T - M\). So sides \(MN\), \(NP\), \(PT\), \(TM\). Then \(MN \parallel PT\), \(NP \parallel TM\). So angles: \(\angle M\) (at \(M\), between \(TM\) and \(MN\)) and \(\angle N\) (at \(N\), between \(MN\) and \(NP\)) are consecutive (supplementary). \(\angle N\) and \(\angle P\) (at \(P\), between \(NP\) and \(PT\)) are consecutive (supplementary). \(\angle P\) and \(\angle T\) (at \(T\), between \(PT\) and \(TM\)) are consecutive (supplementary). \(\angle T\) and \(\angle M\) (at \(M\)) are consecutive (supplementary). Wait, no, that can't be. Wait, in a parallelogram, opposite angles are equal. So \(\angle M = \angle P\), \(\angle N = \angle T\). Ah! That's the key. So \(\angle N\) and \(\angle T\) are opposite angles, so they are equal. Wait, but earlier we found \(m\angle N = 83^\circ\), so \(m\angle T = 83^\circ\)? Wait, no, that contradicts. Wait, no, let's recheck: if \(M\) and \(N\) are consecutive, then \(M\) and \(T\) are opposite? Wait, maybe I had the opposite angles wrong. Let's use the property: in a parallelogram, consecutive angles are supplementary, opposite angles are equal. So if \(\angle M\) and \(\angle N\) are consecutive (supplementary), then \(\angle M\) and \(\angle T\) are opposite? Wait, no, \(\angle M\) is at vertex \(M\), \(\angle T\) at vertex \(T\). So the sides adjacent to \(\angle M\) are \(TM\) and \(MN\), and the sides adjacent to \(\angle T\) are \(PT\) and \(TM\). Since \(MN \parallel PT\) and \(TM\) is a transversal, \(\angle M\) and \(\angle T\) are same-side interior angles? No, that would be supplementary. Wait, I'm confused. Wait, let's take a standard parallelogram \(ABCD\), with \(A-B-C-D-A\). Then \(\angle A\) and \(\angle B\) are consecutive (supplementary), \(\angle B\) and \(\angle C\) (supplementary), \(\angle C\) and \(\angle D\) (supplementary), \(\angle D\) and \(\angle A\) (supplementary). Opposite angles: \(\angle A = \angle C\), \(\angle B = \angle D\). So in \(MNPT\), let's map to \(ABCD\): \(M = A\), \(N = B\), \(P = C\), \(T = D\). Then \(\angle M = \angle A\), \(\angle N = \angle B\), \(\angle P = \angle C\), \(\angle T = \angle D\). So opposite angles: \(\angle M = \angle P\), \(\angle N = \angle T\). Wait, but \(\angle T\) is \(\angle D\), which is opposite to \(\angle B\) (\(\angle N\)). So \(\angle T = \angle N\)? No, \(\angle D\) is opposite to \(\angle B\)? No, in \(ABCD\), \(\angle A\) opposite \(\angle C\), \(\angle B\) opposite \(\angle D\). So \(\angle T\) (as \(\angle D\)) is opposite to \(\angle N\) (as \(\angle B\))? No, \(\angle B\) is opposite \(\angle D\), so yes, \(\angle N = \angle T\). Wait, but earlier we calculated \(m\angle N = 83^\circ\), so \(m\angle T = 83^\circ\)? But that contradicts if \(\angle M\) and \(\angle T\) are related. Wait, no, maybe I mixed up the vertices. Wait, the problem says "how many degrees is \(\angle T\)". Let's re-express: in parallelogram \(MNPT\), \(m\angle M = (6x + 10)^\circ\), \(m\angle N = (5x + 10.5)^\circ\). We found \(x = 14.5\), so \(m\angle M = 97^\circ\), \(m\angle N = 83^\circ\). Now, in a parallelogram, opposite angles are equal. So \(\angle M\) and \(\angle T\): wait, maybe \(\angle M\) and \(\angle T\) are opposite? Wait, maybe the parallelogram is \(M-N-T-P-M\), so \(MN \parallel TP\), \(NT \parallel PM\). Then \(\angle M\) (at \(M\), between \(PM\) and \(MN\)) and \(\angle T\) (at \(T\), between \(TP\) and \(NT\)) are opposite angles, so they are equal. Wait, that would make \(\angle M = \angle T\). But then \(\angle M = 97^\circ\), so \(\angle T = 97^\circ\). Wait, now I'm confused. Let's use the consecutive angles: \(\angle M\) and \(\angle N\) are consecutive (supplementary), \(\angle N\) and \(\angle T\) are consecutive (supplementary), so \(\angle M = \angle T\) (since both are supplementary to \(\angle N\)). Ah! Yes, that's the key. If \(\angle M + \angle N = 180^\circ\) and \(\angle T + \angle N = 180^\circ\), then \(\angle M = \angle T\) (transitive property: both equal to \(180^\circ - \angle N\)). So \(\angle T = \angle M = 97^\circ\). That makes sense. So because \(\angle M\) and \(\angle T\) are both supplementary to \(\angle N\), they are equal. So \(m\angle T = m\angle M = 97^\circ\).

Answer:

\(97\)