Question

solve.
4 in the diagram, \\(\overrightarrow{ea} \perp \overrightarrow{ec}\\), and \\(bd\\) is a straight line.
find the value of \\(x\\).
show your work.
solution:
use the diagram at the right to solve problems 5–6.
5 explain how you would find \\(m\angle stq\\).

6 find \\(m\angle stq\\).
show your work.
solution:

Explanation:

Problem 4

Step1: Identify angle relationships

Since \(\overrightarrow{EA} \perp \overrightarrow{EC}\), \(\angle AEC = 90^\circ\). Also, \(BD\) is a straight line, so \(\angle BED = 180^\circ\). The angles around point \(E\) on line \(BD\) are \(x^\circ\), \(90^\circ\), and \((3x - 10)^\circ\), so their sum is \(180^\circ\).
\[x + 90 + (3x - 10) = 180\]

Step2: Simplify and solve for \(x\)

Combine like terms:
\[4x + 80 = 180\]
Subtract 80 from both sides:
\[4x = 100\]
Divide by 4:
\[x = 25\]

To find \(m\angle STQ\), first note that \(\angle STM\) (which is \((2x + 8)^\circ\)) and \(\angle PTN\) (which is \((71 - x)^\circ\)) are complementary to the right angle at \(T\)? Wait, no—actually, the two angles \((2x + 8)^\circ\) and \((71 - x)^\circ\) are complementary because they form a right angle (the diagram has a right angle symbol between them). So we can set up the equation \((2x + 8) + (71 - x) = 90\) to solve for \(x\). Then, once we have \(x\), we can find \(\angle STQ\) by using the fact that \(\angle STQ\) is supplementary to \(\angle STM\) (since they form a linear pair, or we can use the value of \(x\) to find \(\angle STQ\) directly if we consider the straight line or triangle properties).

Step1: Solve for \(x\) using complementary angles

The angles \((2x + 8)^\circ\) and \((71 - x)^\circ\) are complementary (sum to \(90^\circ\)):
\[2x + 8 + 71 - x = 90\]
Simplify:
\[x + 79 = 90\]
Subtract 79:
\[x = 11\]

Step2: Find \(m\angle STM\)

Substitute \(x = 11\) into \((2x + 8)^\circ\):
\[2(11) + 8 = 22 + 8 = 30^\circ\]

Step3: Find \(m\angle STQ\)

Since \(\angle STM\) and \(\angle STQ\) are supplementary (form a linear pair, sum to \(180^\circ\)):
\[m\angle STQ = 180 - 30 = 150^\circ\]

Answer:

\(x = 25\)

Problem 5