In the figure below, C is the midpoint of EA, E and A are right angles, and x PCE and ACN are ve

step 1 So triangle PEC is congruent to triangle NAC because we have two angles and an included side. So

In the figure below, C is the midpoint of EA, E and A are...

In the figure below, $\mathrm{C}$ is the midpoint of $\overline{\mathrm{EA}}, \Varangle \mathrm{E}$ and $\Varangle \mathrm{A}$ are right angles, and $\mathrm{x} \mathrm{PCE}$ and $\Varangle \mathrm{ACN}$ are vertical angles. (FIGURE CANT COPY) Why is $\triangle \mathrm{PEC} \cong \triangle \mathrm{NAC} ?$

In the figure below, $\mathrm{C}$ is the midpoint of $\overline{\mathrm{EA}}, \Varangle \mathrm{E}$ and $\Varangle \mathrm{A}$ are right angles, and $\mathrm{x} \mathrm{PCE}$
and $\Varangle \mathrm{ACN}$ are vertical angles.
(FIGURE CANT COPY)
Why is $\triangle \mathrm{PEC} \cong \triangle \mathrm{NAC} ?$

Key Concepts

Triangle Congruence

Triangle congruence is a fundamental concept in geometry that states two triangles are congruent if all their corresponding sides and angles are equal. Establishing triangle congruence typically involves verifying one of the criteria such as SSS, SAS, ASA, or HL, which provides a basis to conclude that the triangles have identical geometric properties.

ASA Congruence Criterion

The Angle-Side-Angle (ASA) congruence criterion asserts that if two angles and the included side in one triangle are respectively equal to two angles and the included side in another triangle, then the triangles are congruent. This criterion is crucial when proving the equality of triangles by relying primarily on angle measurements and a shared side length.

Vertical Angles

Vertical angles are the pairs of opposite angles formed when two lines intersect. A key property of vertical angles is that they are always equal. This concept is often used in geometric proofs to establish angle congruence within different parts of a figure.

Midpoint

A midpoint of a segment is the point that divides the segment into two equal portions. Recognizing midpoints is important in geometric problems because they create pairs of equal segments, which can serve as one of the necessary conditions for triangle congruence by providing a congruent side to compare.

Right Angles

Right angles, which measure 90 degrees, play a significant role in geometry as they provide fixed and easily recognizable angle measures. Right angles are particularly useful in congruence proofs as they offer a common, definitive angle measure that can help establish the equality of angles between different triangles.

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