Chapter 5.3, Problem 35E

### Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085

Chapter
Section

### Elementary Geometry For College St...

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:                           Δ D E F ∼ Δ M N P ; D G ¯ and M Q ¯ are altitudesProve:                             D G M Q = D E M N

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 DGMQ=DEMN by using given information.

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 Î”DEFâˆ¼Î”MNP, DGÂ¯ and MQÂ¯ are altitudes.

The given figure is shown below.

Figure

From the given figure, it is observed that DGÂ¯ and MQÂ¯ are altitudes.

It is known that an altitude is a line segment through a vertex and perpendicular to a line of base. That is, DGÂ¯âŠ¥EFÂ¯ and MQÂ¯âŠ¥NPÂ¯.

Since, perpendicular lines form a right angle triangle, âˆ DGE and âˆ MQN are right angle triangles.

Right angle triangles are congruent. So that âˆ DGE=âˆ MQN.

From the above mentioned CASTC definition, âˆ Eâ‰…âˆ N since corresponding angles of similar triangles are congruent.

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, Î”DGEâˆ¼Î”MQN.

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

That is, DGMQ=DEMN

### Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

#### The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started