Question

in δabc, d and e are midpoints of ab and ac, respectively. prove that de || bc. fill in the missing reason.
statement\treason
d and e are midpoints of ab and ac\tgiven
de = ½ bc\tdefinition of midsegment
de || bc\t____

a. midsegment theorem
b. aa similarity criterion
c. alternate interior angles theorem
d. csstp

which of the following is true for all similar triangles?
a. their corresponding angles are equal, and their sides are proportional.
b. their areas are equal.
c. their corresponding sides are equal, and their angles are proportional.
d. their angles are equal, and their sides are proportional.

Explanation:

First Question (Proving \( \overline{DE} \parallel \overline{BC} \))

The Midsegment Theorem (also known as the Midline Theorem) states that the segment connecting the midpoints of two sides of a triangle is parallel to the third side and half its length. Here, \( D \) and \( E \) are midpoints, \( DE=\frac{1}{2}BC \) (from the definition of midsegment), and the parallelism \( \overline{DE} \parallel \overline{BC} \) is a direct result of the Midsegment Theorem. The AA Similarity Criterion is for proving triangle similarity, Alternate Interior Angles Theorem is for proving parallel lines from angle relationships, and CSSTP (Corresponding Sides of Similar Triangles are Proportional) is about side proportions in similar triangles, not directly for this parallelism from midpoints.

By definition, similar triangles have corresponding angles equal (congruent) and corresponding sides proportional. Option b is wrong because similar triangles can have different areas (they are similar, not necessarily congruent). Option c is wrong because corresponding sides are proportional, not equal, and angles are equal (not proportional). Option d is slightly incorrect in wording (it should be "corresponding angles" and "corresponding sides") but option a is more precise. However, re - evaluating: The key properties of similar triangles are that corresponding angles are equal and corresponding sides are proportional. Option a states "their corresponding angles are equal, and their sides are proportional" (implying corresponding sides), which is the correct definition. Option d says "their angles are equal" (not specifying corresponding) and "their sides are proportional" (not specifying corresponding), but in the context of similar triangles, we mean corresponding angles and sides. But the most accurate is option a. Wait, actually, the standard definition is: Two triangles are similar if their corresponding angles are congruent and their corresponding sides are proportional. So option a is correct. Option b: Areas are equal only for congruent triangles, not all similar. Option c: Corresponding sides are proportional, not equal, and angles are equal (not proportional). Option d: "Angles are equal" (should be corresponding angles) and "sides are proportional" (should be corresponding sides), but among the options, option a is the most accurate as it specifies "corresponding angles" and "sides" (implying corresponding sides in the context of similar triangles).

Answer:

a. Midsegment Theorem

Second Question (Properties of Similar Triangles)