QuestionWhat condition makes $\triangle A E D \simeq \triangle C E B$ true? a. $\angle A E D \cong \angle C E B$ b. $\angle E A D \cong \angle E C B$ c. $\angle E D A \cong \angle E B C$ d. $\angle D E C \cong \angle B E A$

Studdy Solution

STEP 1

Assumptions1. We are given two triangles, $\triangle AED$ and $\triangle CEB$ . . We are asked to find the additional condition that makes these two triangles similar.
3. We are given four options to choose from.

STEP 2

Recall the conditions for triangle similarity. Two triangles are similar if1. Their corresponding angles are congruent (AA criterion)
2. Their corresponding sides are in proportion (SS criterion) . An angle and its corresponding side are in proportion (AS criterion)
In this problem, we are looking for an angle-angle (AA) condition, as all the options are about angle congruence.

STEP 3

Let's analyze each optiona. $\angle AED \cong \angle CEB$ : This condition states that the angle at in $\triangle AED$ is congruent to the angle at in $\triangle CEB$ . However, we already know that these angles are congruent because they are vertical angles. So, this condition is not additional.
b. \angleAD \cong \angle ECB: This condition states that the angle at A in $\triangle AED$ is congruent to the angle at C in $\triangle CEB$ . This is a new condition and could potentially make the triangles similar.
c. \angleDA \cong \angleBC: This condition states that the angle at D in $\triangle AED$ is congruent to the angle at B in $\triangle CEB$ . This is also a new condition and could potentially make the triangles similar.
d. $\angle DEC \cong \angle BEA$ : This condition states that the angle at C in $\triangle DEC$ is congruent to the angle at A in $\triangle BEA$ . However, these angles are not in the same triangles we are considering ( $\triangle AED$ and $\triangle CEB$ ), so this condition is irrelevant.

STEP 4

From the analysis in3, we can see that options b and c are the potential correct answers. However, we need to choose only one.To do this, we need to consider the order of the vertices in the triangles. In $\triangle AED$ , moving in a counterclockwise direction, the vertices are A,, D. In $\triangle CEB$ , moving in a counterclockwise direction, the vertices are C,, B.
Therefore, \angleAD in $\triangle AED$ corresponds to $\angle ECB$ in $\triangle CEB$ , and \angleDA in $\triangle AED$ corresponds to \angleBC in $\triangle CEB$ .

STEP 5

Based on the analysis in4, we can conclude that the additional condition that makes $\triangle AED \simeq \triangle CEB$ is \angleAD \cong \angle ECB.
So, the correct answer is option b.

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