Question

1. there are two triangles, triangle pqr with angle at r being 57°, and triangle stu with angle at s being 49° and angle at u being 68°. (the image shows the two triangles with these angle markings.)

Explanation:

Step1: Find angle at T in triangle STU

The sum of angles in a triangle is \(180^\circ\). For \(\triangle STU\), we know two angles: \(\angle S = 49^\circ\) and \(\angle U = 68^\circ\). So, \(\angle T=180^\circ - 49^\circ - 68^\circ\). Calculating that: \(180 - 49 - 68 = 63^\circ\)? Wait, no, wait. Wait, in \(\triangle PRQ\), we have \(\angle R = 57^\circ\). Wait, maybe we need to check if the triangles are similar or find a missing angle? Wait, maybe the problem is to find the missing angle in one of the triangles? Wait, let's re - examine.

Wait, in \(\triangle PRQ\), we know \(\angle R = 57^\circ\). Let's find \(\angle P + \angle Q=180 - 57 = 123^\circ\). In \(\triangle STU\), \(\angle S = 49^\circ\), \(\angle U = 68^\circ\), so \(\angle T=180-(49 + 68)=180 - 117 = 63^\circ\). Wait, maybe the problem is to check if the triangles have a certain relationship or find a missing angle. Wait, maybe the question is to find the measure of an angle, like in \(\triangle PRQ\), if we assume some correspondence? Wait, maybe the original question was to find a missing angle, like \(\angle Q\) or \(\angle P\), or in the other triangle. Wait, since the sum of angles in a triangle is \(180^\circ\), let's suppose we need to find \(\angle Q\) in \(\triangle PRQ\) (assuming it's an isoceles? No, not necessarily). Wait, maybe the problem is to find the third angle in each triangle.

Wait, let's correct the calculation for \(\angle T\): \(49 + 68 = 117\), \(180-117 = 63^\circ\). For \(\triangle PRQ\), if we assume that maybe the triangles are related, but since the problem's image shows two triangles, maybe the task is to find the missing angle in one of them. Let's assume we need to find the third angle in \(\triangle PRQ\). Wait, no, we only know \(\angle R = 57^\circ\). Wait, maybe the question is to find the measure of \(\angle Q\) or \(\angle P\), but we need more info. Wait, maybe the original problem was to find the third angle in each triangle. Let's recast:

For \(\triangle STU\):
Sum of angles in a triangle is \(180^\circ\). So \(\angle T=180^\circ-\angle S - \angle U\)
\(\angle S = 49^\circ\), \(\angle U = 68^\circ\)
\(\angle T = 180 - 49 - 68=63^\circ\)

For \(\triangle PRQ\):
\(\angle R = 57^\circ\), so \(\angle P+\angle Q = 180 - 57 = 123^\circ\). But if we assume that maybe the triangles are similar or have some angle correspondence, but without more info, maybe the problem is to calculate the third angle in each triangle.

Wait, maybe the question is to find the measure of \(\angle T\) in \(\triangle STU\) and \(\angle Q\) (or \(\angle P\)) in \(\triangle PRQ\). Let's proceed with calculating \(\angle T\) first:

Step1: Recall triangle angle sum property

The sum of the interior angles of a triangle is \(180^\circ\). Mathematically, for a triangle with angles \(A\), \(B\), \(C\), \(A + B + C=180^\circ\).

Step2: Calculate \(\angle T\) in \(\triangle STU\)

We know \(\angle S = 49^\circ\) and \(\angle U = 68^\circ\). Let \(\angle T=x\). Then:
\(x + 49^\circ+ 68^\circ=180^\circ\)
\(x=180^\circ-(49^\circ + 68^\circ)\)
\(49 + 68 = 117\), so \(x = 180 - 117=63^\circ\)

If we want to calculate the third angle in \(\triangle PRQ\) (let's say \(\angle Q\), assuming we need to find it, but we only know \(\angle R = 57^\circ\). Wait, maybe the problem was to find the third angle in each triangle. So for \(\triangle PRQ\), let \(\angle P = y\), \(\angle Q=z\), then \(y + z+57^\circ = 180^\circ\), so \(y + z = 123^\circ\). But without more info, maybe the key calculation here is for \(\angle T\) which is \(63^\circ\), or for \(\triangle PRQ\), if we assume some other relation. Wait, maybe the original problem was to find the measure of \(\angle Q\) and we assume that the triangle is such that maybe \(\angle Q\) is equal to \(\angle T\)? No, that's an assumption. Wait, maybe the problem is to find the third angle in each triangle, so:

For \(\triangle STU\): \(\angle T = 63^\circ\)

For \(\triangle PRQ\): Let's say we need to find \(\angle Q\), but we can't without more info. Wait, maybe the problem was mis - presented, but based on the given angles, the calculation for \(\angle T\) is \(63^\circ\).

Answer:

If the question was to find \(\angle T\) in \(\triangle STU\), the answer is \(63^\circ\). If it was to find the third angle in \(\triangle PRQ\) (assuming we need to express it in terms of other angles), but based on the calculation for \(\triangle STU\), the angle \(\angle T = 63^\circ\).