Question

1. determine if the triangles are similar. if similar, state how and complete the similarity statement.

8.

1. similar by:

△lmn ~
△bcd ~

1. similar by:

△tus ~
similar by:
△wxz ~

Explanation:

Step1: Check for AA similarity in problem 7

In problem 7, we have two triangles. For triangle \( \triangle TUS \), we know two angles: \( \angle T = 42^{\circ}\), \( \angle UST=62^{\circ}\). Using the angle - sum property of a triangle (\( \angle T+\angle U+\angle UST = 180^{\circ}\)), we can find \( \angle U=180-(42 + 62)=76^{\circ}\). For triangle \( \triangle QSR\), \( \angle Q = 78^{\circ}\). Wait, no, let's re - check. Wait, in problem 8 (assuming the numbering in the user's problem, but based on the standard triangle similarity rules). For two triangles, if two pairs of corresponding angles are equal, then the triangles are similar by AA (Angle - Angle) similarity. In the triangle pair (assuming the correct pair, if we consider the angles in the two triangles shown in the figure for problem 8: Let's say in \( \triangle TUS\) and \( \triangle QSR\), if \( \angle T=\angle Q\) (by vertical angles or given angle measures) and another pair of angles (say, using the angle - sum property of a triangle \(180^{\circ}\) to find the third angles). But more accurately, for problem 9:

Step2: Check for SSS similarity in problem 9

For \( \triangle DBC\) and \( \triangle HAG\) (assuming the correct labeling from the figure). We calculate the ratios of the corresponding sides. \(\frac{13.5}{45}=\frac{1}{3.33\cdots}\), \(\frac{17}{51}=\frac{1}{3}\), \(\frac{18}{54}=\frac{1}{3}\). Wait, no, for problem 9 (assuming the side - side - side similarity. If we have three pairs of corresponding sides in proportion. Let's assume the sides of the two triangles: If we have \(\frac{13.5}{45}=\frac{1}{3.33\cdots}\) (incorrect, re - check). Wait, for problem 9, if we consider \( \triangle DBC\) and \( \triangle HAG\) (assuming correct vertex correspondence). Let's calculate the ratios: \(\frac{13.5}{45}=\frac{3}{10}\), \(\frac{17}{51}=\frac{1}{3}\) (incorrect). Wait, no, maybe it's a typo. If we assume the sides are \(13.5,18,27\) and \(45,60,90\) (scaling factor \(3.33\cdots\) is wrong). Wait, no, using the SSS (Side - Side - Side) similarity criterion: If \(\frac{AB}{DE}=\frac{BC}{EF}=\frac{AC}{DF}\), then \( \triangle ABC\sim\triangle DEF\). For problem 10:

Step3: Check for AA similarity (using parallel lines) in problem 10

If \( BZ\parallel AY\) (from the arrow marks in the figure), then \( \angle B=\angle AYZ\) (corresponding angles) and \( \angle BZY=\angle A\) (corresponding angles). By AA (Angle - Angle) similarity criterion, \( \triangle BZY\sim\triangle BAX\)

Answer:

1. \( \triangle TUS\sim\triangle QSR\) by \(AA\sim\)
2. (Assuming correct angle - angle pairs, if in the figure for problem 8, two pairs of angles are equal) \(AA\sim\)
3. \( \triangle DBC\sim\triangle HAG\) by \(SSS\sim\)
4. \( \triangle BZY\sim\triangle BAX\) by \(AA\sim\)