Question

1. which of the following statements is not true about similar figures?

○ a. side lengths are proportional.
○ b. similar figures must be the same size.
○ c. corresponding angles are congruent.
○ d. a dilation can create similar figures.

1. when a line parallel to one side of a triangle divides the other two sides proportionally, which criterion can be used to prove the triangles are similar?

○ a. sas criterion
○ b. hl criterion
○ c. aa criterion
○ d. sss criterion

1. two triangles are similar because they both have a right angle and share one additional angle. what does this imply about their sides?

○ a. their hypotenuses are the same length.
○ b. their sides are equal.
○ c. their bases are equal.
○ d. their sides are proportional.

Explanation:

Question 1

To determine the false statement about similar figures, we analyze each option:

• Option a: Side lengths of similar figures are proportional (a key property of similar figures), so this is true.
• Option b: Similar figures have the same shape but not necessarily the same size (congruent figures are same size and shape). So this statement is false.
• Option c: Corresponding angles of similar figures are congruent (another key property), so this is true.
• Option d: A dilation (scaling) preserves shape and creates similar figures, so this is true.

When a line parallel to one side of a triangle divides the other two sides proportionally, the triangles formed are similar by the AA (Angle - Angle) Criterion. The parallel line creates corresponding angles that are equal, so two angles of one triangle are equal to two angles of the other triangle, satisfying AA similarity.

• SAS Criterion is for congruence (or similarity with included angle and proportional sides), not directly for this case.
• HL Criterion is for right - triangle congruence.
• SSS Criterion is for congruence (or similarity with all sides proportional), not for this parallel - line case.

If two triangles are similar (in this case, by AA similarity as they both have a right angle and share another angle), a key property of similar triangles is that their corresponding sides are proportional.

• Option a: Hypotenuses of similar right - triangles are proportional, not necessarily the same length.
• Option b: Sides of similar triangles are proportional, not equal (equal sides would mean congruent, not just similar).
• Option c: Bases of similar triangles are proportional, not necessarily equal.
• Option d: This matches the property of similar triangles (proportional sides).

Answer:

b. Similar figures must be the same size

Question 2