Chapter 5.3, Problem 35E

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7th Edition

Alexander + 2 others

ISBN: 9781337614085

Textbook Problem

Use a two-column proof to prove the following theorem:

“The lengths of the corresponding altitudes of similar triangles have the same ratio as lengths of any pair of corresponding sides.”

*Given:*

*Prove: *

To determine

**To prove:**

The theorem “The lengths of the corresponding altitudes of similar triangles have the same ratio as lengths of any pair of corresponding sides.” By using two-column proof and prove

Explanation

**Definition:**

*AA*:

If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.

**CSSTP:**

Corresponding sides of similar triangles are proportional.

**CASTC:**

Corresponding angles of similar triangles are congruent.

**Description:**

Given that

The given figure is shown below.

Figure

From the given figure, it is observed that

It is known that an altitude is a line segment through a vertex and perpendicular to a line of base. That is,

Since, perpendicular lines form a right angle triangle,

Right angle triangles are congruent. So that

From the above mentioned CASTC definition,

The above mentioned *AA *definition, the two triangles *DGE *and *MQN *are similar since the two angles of one triangle are congruent to two angles of another triangle. Hence,

From the definition of CSSTP, corresponding sides of similar triangles are proportional.

That is,

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