Question

9.
j with 36° angle, triangle jik with 101° angle, triangle gih with sides s, q, r, intersecting at i. find p, q, s.
10.
line ad with b, c; triangle bcf with 65° angle, line eg with f, o, 55° at f, lines hi intersecting. find m, n, o, p, q.
11.
quadrilateral wxyz with 84°, 135°, equal sides wx and zy, xy and wz? find a, b.
12.
parallelogram rstu with 75° at t, 85° at u, line from s to u? find a, b, c, d.

Explanation:

it's a quadrilateral with two sides equal, so \( a = 180 - 135 = 45^\circ \), then \( b = 180 - 84 - 45 = 51^\circ \)? Wait, sum of angles in triangle is \( 180^\circ \). If it's a triangle, then \( 84 + 135 + b = 180 \)? No, that's more than 180. So it's a quadrilateral: \( 84 + a + 135 + b = 360 \), and \( a = 180 - 135 = 45 \) (consecutive angles supplementary), so \( b = 360 - 84 - 45 - 135 = 96 \). But that seems high. Alternatively, \( a = 45^\circ \), \( b = 180 - 84 - 45 = 51^\circ \) (if it's a triangle). Maybe the diagram is a triangle with \( W \), \( X \), \( Z \), and \( Y \) is a point, so \( a = 45^\circ \), \( b = 51^\circ \).

Step1: Find \( a \) (alternate interior angles)

\( ST \parallel RU \), so \( a = 180^\circ - 75^\circ - (180^\circ - 85^\circ) \)? No, \( \angle T = 75^\circ \), \( \angle U = 85^\circ \), so \( \angle a = 180 - 75 - (180 - 85) = 10^\circ \)? Wait, \( ST \parallel RU \), so \( \angle S + \angle R = 180^\circ \), \( \angle T + \angle U = 180^\circ \)? No, \( \angle U = 85^\circ \), so \( \angle b = 180 - 85 = 95^\circ \) (supplementary). \( \angle T = 75^\circ \), so \( \angle a = 180 - 75 - 95 = 10^\circ \)? Then \( c = 75^\circ \) (alternate interior), \( d = 85^\circ \) (alternate interior).

Step2: Find \( b \)

\( b = 180^\circ - 85^\circ = 95^\circ \) (supplementary to \( 85^\circ \)).

Step3: Find \( c \) and \( d \)

\( c = 75^\circ \) (alternate interior with \( \angle T \)), \( d = 85^\circ \) (alternate interior with \( \angle U \)).

Answer:

\( a = 45^\circ \), \( b = 51^\circ \) (or \( 96^\circ \) if quadrilateral, but likely triangle: \( a = 45^\circ \), \( b = 51^\circ \))

Problem 12