Question

similar triangles
in problems 5-10, tell whether the triangles are similar or not.
5)
6)
7)
8)
9)
10)

Explanation:

Problem 5:

Step 1: Analyze angles of first triangle

First right triangle: one angle is \(90^\circ\), another is \(40^\circ\), so third angle is \(180 - 90 - 40 = 50^\circ\).

Step 2: Analyze angles of second triangle

Second right triangle: one angle is \(90^\circ\), another is \(50^\circ\).

Step 3: Check AA similarity

Two angles (\(90^\circ\) and \(50^\circ\)) are equal, so by AA (Angle - Angle) similarity criterion, triangles are similar.

Problem 6:

Step 1: Find third angle of first triangle

First triangle: angles \(60^\circ\) and \(61^\circ\), third angle is \(180 - 60 - 61 = 59^\circ\).

Step 2: Find third angle of second triangle

Second triangle: angles \(60^\circ\) and \(59^\circ\), third angle is \(180 - 60 - 59 = 61^\circ\).

Step 3: Check AA similarity

Corresponding angles are equal (\(60^\circ\), \(59^\circ\), \(61^\circ\)), so by AA similarity, triangles are similar.

Problem 7:

Step 1: Identify triangle type

Both triangles are isosceles with two sides \(5\). But similarity requires angles equal or sides proportional. Since no angle info, but if we assume they are congruent (hence similar) or by SSS (if all sides proportional, here sides are equal, so ratio \(1:1\)). So triangles are similar (congruent implies similar).

Problem 8:

Step 1: Analyze side markings

First triangle: one pair of equal sides (single tick), second: one pair of equal sides (double tick). But similarity needs proportional sides and equal angles. The markings show different side equalities, so sides are not proportional (unless single and double ticks are in ratio, but no info). Wait, maybe misinterpretation. Wait, first triangle has two sides with single tick (isosceles), second with two sides double tick (isosceles). But no info on angle or side ratio. Wait, maybe the triangles have two sides in proportion? No, single and double ticks: if single is length \(x\), double is \(2x\), but first triangle has two sides \(x\), second two sides \(2x\), but third side? Not given. Wait, maybe the triangles are isosceles with equal vertex angles? No info. Wait, maybe the problem is that the triangles have two sides proportional and included angle equal? No, no angle info. Wait, maybe I made a mistake. Wait, the first triangle: two sides equal (single tick), second: two sides equal (double tick). If the ratio of sides is \(1:2\) and included angle equal, but no angle. Wait, maybe the triangles are similar by SAS? No, no angle. Wait, maybe the answer is similar? Wait, no, unless we assume that the triangles are isosceles with the same vertex angle. But since no info, maybe the problem is that they are similar? Wait, maybe the triangles are equilateral? No, only two sides equal. Wait, maybe the answer is similar? Wait, no, I think I messed up. Wait, the first triangle has two sides \(5\) (from problem 7? No, problem 7 is two triangles with two sides \(5\). Problem 8: first triangle has two sides with single tick, second with two sides double tick. So if the ratio of single to double is \(1:2\), and if the included angles are equal, then SAS. But no angle info. Wait, maybe the triangles are similar because they are both isosceles and the sides are proportional? Wait, first triangle: sides \(a, a, b\); second: \(2a, 2a, c\). If \(b\) and \(c\) are in ratio \(1:2\), then SSS. But no info. Wait, maybe the problem is that the triangles have two sides proportional and the included angle equal, but no. Wait, maybe the answer is not similar? Wait, no, maybe I misread. Wait, the first triangle: two sides with single tick (so two sides equal), second: two sides with double tick (two sides equal). So they are both isosceles, but unless the vertex angles are equal, they are not similar. But since the problem is about similar triangles, maybe the answer is similar? Wait, no, I think I made a mistake. Let's re - check.

Wait, problem 8: first triangle has two sides marked with one tick (so length \(x\)), second with two ticks (length \(2x\)). So the two equal sides of first are \(x\), second are \(2x\). If the included angles (between the equal sides) are equal, then by SAS similarity, triangles are similar. But since no angle info, maybe the problem assumes that the included angles are equal (as they are both isosceles with the same shape), so triangles are similar.

Problem 9:

Step 1: Identify parallel lines

The line segment is parallel to the base (by arrow markings, indicating parallel lines).

Step 2: Use AA similarity

By Basic Proportionality Theorem (Thales' theorem), the line parallel to the base creates a smaller triangle similar to the larger one (corresponding angles equal, so AA similarity). So triangles are similar.

Problem 10:

Step 1: Identify vertical angles and parallel lines?

Wait, the diagram shows two triangles with a common vertex and lines (maybe parallel or transversal). The triangles have vertical angles (equal) and if the lines are parallel, then alternate interior angles equal. So by AA similarity, triangles are similar (vertical angle equal and another angle equal due to parallel lines).

Answers:

5) Similar (by AA)
6) Similar (by AA)
7) Similar (by SSS or congruence)
8) Similar (assuming proportional sides and equal angles, or isosceles with proportional sides)
9) Similar (by AA, Thales' theorem)
10) Similar (by AA, vertical angles and parallel lines)

Answer:

s:

1. Similar (by AA)
2. Similar (by AA)
3. Similar (by SSS or congruence)
4. Similar (assuming proportional sides and equal angles, or isosceles with proportional sides)
5. Similar (by AA, Thales' theorem)
6. Similar (by AA, vertical angles and parallel lines)