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$Euclidean Triangle Proofs Short Answer 5. In the diagram of /_\LAC and /_\DNC below, bar(LA)~= bar(DN), bar(CA)~= bar(CN), and bar(DAC)_|_ bar(C_(CN)) a) Prove that /_\LAC~=/_\DNC. b) Describe a sequence of rigid motions that will map /_\LAC onto /_\DNC.$

Euclidean Triangle Proofs $\newline$ Short Answer $\newline$ $5$ . In the diagram of $\triangle L A C$ and $\triangle D N C$ below, $\overline{L A} \cong \overline{D N}, \overline{C A} \cong \overline{C N}$ , and $\overline{D A C} \perp \overline{C_{C N}}$ $\newline$ a) Prove that $\triangle L A C \cong \triangle D N C$ . $\newline$ b) Describe a sequence of rigid motions that will map $\triangle L A C$ onto $\triangle D N C$ .

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Q. Euclidean Triangle Proofs

\newline

Short Answer

\newline

5

. In the diagram of

\triangle L A C

and

\triangle D N C

below,

\overline{L A} \cong \overline{D N}, \overline{C A} \cong \overline{C N}

, and

\overline{D A C} \perp \overline{C_{C N}}

\newline

a) Prove that

\triangle L A C \cong \triangle D N C

\newline

b) Describe a sequence of rigid motions that will map

\triangle L A C

onto

\triangle D N C

Given Information: To prove that triangle $LAC$ is congruent to triangle $DNC$ , we need to use the given information that $LA$ is congruent to $DN$ , $CA$ is congruent to $CN$ , and $DAC$ is perpendicular to $CN$ .
SSS Congruence Postulate: Since $\bar{LA}$ $\approx$ $\bar{DN}$ and $\bar{CA}$ $\approx$ $\bar{CN}$ , we have two sides of each triangle that are congruent. This is part of the Side-Side-Side (SSS) congruence postulate.
Right Angles in Triangles: The information $\overline{DAC}$ _|_ $\overline{C_{(CN)}}$ tells us that angle $DAC$ is a right angle. Since both triangles share this angle, they both have a right angle, which means they are both right triangles.
HL Theorem Application: Since both triangles are right triangles and they have two congruent sides, by the Hypotenuse-Leg (HL) theorem, which is a special case of the SSS postulate for right triangles, we can conclude that triangle $LAC$ is congruent to triangle $DNC$ .
Translate Triangle LAC: To describe a sequence of rigid motions that will map triangle $LAC$ onto triangle $DNC$ , we can first translate triangle $LAC$ so that point $A$ coincides with point $D$ .
Rotate Triangle Around Point $D$ : Next, we can rotate the translated triangle around point $D$ until side $AC$ aligns with side $DN$ .
Reflect Triangle Over Line: Finally, if necessary, we can reflect the triangle over the line that contains $DN$ to ensure that point $C$ coincides with point $N$ .