Question

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Explanation:

Step 1: Analyze Triangle EDF (Problem 33)

In triangle \( EDF \), we know that \( \angle EFD = 34^\circ \), \( \angle DEF = 8x - 3 \), and \( \angle QDE \) is an exterior angle to triangle \( EDF \) at \( D \). Also, \( \angle QDE = 10x + 9 \). By the exterior angle theorem, the exterior angle is equal to the sum of the two non - adjacent interior angles. So, \( 10x + 9=(8x - 3)+34 \).

Step 2: Solve for \( x \) (Problem 33)

Simplify the right - hand side of the equation: \( (8x - 3)+34 = 8x+31 \).
So our equation becomes \( 10x + 9=8x + 31 \).
Subtract \( 8x \) from both sides: \( 10x-8x + 9=8x - 8x+31 \), which gives \( 2x+9 = 31 \).
Subtract 9 from both sides: \( 2x+9 - 9=31 - 9 \), so \( 2x=22 \).
Divide both sides by 2: \( x = 11 \).

Step 3: Find \( m\angle QDE \) (Problem 33)

Substitute \( x = 11 \) into the expression for \( \angle QDE \): \( 10x + 9=10\times11+9=110 + 9=119^\circ \).

Step 4: Analyze Triangle DST (Problem 34)

In triangle \( DST \), \( \angle DST = 12x + 2 \), \( \angle DTS=10x - 228^\circ \)? Wait, that can't be right. Wait, maybe it's a typo and should be \( 10x - 22 \)? Or maybe the angle at \( U \) is a typo. Wait, assuming that the sum of angles in a triangle is \( 180^\circ \) and using the exterior angle or angle - sum property. Wait, maybe the angle at \( U \) is a straight line? Wait, let's re - examine. If we assume that \( \angle DST \) and \( \angle UST \) are related, and \( \angle U = 180^\circ-(10x - 228^\circ) \)? No, this seems incorrect. Wait, maybe the original problem has a typo. But let's proceed with the first problem (33) as the second one may have a typo.

Answer:

(for problem 33):
\( m\angle QDE = 119^\circ \)