In ΔDEF shown below, segment DG is an altitude: Triangle DEF with segment DG drawn from vertex D and intersecting side EF. Which of the following is a step towards proving the similarity of triangles ΔDEF and ΔGED? (6 points) Segment EF is a hypotenuse. Angle E is congruent to itself. Segment ED is shorter than segment EF. Segment EF is intersected by segment DG.
The Correct Answer and Explanation is:
Correct Answer:
Angle E is congruent to itself.
Explanation
To prove that two triangles are similar, we need to establish that their corresponding angles are equal and that their corresponding sides are in proportion. One of the most common methods used to prove similarity is the Angle-Angle (AA) Similarity Postulate, which states that if two angles of one triangle are congruent to two angles of another triangle, the triangles are similar.
In triangle DEF, segment DG is drawn from vertex D and intersects side EF at point G, forming a right angle. This makes DG an altitude, and creates two smaller triangles: triangle GED and triangle DEF.
Since DG is an altitude, angle DGE is a right angle. Triangle DEF also contains this right angle because DG is perpendicular to side EF. Therefore, both triangles GED and DEF share a right angle.
Next, look at angle E. It is a part of both triangle DEF and triangle GED. Since it is the exact same angle in both triangles, it is congruent to itself. This is a valid and important geometric reasoning called the Reflexive Property of angle congruence.
By confirming that triangle DEF and triangle GED share two angles (the right angle at point G and angle E), we have met the conditions of the AA Similarity Postulate. Hence, triangles DEF and GED are similar.
Now, examine the incorrect options:
- “Segment EF is a hypotenuse” does not help prove similarity directly.
- “Segment ED is shorter than segment EF” speaks about lengths, but not ratios or angles.
- “Segment EF is intersected by segment DG” is just a fact about the diagram, not a step in proving similarity.
Thus, the correct reasoning step is: Angle E is congruent to itself.
