Thread: Congruence and Similarity I Know They Are In There

Given: ÎABC is an isosceles triangle with AB = AC.
D is the midpoint of AB.
E is the midpoint of AC.
DE is parallel to BC.

1. Find two pairs (two sets of two triangles )of
similar triangles and then explain why they are
similar.
2. Find two pairs (two sets of two triangles )of congruent triangles that are different from
those that were used in question #1 and explain why they are congruent.
3. Given the statement: “If two triangles are congruent then the triangles are similar.”. Do
you agree /disagree with this statement. Explain.

I am lost for this review help please so I can study it!

What, exactly, is your difficulty? Do you know what "similar" and "congruent" mean?

a little help:

congruent means (in somewhat imprecise terms): "same shape, same size" (might be "turned around", or "flipped over", though).

similar means: same shape (might not be the same size. might be. hard to say).

you might find this web-page helpful: Similarity and Congruence

Refer to your diagram:

DE || BC [ Line joining the midpoints of two sides of a triangle is parallel to the third side ]
∴ ∠ ADE= ∠ABC and ∠AED= ∠ ACB
Thus ∆ ADE ~ ∆ABC by AA criterion.
Also since DE || BC we have
∠ DEF= ∠FBC and ∠EDF= ∠ BFC [ pairs of alternate angles ]
Thus ∆ DEF ~ ∆BCF by AA criterion
Now for congruence
AB = AC [ given]
∴ ∠ ABC= ∠ACB [ Angles opposite equal sides are equal ]
Now in ∆ DBC and ∆ ECB we have
BC = CB [ Common ]
∠ DBC= ∠ECB
DB = EC [ ∵ AB = AC and DB = ½ AB & EC = ½ AC ]
Thus ∆ DBC ≅∆ ECB by SAS criterion.
∆ ADC ≅∆ AEB by AAS criterion.