Question

1. triangle and quadrilateral with angle 64°, 6x - 4, 72° at vertex l, and vertices k, j, m. 8) quadrilateral with angle 85°, 11x + 1, 50° at vertex u, and vertices t, s, v. 9) quadrilateral with right angle at e, angle 23°, side 11x + 1, and vertices d, g, f. 10) quadrilateral with angle 35°, 81°, side 22x - 2, and vertices d, c, f, e.

Explanation:

Problem 7)

Step1: Set up angle sum equation

The angles at point \(L\) form a straight angle (180°), so:
\(64^\circ + (6x-4)^\circ + 72^\circ = 180^\circ\)

Step2: Simplify and solve for \(x\)

Combine constants: \(64 + 72 - 4 + 6x = 180\)
\(132 + 6x = 180\)
\(6x = 180 - 132 = 48\)
\(x = \frac{48}{6} = 8\)

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Problem 8)

Step1: Find angle in right triangle

In right triangle \(TUV\), \(\angle TUV = 90^\circ - 85^\circ = 5^\circ\)

Step2: Set equal alternate interior angle

\(\angle SVU = \angle TUV = 5^\circ\), so \(11x + 1 = 5\)

Step3: Solve for \(x\)

\(11x = 5 - 1 = 4\)
\(x = \frac{4}{11}\)

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Problem 9)

Step1: Find angle in right triangle

In right triangle \(DEF\), \(\angle DFE = 90^\circ - 23^\circ = 67^\circ\)

Step2: Set equal alternate interior angle

\(\angle GFD = \angle DFE = 67^\circ\), so \(11x + 1 = 67\)

Step3: Solve for \(x\)

\(11x = 67 - 1 = 66\)
\(x = \frac{66}{11} = 6\)

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Problem 10)

Step1: Find angle in triangle \(DEF\)

\(\angle DFE = 180^\circ - 35^\circ - 81^\circ = 64^\circ\)

Step2: Set equal alternate interior angle

\(\angle FEC = \angle DFE = 64^\circ\), so \(22x - 2 = 64\)

Step3: Solve for \(x\)

\(22x = 64 + 2 = 66\)
\(x = \frac{66}{22} = 3\)

Answer:

1. \(x = 8\)
2. \(x = \frac{4}{11}\)
3. \(x = 6\)
4. \(x = 3\)