Question

ure of each angle.
60°, ( mangle u = (7 + 5x)^circ ) and ( mangle p = (1 + 3x)^circ ). find t
ngles that each measure 42°. if ( angle p ) is obtuse, what
h diagram.
17.
terior angles of a triangle is 66°. the exterior ang
other remote interior angle?

Explanation:

First Triangle (Right Triangle \( \triangle AMT \)):

Step1: Sum of angles in triangle is \( 180^\circ \), and \( \angle M = 90^\circ \). So, \( 2x + (x - 6) + 90 = 180 \).

\( 2x + x - 6 + 90 = 180 \)

Step2: Combine like terms.

\( 3x + 84 = 180 \)

Step3: Subtract 84 from both sides.

\( 3x = 180 - 84 = 96 \)

Step4: Divide by 3.

\( x = \frac{96}{3} = 32 \)

Step5: Find each angle.

\( \angle A = 2x = 2 \times 32 = 64^\circ \)
\( \angle T = x - 6 = 32 - 6 = 26^\circ \)
\( \angle M = 90^\circ \)

Problem with \( \angle 60^\circ \), \( \angle U = (7 + 5x)^\circ \), \( \angle P = (1 + 3x)^\circ \) (Assuming it's a triangle, sum of angles \( 180^\circ \)):

Step1: Sum of angles: \( 60 + (7 + 5x) + (1 + 3x) = 180 \)

\( 60 + 7 + 5x + 1 + 3x = 180 \)

Step2: Combine like terms.

\( 68 + 8x = 180 \)

Step3: Subtract 68.

\( 8x = 180 - 68 = 112 \)

Step4: Divide by 8.

\( x = \frac{112}{8} = 14 \)

Step5: Find angles.

\( \angle U = 7 + 5 \times 14 = 7 + 70 = 77^\circ \)
\( \angle P = 1 + 3 \times 14 = 1 + 42 = 43^\circ \)

Problem 17 (Triangle with exterior angle \( 133^\circ \) and interior angle \( 111^\circ \)):

Step1: The adjacent interior angle to \( 133^\circ \) is \( 180 - 133 = 47^\circ \) (linear pair).

Step2: Sum of angles in triangle: \( x + 47 + 111 = 180 \)

\( x + 158 = 180 \)

Step3: Subtract 158.

\( x = 180 - 158 = 22 \)

Last Problem (Interior angle \( 66^\circ \), exterior angle... Assuming exterior angle is equal to sum of two remote interior angles, but need more info. If we assume one remote interior is \( 66^\circ \), let the other be \( y \), and exterior angle \( E \), then \( E = 66 + y \). But since info is partial, maybe a typo. If we assume it's a triangle with one interior \( 66^\circ \), and exterior angle related, but need more context.

Answer:

s (for solved parts):

• For \( \triangle AMT \): \( \angle A = 64^\circ \), \( \angle T = 26^\circ \), \( \angle M = 90^\circ \)
• For \( \angle U, \angle P \) problem: \( x = 14 \), \( \angle U = 77^\circ \), \( \angle P = 43^\circ \)
• For Problem 17: \( x = 22^\circ \)