Question

In triangle MNO shown below, segment NP is an altitude:

The figure is triangle MNO (a right triangle) and NP is the altitude from the right angle.
Which of the following is a justification used while proving the similarity of triangles MPN and NPO? A. Transitive Property of Equality
B. Addition Property of Equality
C. Definition of an Altitude D. Definition of Supplementary Angles

Answer 1

Answer: C. Definition of an Altitude

Explanation:

Given: In triangle MNO shown below, segment NP is an altitude from the right angle.

Let ∠MNP=x

Then ∠PNO=90°-x

Therefore in triangle MNO,

∠MPN=∠NPO =90° [by definition of Altitude]

[Definition of altitude : A line which passes through a vertex of a triangle, and joins the opposite side forming right angles. ]

Now using angle sum property in ΔMNP

∠MNP+∠MPN+∠PMN=180°

⇒x+90°+∠PMN=180°

⇒∠PMN=180°-90°-x

⇒∠PMN=90°-x

Now, in ΔMNO and ΔPNO

∠PMN=∠PNO=90°-x

and ∠MPN=∠NPO =90° [by definition of altitude]

Therefore by AA similarity postulate, we have

ΔMNO ≈ ΔPNO